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# On the green's functions and martin boundary structure of a planar diffusion in a discontinuous layered medium

Sandro Franceschi, Irina Kourkova, Maxence Petit

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Sandro Franceschi, Irina Kourkova, Maxence Petit. On the green's functions and martin boundary structure of a planar diffusion in a discontinuous layered medium. 2025. hal-05497599

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# ON THE GREEN'S FUNCTIONS AND MARTIN BOUNDARY STRUCTURE OF A PLANAR DIFFUSION IN A DISCONTINUOUS LAYERED MEDIUM

SANDRO FRANCESCHI, IRINA KOURKOVA, AND MAXENCE PETIT

**ABSTRACT.** We consider a two-dimensional diffusion process in a two-layered plane, governed by distinct covariance matrices in the upper and lower half-planes and by two drift vectors pointed away from the x-axis. We first analyze the case where the generator of the process is in divergence form, that is, when the flux is continuous across the interface. Then we extend the study to a broader class of processes whose behavior at the interface forms an oblique two-dimensional analogue of the skew Brownian motion.

We provide a detailed theoretical analysis of this transient process. Our main results are as follows: (i) we derive explicit Laplace transforms of the Green's functions; (ii) we compute exact asymptotics of the Green's functions along all possible trajectories in the plane; (iii) We determine all positive harmonic functions, identifying the full and minimal Martin boundaries, which turn out to be distinct. The nonminimality of the Martin boundary is a noteworthy phenomenon for diffusions.

To obtain an analytical description of the process, we fully develop a three-variable version of the so-called kernel method by deriving and exploiting a functional equation involving unknown Laplace transforms of Green's functions and two known kernels $\gamma_+(x, y)$ and $\gamma_-(x, z)$. The introduction of independent auxiliary variables $y$ and $z$, associated with each half-plane, is a key idea.

## CONTENTS

<table><tr><td>1. Introduction and main results</td><td>1</td></tr><tr><td>2. Process and functional equation</td><td>15</td></tr><tr><td>3. Proof of Theorem 1 and 2: Laplace transforms and asymptotics on the axis</td><td>22</td></tr><tr><td>4. Proofs of Theorem 3 to 6: asymptotics along all directions</td><td>24</td></tr><tr><td>5. Proof of Theorem 7: full and minimal Martin boundary</td><td>40</td></tr><tr><td>6. Generalization to skew diffusions acting in any oblique direction q</td><td>42</td></tr><tr><td>Appendix A. Two-sided Tauberian theorem</td><td>49</td></tr><tr><td>Appendix B. An integral asymptotic lemma</td><td>49</td></tr><tr><td>References</td><td>51</td></tr></table>

## 1. INTRODUCTION AND MAIN RESULTS

### 1.1. Context, goals and strategy.
The study of diffusion processes in multi-dimensional discontinuous or layered media is a challenging problem. Diffusion processes with discontinuous coefficients arise naturally in various applied contexts, such as geophysics, ecology, and astrophysics, where the medium may be composed of layers with different diffusion and advection properties (see [25, 26] and references therein). In such models, discontinuities of the diffusion matrix act as permeable interfaces, leading to complex boundary behaviors.

In [26], Lejay proposed a simulation algorithm for 2-dimensional diffusion processes in layered media. More precisely, the process is governed by different diagonal covariance matrices in the upper and lower half-plane of $\mathbb{R}^2$, the value of the singular drift on the axis is chosen to have a generator in the divergence form. In the present paper, we consider the generalisation of this process with arbitrary covariance matrices $\Sigma^+$ and $\Sigma^-$ in the upper and lower half-plane and drifts $\mu^+$ and $\mu^-$ directed away from the

<footnote>This project has received funding from Agence Nationale de la Recherche, ANR JCJC programme under the Grant Agreement ANR-22-CE40-0002 (ANR RESYST)..</footnote>
x-axis. This model can be interpreted as a “bang-bang” type process, switching between two regimes, a structure reminiscent of stochastic control models [37]. Our main focus is on the case where the generator of the process is in divergence form, that is, when the flux is continuous across the interface. In the last section of this paper, we also consider a broader class of processes whose behavior can be viewed as a natural two-dimensional generalization of a Skew Bang-Bang Brownian Motion (SBBBM), with oblique interaction at the interface.

A rich literature exists for skew Brownian motion in dimension one, see the foundational paper by Harrison and Shepp [15] and the survey by Lejay [25]. See also Fernholz et al. for a detailed introduction and study of one-dimensional SBBBM [13] and Lenotre who studies and simulates this type of process [27]. Some results also exist in higher dimensions, a recent proof of weak existence and uniqueness was provided by Atar and Budhiraja [2]. The first study of multidimensional skew Brownian motion was conducted by Portenko who constructed a diffusion process whose infinitesimal generator involves a singular drift concentrated on a smooth hypersurface S [34, 35]. Zaitseva studied a stochastic differential equation describing a multidimensional Brownian motion with an oblique skew penetration through a hyperplane, under the assumption that both the skewness coefficient and the direction are constant [41]. The extension to the general case of an oblique skew diffusion with non-constant coefficients remains a challenging open question [1, 36].

The skew Brownian motion is deeply linked to diffusion processes generated by

$$ (1.1) \qquad \mathcal{L} = \frac{\rho(x)}{2} \frac{d}{dx} \left( a(x) \frac{d}{dx} \right) + \mu(x) \frac{d}{dx}, $$

where a and ρ have discontinuities [10]. Indeed, the diffusion process X generated by L can be described as the solution of an SDE with local time:

$$ (1.2) \qquad dX_t = \sigma(X_t) dW_t + \mu(X_t) dt + \int_{\mathbb{R}} \nu(dx) dL_t^x(X), $$

where σ² = ρa and ν is a finite signed measure whose mass is concentrated at the points where a or ρ are discontinuous. See Legall's work [24] in dimension one and Stroock's work [38] in higher dimension. The process studied in this paper satisfies an equation of type (1.2).

From a structural viewpoint, our model also connects with discrete analogues such as random walks or queueing processes evolving under different dynamics in distinct regions. The semi-martingale reflecting Brownian motion (SRBM) in a cone and its discrete counterparts are now well understood in dimension two and beyond (see, e.g., [11, 39]), but models involving different generators in several cones sharing a common boundary have been much less studied. Among discrete examples, the joining-the-shortest-queue (JSQ) model (studied by Kurkova and Suhov [22], Kobayashi et al. [18]) and the doubly two-sided quasi-birth-and-death (DQBD) process (studied by Miyazawa [29, 30, 31]) display related features, although symmetries sometimes reduce them to single-kernel functional equations. In contrast, our diffusion model genuinely involves two distinct kernels corresponding to the two half-planes. Some time-space continuous models for the multi-level single server queue are also studied in [19, 32].

Our aim is to provide a detailed theoretical analysis of the transient process presented above. The main results of the present paper are as follows:

(i) We make explicit the Laplace transforms of the Green's functions.

(ii) We derive the exact asymptotics of the Green's functions along all possible trajectories in $\mathbb{R}^2$.

(iii) As a consequence, we obtain explicit expressions for all positive harmonic functions and determine the full Martin boundary. We also identify the minimal Martin boundary, which turns out to be strictly smaller than the full one.

The nonminimality of the full Martin boundary reveals a genuinely new phenomenon, which contrasts with many known results for diffusions with regular coefficients, whether free, killed or confined in cones or in other domains, see [9, 14] and DeBlassie's papers [5, 6, 7]. This phenomenon can be also expected for diffusions in two-dimensional cones with special assumptions on drifts making the process escape to infinity along the boundaries, see [16, 21] for the proof in the case of the analogous discrete random walks. However, the minimal Martin boundary in the last example is reduced to only two points, while for our process it is homeomorphic to a union of arcs of a circle.
The main tool of our analysis is a version of the so-called kernel method usually used to study random processes in cones; see Zhao [40] for a survey. The kernel $\gamma(x, y)$ of the SRBM is usually given by the sum of one half of the quadratic form associated with the covariance matrix and the linear form corresponding to the drift. In our model, we consider two kernels, $\gamma^+(x, y)$ and $\gamma^-(x, z)$, which share the same first variable and have distinct second variables. The crucial idea of our method is to introduce the independent variables $y$ and $z$, which correspond to the two half-planes.

This approach was first fruitfully applied in the large deviation analysis of random walks in $Z^{d+1}$ with a discontinuity on a hyperplane; as discussed by Ignatyuk et al. [17]. It was also employed in [30], where the exponential decay rate of the stationary distribution for a DQBD process along the axes is determined by certain extremal points of the domains bounded by two kernels.

In this paper, we extend the kernel method to three variables by fully developing it through the derivation and exploitation of a functional equation involving the Laplace transforms of Green's functions, see (1.13) below. To the best of our knowledge, this is the first example of a three-variable implementation of the method, leading to a complete analytical characterization of the model.

**Detailed outline of the article** In Section 1.2, we introduce the main process. In Section 1.3, we define the Green's functions and state the three-variable kernel functional equation for their Laplace transforms. In Section 1.4 we state our main results. In Section 1.4.1, we first present result (i), providing explicit formulas for the Laplace transforms of the Green's functions in Theorem 1. Then, in Section 1.4.2, we state result (ii), which provides the exact asymptotics of the Green's functions $g^{z0}$ along all trajectories in the plane $\mathbb{R}^2$. In other words, we consider the limit of $g^{z0}(r \cos \alpha, r \sin \alpha)$ as $r \to \infty$ and $\alpha \to \alpha_0$, where $\alpha_0 \in [0, 2\pi)$ and $z_0 \in \mathbb{R}^2$ denotes the initial state. Theorem 2 states the asymptotics along the $x$-axis, when $\alpha = 0$ or $\pi$. We obtain asymptotics of the form

$$g^{z0}(r, 0) \underset{r \to \infty}{\sim} C_0 f_0(z_0) e^{-rx_b} r^{-3/2}$$

where $C_0$ is a constant, $f_0$ a harmonic function, and $x_b$ an explicit branching point of one of the two-valued functions $Y^\pm(x)$ or $Z^\pm(x)$ defined by the kernels as $\gamma_+(x, Y^\pm(x)) = 0$ and $\gamma_-(x, Z^\pm(x)) = 0$ respectively, see Definition (1.16). We then highlight two key explicit angles for the asymptotic study,

$$\alpha_b \text{ and } \tilde{\alpha}_b \in [0, 2\pi]$$

which are related to branching points of $Y^\pm(x)$ and $Z^\pm(x)$ as in Definitions (1.21) and (1.22). These angles define the domain $\mathcal{M}$ which takes the following form depending on the configuration (see Figure 5 below) :

$$\mathcal{M} = [\alpha_b, \tilde{\alpha}_b] \cup [\pi, 2\pi] \text{ or } [0, \tilde{\alpha}_b] \cup [\pi, \alpha_b] \text{ or } [0, \pi] \cup [\tilde{\alpha}_b, \alpha_b] \text{ or } [\alpha_b, \pi] \cup [\tilde{\alpha}_b, 2\pi].$$

Theorem 3 states the typical asymptotic result: when $\alpha_0 \in \mathring{\mathcal{M}}$, we have

$$g^{z0}(r \cos(\alpha), r \sin(\alpha)) \underset{\alpha \to \alpha_0}{\sim} C(\alpha_0) h_{\alpha_0}(z_0) r^{-1/2} e^{-r \left( \cos(\alpha) x(\alpha) + \sin(\alpha) \times \begin{array}{c} y(\alpha) \\ z(\alpha) \end{array} \right) \begin{array}{c} \text{when } \alpha_0 \in (0, \pi) \\ \text{or} \\ \text{when } \alpha_0 \in (\pi, 2\pi) \end{array}}}$$

where $C(\alpha)$ is a constant depending on $\alpha$, $h_\alpha$ is a harmonic function, and $x(\alpha)$, $y(\alpha)$ and $z(\alpha)$ are explicit points depending on $\alpha$. Theorem 4 concerns the asymptotics as $\alpha \in \mathcal{M}$ tends to $\alpha_0 = 0$ or $\pi$. In this case a competition between $\alpha$ and $1/r$ appears in the asymptotics, which take the form

$$g^{z0}(r \cos(\alpha), r \sin(\alpha)) \underset{\alpha \to 0, \alpha \in \mathcal{M}}{\sim} C_0 f_0(z_0) (\kappa\alpha + r^{-1}) r^{-1/2} e^{-r \left( \cos(\alpha)x(\alpha) + \sin(\alpha) \times \begin{array}{c} y(\alpha) \\ z(\alpha) \end{array} \right) \begin{array}{c} \text{when } \alpha > 0 \\ \text{or} \\ \text{when } \alpha < 0 \end{array}}}.$$

Theorem 5 gives the asymptotics for $\alpha_0 = \alpha_b$ or $\tilde{\alpha}_b$, which involve a linear combination of two harmonic functions, $h_{\alpha_b}$ and $f_0$, with coefficients $C$ depending on the rate at which $\alpha$ converges to $\alpha_0$ as $r \to \infty$. We obtain an asymptotic expression of the form

$$g^{z0}(r \cos(\alpha), r \sin(\alpha)) \underset{\alpha \to \alpha_b}{\sim} (C(\alpha_b)h_{\alpha_b}(z_0) + C f_0(z_0)) r^{-1/2} e^{-r \left( \cos(\alpha)x(\alpha) + \sin(\alpha) \times \begin{array}{c} y(\alpha) \\ z(\alpha) \end{array} \right) \begin{array}{c} \text{when } \alpha_b \in (0, \pi) \\ \text{or} \\ \text{when } \alpha_b \in (\pi, 2\pi) \end{array}}}.$$
This is where the difference between the full Martin boundary and the minimal one appears. Theorem 6 deals with the asymptotics when $\alpha_0 \notin \mathcal{M}$ (or $\alpha_0 = 0, \pi$ and $\alpha \to \alpha_0$ while $\alpha \notin \mathcal{M}$). We obtain an asymptotic expression of the form

$$g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\alpha \to \alpha_0, \alpha \notin \mathcal{M}}{\sim} C'_{br}(\alpha) f_0(z_0) r^{-3/2} e^{-r(\cos(\alpha)x(\alpha_b)+\sin(\alpha)\times\frac{y(\alpha_b)}{z(\alpha_b)} \text{ when } \alpha \in (0,\pi) \text{ or } \alpha \in (\pi,2\pi))}$$

In Section 1.4.3, we state result (iii) in Theorem 7, concerning the harmonic functions, the full Martin boundary, and the minimal Martin boundary which is homeomorphic to $\mathcal{M}$.

In Section 2, we study the process in greater detail, establish its existence and uniqueness in the case where the generator is in divergence form and prove the main functional equation. In Section 3, we prove Theorem 1 using the functional equation from which we derive Theorem 2 applying Tauberian theorems; in Section 4, we prove Theorems 3-6, the proofs use variants of the saddle-point method; and in the short Section 5, we prove Theorem 7 which follows from the asymptotic results. Finally, in Section 6, we extend our analysis to a two-dimensional oblique skew diffusion, where the flux is not necessarily continuous: substantially new results under this assumption are found in Theorems 8, 9 and 10.

## 1.2. Planar skew bang-bang diffusion with piecewise constant coefficients.
We now define a two-dimensional generalization of the SBBBM, which acts obliquely at the interface between two half-planes. The diffusion coefficients and the drift vector are piecewise constant, taking different values in the upper and lower half-planes. Let $\Sigma^+, \Sigma^-$ be covariance matrices and $\mu^+, \mu^-$ vectors in $\mathbb{R}^2$. We define the matrices $\Sigma(y)$ and $\sigma(y)$ and the vector $\mu(y)$ such that

$$\Sigma(y) = \sigma(y)\sigma^{\top}(y) = \Sigma^{-1}\mathbf{1}_{y<0} + \Sigma^{+}\mathbf{1}_{y\ge0} \quad \text{and} \quad \mu(y) = \mu^{-1}\mathbf{1}_{y<0} + \mu^{+}\mathbf{1}_{y\ge0}.$$

Let $q = (q_1, q_2) \in \mathbb{R} \times (-1, 1)$. This vector represents a singular drift on the axis $\{y = 0\}$. The parameter $q_2 = 2\beta - 1$ corresponds to the classical skewness intensity at the boundary, when the process hits the axis $y = 0$, it restarts on the $y > 0$ side with probability $\beta = (1 + q_2)/2$, and on the $y < 0$ side with probability $1 - \beta = (1 - q_2)/2$. The parameter $q_1$ fixes the singular drift component along the axis and thus introduces the oblique direction of the skew behaviour at the interface.

**Definition 1.1** (Planar skew bang-bang diffusion with piecewise constant coefficients). We define a planar skew bang-bang diffusion associated with coefficients $(\Sigma^+, \Sigma^-, \mu^+, \mu^-)$ and with the skew vector $q$, as a continuous adapted process $(Z_t)_{t \ge 0} = (A_t, B_t)_{t \ge 0}$ on $\mathbb{R}^2$, such that $Z$ can be expressed almost surely as

$$(1.3) \qquad Z_t = (A_t, B_t) = (a_0, b_0) + \int_0^t \sigma(B_s) dW_s + \int_0^t \mu(B_s) ds + qL_t^0(B), \quad t \ge 0,$$

where:

(i) $z_0 = (a_0, b_0) \in \mathbb{R}^2$ is the starting point;

(ii) $(W_t)_{t \ge 0}$ is an adapted two-dimensional standard Brownian motion;

(iii) $(L_t^0(B))_{t \ge 0}$ is the symmetric local time at 0 of $B$ defined by

$$(1.4) \qquad L_t^0(B) = \lim_{\epsilon \to 0} \frac{1}{2\epsilon} \int_0^t \mathbf{1}_{[-\epsilon, \epsilon]}(B_s) ds.$$

Let $q_0$ be the vector defined by

$$(1.5) \qquad q_0 = \left( \frac{\Sigma_{12}^{+} - \Sigma_{12}^{-}}{\Sigma_{22}^{+} + \Sigma_{22}^{-}}, \frac{\Sigma_{22}^{+} - \Sigma_{22}^{-}}{\Sigma_{22}^{+} + \Sigma_{22}^{-}} \right).$$

In all sections of the article, except Section 6, we will assume that

$$q = q_0.$$

In this case, the generator, understood in the sense of Strook [38, Theorem II.3.1], can be written in divergence form; see (1.6) below. This form is relevant in physical applications [26]. The transmission conditions then ensure the continuity of the probability flux across the boundary, see the partial differential equation (1.41). The following proposition is proved in Section 2.1.
<figure><img src="image_3.png" /><figcaption>FIGURE 1. Parameters and path of the process. Locally, the process Z behaves like a planar Brownian motion with covariance Σ⁺ (resp. Σ⁻) and drift μ⁺ (resp. μ⁻) in the upper (resp. lower) half-plane. When the process hits the axis {y = 0}, it is subject to a singular drift in the direction of q.</figcaption></figure>

**Proposition 1.2** (Existence, uniqueness and divergence form). There exists a unique solution Z_t (both pathwise and in law) to the stochastic differential equation (1.3) associated with the skew vector q = q_0. This process defines a strong Markov process and its generator can be expressed as follows:

$$ (1.6) \qquad \mathcal{L} = \frac{1}{2} \nabla \cdot (\Sigma \nabla) + \mu \cdot \nabla. $$

For the remainder of this article, we will work under the following hypothesis regarding the drift:

$$ (1.7) \qquad \mu_1^+ > 0, \quad \mu_2^+ > 0, \quad \mu_1^- > 0, \quad \mu_2^- < 0, $$

see Figure 1. In particular, this hypothesis means that the process $Z_t = (A_t, B_t)$ is transient and eventually no longer touches the horizontal axis. The process B is a one dimensional skewed bang-bang process of parameter $q_2$ at 0 with positive drift on the nonnegative real axis and negative drift on the negative real axis.

1.3. **Green's functions and functional equation.** We have just seen that, when $q = q_0$, the generator can be expressed in divergence form. From Strook [38, Theorem II.3.8], we deduce that for all $t > 0$, the law of $Z_t$ has a density $p_t^{z_0}(z)$ with respect to the Lebesgue measure on $\mathbb{R}^2$. Furthermore, the function $(t, z_0, z) \mapsto p_t^{z_0}(z)$ is continuous on $(0, +\infty) \times \mathbb{R}^2 \times \mathbb{R}^2$.

**Definition 1.3** (Green's measures and Laplace transforms). For $z_0 = (a_0, b_0) \in \mathbb{R}^2$, we define the Green's measure on the plane $\mathbb{R}^2$ by

$$ (1.8) \qquad G(z_0, S) = \mathbb{E}_{z_0} \left[ \int_0^{+\infty} \mathbb{1}_S(Z_t) dt \right] = \iint_S g^{z_0}(a, b) da db, \quad \forall S \in \mathcal{B}(\mathbb{R}^2) $$

where its density $g^{z_0}$ is called the Green's function and is equal to

$$ (1.9) \qquad g^{z_0}(a, b) = \int_0^{+\infty} p_t^{z_0}(a, b) dt, \quad \forall (a, b) \in \mathbb{R}^2. $$

For $x, y, z \in \mathbb{R}$, we also define the Laplace transforms of $G(z_0, \cdot)$ restricted to the half-planes $\mathbb{R} \times (0, +\infty)$ and $\mathbb{R} \times (-\infty, 0)$ by

$$ \varphi_+^{z_0}(x, y) = \mathbb{E}_{z_0} \left[ \int_0^{+\infty} e^{xA_t+yB_t} \mathbb{1}_{B_t>0} dt \right] = \int_{\mathbb{R}} \int_0^{+\infty} g^{z_0}(a, b) e^{xa+yb} da db, $$

$$ \varphi_-^{z_0}(x, z) = \mathbb{E}_{z_0} \left[ \int_0^{+\infty} e^{xA_t+zB_t} \mathbb{1}_{B_t<0} dt \right] = \int_{\mathbb{R}} \int_{-\infty}^0 g^{z_0}(a, b) e^{xa+zb} da db. $$

Finally, we define the Green's measure on the axis $\mathbb{R} \times \{0\}$ by

$$ (1.10) \qquad H(z_0, S) = \mathbb{E}_{z_0} \left[ \int_0^{+\infty} \mathbb{1}_S(A_t) dL_t^0(B) \right], \quad \forall S \in \mathcal{B}(\mathbb{R}) $$
and its Laplace transform $\varphi^{z_0}$ as

$$ (1.11) \qquad \varphi^{z_0}(x) = \mathbb{E}_{z_0} \left[ \int_{0}^{+\infty} e^{xA_t} dL_t^0(B) \right]. $$

For $x, y, z \in \mathbb{C}$, we set:

$$ (1.12) \qquad \begin{cases} \gamma_+(x, y) = \frac{1}{2}(x, y) \cdot \Sigma^+(x, y) + (x, y) \cdot \mu^+ = \frac{1}{2}(\Sigma_{11}^+ x^2 + 2\Sigma_{12}^+ xy + \Sigma_{22}^+ y^2) + \mu_1^+ x + \mu_2^+ y \\ \gamma_-(x, z) = \frac{1}{2}(x, z) \cdot \Sigma^-(x, z) + (x, z) \cdot \mu^- = \frac{1}{2}(\Sigma_{11}^- x^2 + 2\Sigma_{12}^- xz + \Sigma_{22}^- z^2) + \mu_1^- x + \mu_2^- z \\ \gamma(x, y, z) = q_1 x + \frac{1}{2}(y(1 + q_2) + z(q_2 - 1)). \end{cases} $$

These polynomials are the coefficients of the following three-variable kernel functional equation, which is at the heart of this article. The following proposition is proven in Sections 2.3 and 2.4.

**Proposition 1.4 (Functional Equation).** Suppose that (1.7) holds. Then, there exists $\eta > 0$ such that for all $x \in (-\eta, 0)$, $y < 0$, and $z > 0$, the Laplace transforms $\varphi_{-}^{z_0}(x, z)$, $\varphi_{+}^{z_0}(x, y)$, and $\varphi^{z_0}(x)$ are finite and satisfy the following functional equation:

$$ (1.13) \qquad \gamma_-(x, z)\varphi_{-}^{z_0}(x, z) + \gamma_+(x, y)\varphi_{+}^{z_0}(x, y) + \gamma(x, y, z)\varphi^{z_0}(x) = -e^{xa_0+yb_01_{b_0>0}+zb_01_{b_0<0}}. $$

The polynomials $\gamma_+(x, y)$ and $\gamma_-(x, z)$ are called the kernels. We are now going to define some important quantities that will be useful for stating the results. We first determine the complex branches $Y^{\pm}(x)$ and $Z^{\pm}(x)$ which satisfy $\gamma_+(x, Y^{\pm}(x)) = 0$ and $\gamma_-(x, Z^{\pm}(x)) = 0$. By elementary considerations, these branches are given by

$$ (1.14) \qquad Y^{\pm}(x) = \frac{1}{\Sigma_{22}^{+}} \left( -\Sigma_{12}^{+}x - \mu_{2}^{+} \pm \sqrt{(\Sigma_{12}^{+})^{2} - (\Sigma_{11}^{+}\Sigma_{22}^{+})x^{2} + 2(\mu_{2}^{+}\Sigma_{12}^{+} - \mu_{1}^{+}\Sigma_{22}^{+})x + \mu_{2}^{+2}} \right) $$

and

$$ (1.15) \qquad Z^{\pm}(x) = \frac{1}{\Sigma_{22}^{-}} \left( -\Sigma_{12}^{-}x - \mu_{2}^{-} \pm \sqrt{(\Sigma_{12}^{-})^{2} - (\Sigma_{11}^{-}\Sigma_{22}^{-})x^{2} + 2(\mu_{2}^{-}\Sigma_{12}^{-} - \mu_{1}^{-}\Sigma_{22}^{-})x + \mu_{2}^{-2}} \right). $$

Furthermore, $Y^{\pm}(x)$ have branching points $x_{min}^{+} < 0$ and $x_{max}^{+} > 0$ given by

$$ x_{min}^{+} = \frac{\mu_{2}^{+}\Sigma_{12}^{+} - \mu_{1}^{+}\Sigma_{22}^{+} - \sqrt{D_{1}}}{\det(\Sigma^{+})}, \qquad x_{max}^{+} = \frac{\mu_{2}^{+}\Sigma_{12}^{+} - \mu_{1}^{+}\Sigma_{22}^{+} + \sqrt{D_{1}}}{\det(\Sigma^{+})} $$

with $D_1 = (\mu_2^+ \Sigma_{12}^+ - \mu_1^+ \Sigma_{22}^+)^2 + \mu_2^{+2} \det(\Sigma^+)$. Similarly, $Z^{\pm}(x)$ have branching points $x_{min}^{-} < 0$ and $x_{max}^{-} > 0$ with symmetric formulas. Finally, we denote the maximin and the minimax values of the branching points as follows:

$$ (1.16) \qquad \tilde{x}_b = \max(x_{min}^{+}, x_{min}^{-}) < 0, \quad x_b = \min(x_{max}^{+}, x_{max}^{-}) > 0. $$

These points are illustrated in Figure 2.

## 1.4. Main results.

1.4.1. *Explicit expression of the Laplace transforms.* The first theorem provides explicit formulas for the Laplace transforms $\varphi^{z_0}$ and $\varphi_{\pm}^{z_0}$. The proof is given in Section 3 by substituting $y = Y^{-}(x)$ and $z = Z^{+}(x)$ into the functional equation (1.13) to cancel the kernels.

**Theorem 1.** The Laplace transform $\varphi^{z_0}(x)$ extends to a holomorphic function on $\mathbb{C}\setminus((-\infty, \tilde{x}_b] \cup [x_b, +\infty))$ and can be expressed as follows:

$$ (1.17) \qquad \varphi^{z_0}(x) = -\frac{e^{xa_0+Y^{-}(x)b_01_{b_0>0}+Z^{+}(x)b_01_{b_0<0}}}{\gamma(x, Y^{-}(x), Z^{+}(x))}. $$

The points $x_b$ and $\tilde{x}_b$ are the branching points of $\varphi^{z_0}$. Furthermore,

$$ (1.18) \qquad \varphi_{+}^{z_0}(x, y) = \frac{-\gamma(x, y, Z^{+}(x))\varphi_{-}^{z_0}(x) - e^{xa_0+yb_01_{b_0>0}+zb_01_{b_0<0}}}{\gamma_{+}(x, y)}. $$

Similarly, a symmetric expression holds for $\varphi_{-}^{z_0}$.
<figure><img src="image_4.png" /><figcaption>FIGURE 2. Ellipses $\{(x, y) \in \mathbb{R}^2 : \gamma_+(x, y) = 0\}$ and $\{(x, z) \in \mathbb{R}^2 : \gamma_-(x, z) = 0\}$.</figcaption></figure>

1.4.2. Asymptotics of Green's functions. Before stating the main asymptotic theorems, a bit more notation needs to be introduced. For $\alpha \in (0, \pi)$, we define

$$ (1.19) \qquad (x(\alpha), y(\alpha)) = \underset{\gamma_+(x,y)=0}{\operatorname{argmax}} (\cos(\alpha)x + \sin(\alpha)y), $$

and for $\alpha \in (\pi, 2\pi)$, we set

$$ (1.20) \qquad (x(\alpha), z(\alpha)) = \underset{\gamma_-(x,z)=0}{\operatorname{argmax}} (\cos(\alpha)x + \sin(\alpha)z). $$

See Figure 3 for a geometric interpretation. In the rest of this article, these will be the key saddle points we look at. Note that the functions

$$ \alpha \in (0, \pi) \mapsto (x(\alpha), y(\alpha)) \in \{(x, Y^+(x)) : x_{\min}^+ < x < x_{\max}^+\} $$

and

$$ \alpha \in (\pi, 2\pi) \mapsto (x(\alpha), z(\alpha)) \in \{(x, Z^-(x)) : x_{\min}^- < x < x_{\max}^-\} $$

are $C^\infty$ diffeomorphisms. Note also that $\mu^+$ is orthogonal to the ellipse $\{(x, y) : \gamma_+(x, y) = 0\}$ at $(0, 0)$, see Figure 2b. Hence, the direction $\alpha_{\mu^+} = \arctan(\mu_2^+ / \mu_1^+) \in (0, \pi/2)$ of the drift $\mu^+$ satisfies $(x(\alpha_{\mu^+}), y(\alpha_{\mu^+})) = (0, 0)$. Similarly, if $\alpha_{\mu^-} := \arctan(\mu_2^- / \mu_1^-) + 2\pi \in (3\pi/2, 2\pi)$ is the direction of the drift $\mu^-$, then $(x(\alpha_{\mu^-}), z(\alpha_{\mu^-})) = (0, 0)$.

If $x_{\max}^+ > x_{\max}^-$, we define $\alpha_b$ as the unique angle $\alpha_b \in (0, \pi)$ such that

$$ (1.21) \qquad (x(\alpha_b), y(\alpha_b)) = (x_b, Y^+(x_b)), $$

see Figure 4a. In this case, we have $0 < \alpha_b < \alpha_{\mu^+}$ since $0 = x(\alpha_{\mu^+}) < x_b$.

If $x_{\max}^+ < x_{\max}^-$, we define $\alpha_b$ as the unique angle $\alpha_b \in (\pi, 2\pi)$ such that

$$ (1.22) \qquad (x(\alpha_b), z(\alpha_b)) = (x_b, Z^-(x_b)). $$

In this case, we have $\alpha_{\mu^-} < \alpha_b < 2\pi$ since $0 = x(\alpha_{\mu^-}) < x_b$.
<figure><img src="image_5.png" /><figcaption>FIGURE 3. Geometric construction of $(x(\alpha), y(\alpha))$ and $(x(\alpha), z(\alpha))$. The vector $e_\alpha$ is defined by $e_\alpha = (\cos(\alpha), \sin(\alpha))$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIGURE 4. Definitions of $\alpha_b$ and $\tilde{\alpha}_b$ in the cases $x_b = x_{max}^{-}$ and $\tilde{x}_b = x_{min}^{+}$.</figcaption></figure>
Next, considering $\tilde{x}_b$, we similarly define $\tilde{\alpha}_b$ by the condition $x(\tilde{\alpha}_b) = \tilde{x}_b$, where $\tilde{\alpha}_b$ belongs to $(0, \pi)$ if $x_{min}^{+} < x_{min}^{-}$ and to $(\pi, 2\pi)$ if $x_{min}^{+} > x_{min}^{-}$, see Figure 4b. Figure 5 provides an overview of all possible cases: A, B, C, and D. Then, we define

$$ \mathcal{M} = \{\alpha \in [0, 2\pi] : \tilde{x}_b \le x(\alpha) \le x_b\} $$

which can be expressed as follows for the cases shown in Figure 5:

$$ \mathcal{M} = \begin{cases} [\alpha_b, \tilde{\alpha}_b] \cup [\pi, 2\pi] & \text{in the case A: } x_{max}^{+} > x_{max}^{-} \text{ and } x_{min}^{+} < x_{min}^{-}, \\ [0, \tilde{\alpha}_b] \cup [\pi, \alpha_b] & \text{in the case B: } x_{max}^{+} < x_{max}^{-} \text{ and } x_{min}^{+} < x_{min}^{-}, \\ [0, \pi] \cup [\tilde{\alpha}_b, \alpha_b] & \text{in the case C: } x_{max}^{+} < x_{max}^{-} \text{ and } x_{min}^{+} > x_{min}^{-}, \\ [\alpha_b, \pi] \cup [\tilde{\alpha}_b, 2\pi] & \text{in the case D: } x_{max}^{+} > x_{max}^{-} \text{ and } x_{min}^{+} > x_{min}^{-}. \end{cases} $$

The set $\mathcal{M}$ is represented by the red arcs in Figure 5. As we will see in Theorem 7, this is the minimal Martin boundary.

<figure><img src="image_7.png" /><figcaption>FIGURE 5. Angles $\alpha_b$ and $\tilde{\alpha}_b$ depending on the parameters. The red arcs $\mathcal{M}$ correspond to the minimal Martin boundary.</figcaption></figure>

**Main asymptotics theorems.** We now state the asymptotic behaviour of the Green's functions, $g^{z_0}(r \cos(\alpha), r \sin(\alpha))$, as $r \to +\infty$ and $\alpha \to \alpha_0$. Figure 6 summarises the different cases covered by each theorem and indicates the associated colours. In Theorem 2 (brown arrows), we consider the case where the angle $\alpha$ is fixed at $\alpha_0 \in \{0, \pi\}$. In Theorem 3 (red arrows), we focus on directions where $\alpha \to \alpha_0$ with $\alpha_0 \in \mathring{M}$. In Theorem 4 (purple arrows), we study the case when $\alpha \to 0$ or $\pi$ but remaining within the red arc zones $\mathring{M}$. Theorem 5 (blue arrows) treats the asymptotic behavior as $\alpha \to \alpha_b$ or $\alpha \to \tilde{\alpha}_b$. Finally, Theorem 6 (green arrows) provides asymptotics as $\alpha \to \alpha_0$ with $\alpha_0 \in ([0, 2\pi] \setminus \mathcal{M})$, and for $\alpha \to 0$ or $\pi$ while remaining outside $\mathcal{M}$.

The first theorem follows from a Tauberian theorem applied to Laplace transforms, based on the explicit expression (1.18). The proof is given in Section 3.

**Theorem 2 (Asymptotics, case $\alpha = 0, \pi$).** Let $z_0 \in \mathbb{R}^2$. The following asymptotics hold:

$$ (1.23) \qquad g^{z_0}(r, 0) \underset{r \to +\infty}{\sim} C_0 f_0(z_0) \frac{e^{-rx_b}}{r^{3/2}}, \qquad g^{z_0}(-r, 0) \underset{r \to +\infty}{\sim} C_\pi f_\pi(z_0) \frac{e^{-r\tilde{x}_b}}{r^{3/2}}, $$

where $C_0, C_\pi > 0$ and $f_0(z_0), f_\pi(z_0) > 0$ are defined as follows. The constant $C_0$ is given by:

$$ (1.24) \qquad C_0 = \begin{cases} -\frac{\sqrt{\det(\Sigma^{+})(x_b - x_{min}^{+})}}{\sqrt{\pi \Sigma_{22}^{+} (\Sigma_{22}^{+} + \Sigma_{22}^{-})} \gamma(x_b, Y^{-}(x_b), Z^{+}(x_b))} > 0 & \text{if } x_b = x_{max}^{+} < x_{max}^{-} \\ -\frac{\sqrt{\det(\Sigma^{-})(x_b - x_{min}^{-})}}{\sqrt{\pi \Sigma_{22}^{-} (\Sigma_{22}^{+} + \Sigma_{22}^{-})} \gamma(x_b, Y^{-}(x_b), Z^{+}(x_b))} > 0 & \text{if } x_b = x_{max}^{-} < x_{max}^{+}. \end{cases} $$
<figure><img src="image_8.png" /><figcaption>FIGURE 6. Direction of asymptotics: Theorems 2 to 6 (case D). Rounded arrows indicate that the angle $\alpha$ approaches the corresponding direction $\alpha_0$ only from one side, i.e., either $\alpha > \alpha_0$ or $\alpha < \alpha_0$, depending on the arrow. For example, the green rounded arrow on the right means $\alpha \to 0$ with $\alpha > 0$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIGURE 7. Type of asymptotics according to directions: saddle-point type or branching point.</figcaption></figure>

Moreover, $f_0(z_0)$ is defined as follows:

* If $x_b = x_{max}^+ < x_{max}^-$ (cases B and C of Figure 5), then

$$ (1.25) \qquad f_0(a_0, b_0) = \begin{cases} \left(b_0 - \frac{\Sigma_{22}^+}{(\Sigma_{22}^+ + \Sigma_{22}^-)\gamma(x_b, Y^-(x_b), Z^+(x_b))}\right) e^{x_b a_0 + b_0 Y^+(x_b)} & \text{if } b_0 \ge 0 \\ \frac{-\Sigma_{22}^+}{(\Sigma_{22}^+ + \Sigma_{22}^-)\gamma(x_b, Y^-(x_b), Z^+(x_b))} e^{x_b a_0 + b_0 Z^+(x_b)} & \text{if } b_0 < 0 \end{cases} $$
• If $x_b = x_{max}^{-} < x_{max}^{+}$ (cases A and D of Figure 5), then

$$ (1.26) \qquad f_0(a_0, b_0) = \begin{cases} \frac{-\Sigma_{22}^{-}}{(\Sigma_{22}^{+} + \Sigma_{22}^{-})\gamma(x_b, Y^{-}(x_b), Z^{+}(x_b))} e^{x_b a_0 + b_0 Y^{-}(x_b)} & \text{if } b_0 \ge 0 \\ \left(-b_0 - \frac{\Sigma_{22}^{-}}{(\Sigma_{22}^{+} + \Sigma_{22}^{-})\gamma(x_b, Y^{-}(x_b), Z^{+}(x_b))}\right) e^{x_b a_0 + b_0 Z^{\pm}(x_b)} & \text{if } b_0 < 0 \end{cases} $$

The expressions for $C_\pi$ and $f_\pi(z_0)$ are symmetric.

The following theorem gives the asymptotics in the most typical case when $\alpha \in \mathcal{M}$. It is proved at the end of Section 4.3 using the saddle point method on a Riemann surface after extending the Laplace transforms.

**Theorem 3** (Asymptotics for $\alpha \to \alpha_0$, $\alpha_0 \in \mathcal{M}$). Let $z_0 \in \mathbb{R}^2$. Then, if $\alpha_0 \in \mathcal{M} \cap (0, \pi)$, we have

$$ (1.27) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\alpha \to \alpha_0}{\sim} C^{+}(\alpha_0) h_{\alpha_0}(z_0) \frac{e^{-r(\cos(\alpha)x(\alpha) + \sin(\alpha)y(\alpha))}}{\sqrt{r}} $$

and if $\alpha_0 \in \mathcal{M} \cap (\pi, 2\pi)$, the following asymptotics hold

$$ (1.28) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\alpha \to \alpha_0}{\sim} C^{-}(\alpha_0) h_{\alpha_0}(z_0) \frac{e^{-r(\cos(\alpha)x(\alpha) + \sin(\alpha)z(\alpha))}}{\sqrt{r}} $$

where $C^{\pm}(\alpha)$ and $h_{\alpha_0}(z_0)$ are defined as follows. The coefficient $C^{\pm}(\alpha)$ is given by:

$$ (1.29) \qquad C^{\epsilon}(\alpha) = \frac{1}{\sqrt{2\pi \left(\Sigma_{11}^{\epsilon} \sin^2(\alpha) - 2\Sigma_{12}^{\epsilon} \sin(\alpha) \cos(\alpha) + \Sigma_{22}^{\epsilon} \cos^2(\alpha)\right) \sqrt{\frac{\sin(\alpha)}{\partial_y \gamma_{\epsilon}(x(\alpha), y(\alpha))}}}}, \quad \epsilon \in \{+, -\}. $$

Furthermore, $h_\alpha(z_0)$ is defined as follows:

• If $\alpha \in \mathcal{M} \cap (0, \pi)$, then

$$ (1.30) \qquad h_{\alpha}(z_0) = \begin{cases} e^{a_0 x(\alpha) + b_0 y(\alpha)} - \frac{\gamma(x(\alpha), y(\alpha), Z^{+}(x(\alpha)))}{\gamma(x(\alpha), Y^{-}(x(\alpha)), Z^{+}(x(\alpha)))} e^{a_0 x(\alpha) + b_0 Y^{-}(x(\alpha))} & \text{if } b_0 \ge 0 \\ \left(1 - \frac{\gamma(x(\alpha), y(\alpha), Z^{+}(x(\alpha)))}{\gamma(x(\alpha), Y^{-}(x(\alpha)), Z^{+}(x(\alpha)))}\right) e^{a_0 x(\alpha) + b_0 Z^{+}(x(\alpha))} & \text{if } b_0 < 0 \end{cases} $$

• If $\alpha \in \mathcal{M} \cap (\pi, 2\pi)$, then

$$ (1.31) \qquad h_{\alpha}(z_0) = \begin{cases} \left(1 - \frac{\gamma(x(\alpha), Y^{-}(x(\alpha)), z(\alpha))}{\gamma(x(\alpha), Y^{-}(x(\alpha)), Z^{+}(x(\alpha)))}\right) e^{a_0 x(\alpha) + b_0 Y^{-}(x(\alpha))} & \text{if } b_0 \ge 0 \\ e^{a_0 x(\alpha) + b_0 z(\alpha)} - \frac{\gamma(x(\alpha), Y^{-}(x(\alpha)), z(\alpha))}{\gamma(x(\alpha), Y^{-}(x(\alpha)), Z^{+}(x(\alpha)))} e^{a_0 x(\alpha) + b_0 Z^{+}(x(\alpha))} & \text{if } b_0 < 0 \end{cases} $$

The functions $h_\alpha$ are in fact harmonic functions, see Theorem 7 later.

The next theorem states the asymptotics of Green's functions as $\alpha \to 0, \pi$ and $\alpha \in \mathcal{M}$. It establishes the missing connection between Theorem 2 and Theorem 3. To clarify this point further, let us note that in (1.27) the constant term $C(\alpha_0)h_{\alpha_0}(z) \to 0$ as $\alpha_0 \to 0$. Hence, in the case $\alpha_0 = 0$, we need to expand the asymptotics (1.27) up to the second term. The saddle-point method yields:

$$ g(r \cos(\alpha), r \sin(\alpha)) \underset{\alpha \to 0, \alpha \ge 0}{\sim} \left( C^{+}(\alpha) h_{\alpha}(z_0) + \frac{C_0 f_0(z_0)}{r} \right) \frac{e^{-r(\cos(\alpha)x(\alpha) + \sin(\alpha)y(\alpha))}}{\sqrt{r}}, $$

where

$$ C^{+}(\alpha) h_{\alpha}(z_0) \underset{\alpha \to 0}{\sim} \kappa C_0 f_0(z_0) \alpha $$

for some constant $\kappa > 0$ (see (4.28)). The first term corresponds to Theorem 3, while the second corresponds to Theorem 2. As $r \to \infty$ and $\alpha \to 0$, we observe a competition between these two
contributions. It is this competition that is established in the following theorem, proved at the end of Section 4.3.

**Theorem 4** (Asymptotics as $\alpha \to 0, \pi$ along $M$). Let $z_0 \in \mathbb{R}^2$ and let $\alpha_0 = 0$.

* Assume that $x_{max}^{+} < x_{max}^{-}$ (cases B and C of Figure 5). Then, for $\alpha \ge 0$ we have

$$ (1.32) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\alpha \to 0, \alpha \ge 0}{\sim} C_0 f_0(z_0) \left( \kappa\alpha + \frac{1}{r} \right) \frac{e^{-r(\cos(\alpha)x(\alpha)+\sin(\alpha)y(\alpha))}}{\sqrt{r}} $$

where $\kappa > 0$ is an explicit constant (cf. (4.29) below), $C_0$ is given by (1.24), and $f_0(z_0)$ is given by (1.25). Moreover,

$$ \partial_{\alpha} (C(\alpha)h_{\alpha}(z_0))|_{\alpha=0^+} = \kappa C_0 f_0(z_0). $$

* Assume now that $x_{max}^{+} > x_{max}^{-}$ (cases A and D of Figure 5). Then, for $\alpha \le 0$,

$$ (1.33) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\alpha \to 0, \alpha \le 0}{\sim} C_0 f_0(z_0) \left( \kappa|\alpha| + \frac{1}{r} \right) \frac{e^{-r(\cos(\alpha)x(\alpha)+\sin(\alpha)y(\alpha))}}{\sqrt{r}}, $$

where $f_0(z_0)$ is given by (1.26). Moreover,

$$ \partial_{\alpha} (C(\alpha)h_{\alpha}(z_0))|_{\alpha=0^{-}} = \kappa C_0 f_0(z_0). $$

The symmetric result holds for $\alpha_0 = \pi$.

Now, let us consider the asymptotics as $\alpha \to \alpha_b$ (and $\tilde{\alpha}_b$). If $\alpha$ remains within $M$, then the asymptotics are given by those of Theorem 3. However, this is no longer the case in the parts outside $M$. The asymptotic behaviour arises from the competition between the saddle point $(x(\alpha), y(\alpha))$ and the branching point $(x(\alpha_b), y(\alpha_b))$ and thus depends on the speed of convergence of $\alpha \to \alpha_b$ relative to $r \to \infty$. The proof can be found at the end of Section 4.4.

**Theorem 5** (Asymptotics for $\alpha \to \alpha_b, \tilde{\alpha}_b$). Let $z_0 \in \mathbb{R}^2$ and let $\alpha_0 = \alpha_b$. Assume $x_{max}^{+} > x_{max}^{-}$, i.e., $\alpha_b \in (0, \pi)$ (cases A and D of Figure 5). Define

$$ (1.34) \qquad K_{+}^{2} = \frac{\Sigma_{22}^{+} (y(\alpha_b) - Y^{-}(x(\alpha_b)))}{4|\sin(\alpha_b)| (\Sigma_{11}^{+} \sin^2(\alpha_b) - 2\Sigma_{12}^{+} \sin(\alpha_b) \cos(\alpha_b) + \Sigma_{22}^{+} \cos^2(\alpha_b))}. $$

Then:

(i) If $\alpha > \alpha_b$, then (1.27) holds.

(ii) If $\alpha < \alpha_b$ and $r(\alpha_b - \alpha)^2$ is bounded, then (1.27) holds.

(iii) If $\alpha < \alpha_b$, $r(\alpha - \alpha_b)^2 \to +\infty$, and $\frac{e^{K_+^2 r(\alpha_b - \alpha)^2}}{r(\alpha_b - \alpha)^{3/2}} \to 0$, then (1.27) holds.

(iv) If $\alpha < \alpha_b$, $r(\alpha - \alpha_b)^2 \to +\infty$, and $\frac{e^{K_+^2 r(\alpha_b - \alpha)^2}}{r(\alpha_b - \alpha)^{3/2}} \to +\infty$, then

$$ (1.35) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\alpha \to \alpha_b}{\sim} C_{br} f_0(z_0) \frac{e^{-r(\cos(\alpha)x(\alpha)+\sin(\alpha)y(\alpha))}}{r^{3/2}(\alpha_b - \alpha)^{3/2}}, $$

where $f_0(z_0) > 0$ is given by (1.26) and $C_{br} = |\sin(\alpha_b)|^{3/2}C_0$ (see (1.24)).

(v) If $\alpha < \alpha_b$ (resp. $\alpha > \alpha_b$), $r(\alpha - \alpha_b)^2 \to +\infty$, and

$$ \frac{e^{K_+^2 r(\alpha_b - \alpha)^2}}{r(\alpha_b - \alpha)^{3/2}} \to c > 0, $$

then

$$ (1.36) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\alpha \to \alpha_b}{\sim} (C(\alpha_b)h_{\alpha_b}(z_0) + cC_{br}f_0(z_0)) \frac{e^{-r(\cos(\alpha)x(\alpha)+\sin(\alpha)y(\alpha))}}{\sqrt{r}}, $$

where $h_{\alpha_b}(z_0)$ is given by (1.30).

Now assume that $x_{max}^{+} < x_{max}^{-}$ (cases B and C of Figure 5). Then, the analogous cases (i) to (v) hold. More precisely, let

$$ K_{-}^{2} = \frac{\Sigma_{22}^{-}(z(\alpha_b) - Z^{+}(x(\alpha_b)))}{4|\sin(\alpha_b)| (\Sigma_{11}^{-} \sin^2(\alpha_b) - 2\Sigma_{12}^{-} \sin(\alpha_b) \cos(\alpha_b) + \Sigma_{22}^{-} \cos^2(\alpha_b))}. $$
(i)-(iii) In cases (i) to (iii), replace $K_+^2$ by $K_-^2$ and the conditions $\alpha < \alpha_b$ (resp. $\alpha > \alpha_b$) by $\alpha > \alpha_b$ (resp. $\alpha < \alpha_b$). Then, (1.28) holds in these cases.

(iv)-(v) In cases (iv) to (v), replace $K_+^2$ by $K_-^2$ and the condition $\alpha < \alpha_b$ by $\alpha > \alpha_b$. Then, the corresponding asymptotics hold replacing $y(\alpha)$ by $z(\alpha)$. Namely, in case (iv),

$$ (1.37) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\substack{r \to \infty \\ \alpha \to \alpha_b}}{\sim} C_{br} f_0(z_0) \frac{e^{-r(\cos(\alpha)x(\alpha_b)+\sin(\alpha)z(\alpha_b))}}{r^{3/2}|\alpha_b - \alpha|^{3/2}}, $$

where $f_0(z_0) > 0$ is given by (1.25). In case (v),

$$ (1.38) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\substack{r \to \infty \\ \alpha \to \alpha_b}}{\sim} (C(\alpha_b)h_{\alpha_b}(z_0) + c C_{br} f_0(z_0)) \frac{e^{-r(\cos(\alpha)x(\alpha_b)+\sin(\alpha)z(\alpha_b))}}{\sqrt{r}}, $$

where $h_{\alpha_b}(z_0)$ is given by (1.31).

The symmetric result holds for $\alpha_0 = \tilde{\alpha}_b$.

The direction $\alpha_b$ gives rise to two harmonic functions. The first of these is $h_{\alpha_b}(z_0)$, which is defined in (1.30) when $\alpha = \alpha_b$. This function can also be viewed as the limit of the functions $h_\alpha(z_0)$ obtained in Theorem 3 as $\alpha \to \alpha_b$. The second function is $f_0(z_0)$, given by (1.25), and corresponds to the direction $\alpha_0 = 0$. When $\alpha \to \alpha_b$ and $r \to \infty$ at the rate described in case (v), linear combinations of these two functions determine the asymptotic constants as in (1.36).

The last asymptotic theorem deals with the remaining cases where $\alpha_0 \in [0, 2\pi] \setminus \mathcal{M}$ or $\alpha \to 0, \pi$ as $\alpha \notin \mathcal{M}$. The asymptotics are then determined by the branching points. The proof is given at the end of Section 4.5.

**Theorem 6** (Asymptotics for $\alpha \to \alpha_0$, $\alpha_0 \in [0, 2\pi] \setminus \mathcal{M}$, or $\alpha \to 0, \pi$ with $\alpha \in [0, 2\pi] \setminus \mathcal{M}$). Assume $x_{max}^+ > x_{max}^-$ (cases A and D of Figure 5). Let $z_0 \in \mathbb{R}^2$ and let $0 \le \alpha_0 < \alpha_b$. Then:

$$ (1.39) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\substack{r \to \infty \\ \alpha \to \alpha_0, \alpha \ge 0}}{\sim} C'_{br}(\alpha_0) f_0(z_0) \frac{e^{-r(\cos(\alpha)x(\alpha_b)+\sin(\alpha)y(\alpha_b))}}{r^{3/2}}, $$

where $C'_{br}(\alpha_0) = \left( \frac{\sin(\alpha_b)}{\sin(\alpha_b - \alpha_0)} \right)^{3/2} C_0$ (see (1.24)) and where $f_0(z_0)$ is given by (1.26).

Assume now $x_{max}^+ < x_{max}^-$ (cases B and C of Figure 5) and let $\alpha_b < \alpha_0 \le 2\pi$. Then,

$$ (1.40) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\substack{r \to \infty \\ \alpha \to \alpha_0, \alpha \le 2\pi}}{\sim} C'_{br}(\alpha_0) f_0(z_0) \frac{e^{-r(\cos(\alpha)x(\alpha_b)+\sin(\alpha)z(\alpha_b))}}{r^{3/2}}, $$

where $C'_{br}(\alpha_0)$ has the same expression and where $f_0(z_0)$ is given by (1.25).

The symmetric results hold for $\tilde{\alpha}_b < \alpha_0 \le \pi$ in the cases A and B and for $\pi \le \alpha_0 < \tilde{\alpha}_b$ in the cases C and D.

Note that the asymptotics (1.39) formally match case (iv) of Theorem 5 in the direction $\alpha_b$, since $\sin(\alpha_b - \alpha) \sim \alpha_b - \alpha$ as $\alpha \to \alpha_b$. Since $C'_{br}(\alpha)|_{\alpha=0} = C_0$, the asymptotics given by (1.39) also formally match (1.23) for $\alpha = 0$. Furthermore, we observe that a single harmonic function, $f_0(z_0)$ (arising from the asymptotics at $\alpha = 0$ obtained in Theorem 2) corresponds to all directions described in Theorem 6.

### 1.4.3. Martin Boundary and positive harmonic functions.
Before stating the results on the Martin boundary and harmonic functions, let us recall the definition of harmonic function.

**Definition 1.5.** A measurable non-negative function $h$ is called harmonic with respect to the Markov process $Z$ if for all open set $U$ relatively compact in $\mathbb{R}^2$ and $z_0 \in \mathbb{R}^2$,

$$ h(z_0) = \mathbb{E}_{z_0} [h(Z_{TU}) \mathbf{1}_{T_U < +\infty}] $$

where $T_U = \inf\{t \ge 0 \mid X_t \in U\}$.

The general theory of Martin boundary [20, Theorem 3] justifies that the limits of the quotient of Green's functions $\frac{g(z_0, z)}{g(0, z)}$ as $|z| \to +\infty$ – called Martin functions – are generally excessive in $z_0$ (see [20, Section 3]), which is weaker than harmonic. In Theorems 2 to 6, we obtained all Martin functions. In particular, we obtained:
* For each direction $\alpha_0 \in \mathcal{M}$, one function $h_{\alpha_0}(z_0)$ (see Theorem 3);

* For the directions 0 and $\pi$, two functions $f_0(z_0)$ and $f_\pi(z_0)$ (see Theorems 2 and 4);

* For $\alpha_0 = \alpha_b$, all the convex combinations of $h_{\alpha_b}(z_0)$ and $f_0(z_0)$, see (1.36) (Theorem 5);

* Similarly, for $\alpha_0 = \tilde{\alpha}_b$, with $h_{\tilde{\alpha}_b}(z_0)$ and $f_\pi(z_0)$ (see Theorem 5);

* For each direction $\alpha_0 \in [0, 2\pi] \setminus \mathcal{M}$, the functions $f_0(z_0)$ and $f_\pi(z_0)$ (see Theorem 6).

We prove in Section 5 that those functions are harmonic.

**Remark 1.6 (Partial differential equation).** The partial differential equation associated with the harmonic of a function $h$ with respect to our process is given by

$$ (1.41) \qquad \begin{cases} \left(\frac{1}{2}\nabla \cdot \Sigma^+ \nabla + \mu^+ \cdot \nabla\right) h(z) = 0 & \text{for } z \in \mathbb{R} \times (0, +\infty), \\ \left(\frac{1}{2}\nabla \cdot \Sigma^- \nabla + \mu^- \cdot \nabla\right) h(z) = 0 & \text{for } z \in \mathbb{R} \times (-\infty, 0), \\ (q_1, 1+q_2)\nabla h(x, 0^+) = (-q_1, 1-q_2)\nabla h(x, 0^-) & \text{for } x \in \mathbb{R}, \\ h(x, 0^+) = h(x, -) \text{ and } (1,0) \cdot \nabla h(x, 0^+) = (1,0) \cdot \nabla h(x, 0^-) & \text{for } x \in \mathbb{R}. \end{cases} $$

It is straightforward to verify that the harmonic functions $f_0, f_\pi$, and $h_\alpha, \alpha \in \mathcal{M} \setminus \{0, \pi\}$ obtained in our main results satisfy this equation.

Then, the structures of the Martin boundary and the minimal Martin boundary follow: the proof is detailed in Section 5. Let us just notice the fact that the non-minimality of the full Martin boundary comes from the Green's functions asymptotics (1.36) in Theorem 5.

**Theorem 7 (Martin Boundary, minimal Martin Boundary and positive harmonic functions).** All of the functions $h_\alpha, \alpha \in \mathcal{M} \setminus \{0, \pi\}$, $f_0$ and $f_\pi$ are minimal harmonic functions. Furthermore, the Martin boundary $\Gamma$ of the process is given by

$$ (1.42) \qquad \Gamma = [\{e^{i\alpha}\}_{\alpha \in \mathcal{M}} \cup \{ue^{i\alpha}\}_{u \in [1,2], \alpha=\alpha_b, \tilde{\alpha}_b}] / \mathcal{R} \sim \mathbb{S}^1 $$

where $\mathcal{R}$ is the equivalence relation defined by $e^{i \cdot 0} \mathcal{R} 2e^{i \alpha_b}$, $e^{i \pi} \mathcal{R} 2e^{i \tilde{\alpha}_b}$ (see Figure 8).

The minimal Martin boundary, denoted by $\Gamma_{min}$, is given by

$$ (1.43) \qquad \Gamma_{min} = \{e^{i\alpha}\}_{\alpha \in \mathcal{M}} $$

(see Figure 5). Therefore, according to Martin boundary theory, for each positive harmonic function $h$ (with respect to the process), there exists a unique Radon measure $\nu$ on $\mathcal{M}$ such that:

$$ (1.44) \qquad \forall z \in \mathbb{R}^2, \quad h(z) = \int_{\mathcal{M}} \frac{h_\alpha(z)}{h_\alpha(0)} d\nu(\alpha). $$

where we have used the notation $\frac{h_0(z_0)}{h_0(0)} = \frac{f_0(z_0)}{f_0(0)}$ and $\frac{h_\pi(z_0)}{h_\pi(0)} = \frac{f_\pi(z_0)}{f_\pi(0)}$ since

$$ \frac{h_\alpha(z_0)}{h_\alpha(0)} \underset{\alpha \in \mathcal{M}}{\longrightarrow} \frac{f_0(z_0)}{f_0(0)} \quad \text{and} \quad \frac{h_\alpha(z_0)}{h_\alpha(0)} \underset{\alpha \in \mathcal{M}}{\longrightarrow} \frac{f_\pi(z_0)}{f_\pi(0)}. $$

Using the preceding representation theorem, we can derive the well-known one-dimensional result concerning the probabilities of escaping into the upper or lower half-plane. By assumption (1.7) on the drift (see Figure 1), the second component $B$ of the process naturally satisfies $B_t \underset{t \to +\infty}{\longrightarrow} \pm\infty$. By standard arguments, the mappings

$$ z_0 \longmapsto \mathbb{P}_{z_0}(B_t \underset{t \to +\infty}{\longrightarrow} +\infty) \quad \text{and} \quad z_0 \longmapsto \mathbb{P}_{z_0}(B_t \underset{t \to +\infty}{\longrightarrow} -\infty) $$

are harmonic functions. Looking at the bounded non negative harmonic functions (which reduces to convex combinations of $h_{\alpha_{\mu^+}}$ and $h_{\alpha_{\mu^-}}$), we deduce in Section 5 the following classical one-dimensional result.

**Corollary 1.7 (Upper and lower escape probabilities).** Let $(a_0, b_0) \in \mathbb{R}^2$. Then:

$$ \mathbb{P}_{(a_0, b_0)}(B_t \underset{t \to +\infty}{\longrightarrow} +\infty) = h_{\alpha_{\mu^+}}(b_0), \quad \text{and} \quad \mathbb{P}_{(a_0, b_0)}(B_t \underset{t \to +\infty}{\longrightarrow} -\infty) = h_{\alpha_{\mu^-}}(b_0). $$
<figure><img src="image_10.png" /><figcaption>FIGURE 8. Representation of the Martin boundary $\Gamma$ depending on the parameters.</figcaption></figure>

In particular,

$$ (1.45) \qquad \mathbb{P}_{(a_0, b_0)}(B_t \xrightarrow{t \to +\infty} +\infty) = 1 - \mathbb{P}_{(a_0, b_0)}(B_t \xrightarrow{t \to +\infty} -\infty) $$

$$ (1.46) \qquad = \begin{cases} 1 + \frac{\mu_2^-}{\mu_2^+ - \mu_2^-} e^{-2b_0 \mu_2^+ / \Sigma_{22}^+} & \text{if } b_0 \ge 0, \\ \frac{\mu_2^+}{\mu_2^+ - \mu_2^-} e^{-2b_0 \mu_2^- / \Sigma_{22}^-} & \text{if } b_0 < 0. \end{cases} $$

## 2. PROCESS AND FUNCTIONAL EQUATION

In Section 2.1, we prove Proposition 1.2. Then, we present some preliminary lemmas in Section 2.2, which are necessary to establish the functional equation (1.13). A short intuitive proof of the functional equation is given in Section 2.3, followed by a detailed one in Section 2.4. As a first reading, the reader may focus on the short proof.

### 2.1. Construction of the process.
In this section, we prove Proposition 1.2. The proof is adapted from [26, Proposition 1], where the covariance matrix is assumed to be diagonal. We first establish the existence and pathwise uniqueness of the solution to equation (1.3), and then identify the corresponding generator.
Existence and uniqueness of (1.3). First, note that (1.3) can be written as the following SDE:

$$ (2.1) \qquad \begin{cases} dA_t = \sqrt{\Sigma_{11}(B_t) - \frac{\Sigma_{12}^2(B_t)}{\Sigma_{22}(B_t)}} dW_t^1 + \frac{\Sigma_{12}(B_t)}{\sqrt{\Sigma_{22}(B_t)}} dW_t^2 + \mu_1(B_t) dt + \frac{\Sigma_{12}^+ - \Sigma_{12}^-}{\Sigma_{22}^+ + \Sigma_{22}^-} dL_t^0(B), & (1) \\ dB_t = \sqrt{\Sigma_{22}(B_t)} dW_t^2 + \mu_2(B_t) dt + \frac{\Sigma_{22}^+ - \Sigma_{22}^-}{\Sigma_{22}^+ + \Sigma_{22}^-} dL_t^0(B), & (2) \end{cases} $$

Here, $((W_t^1, W_t^2))_{t \ge 0}$ is a standard two-dimensional Brownian motion, and $L^0(B)$ denotes the symmetric local time of $B$ at 0. Indeed, the matrix

$$ \sigma(B_t) = \begin{pmatrix} \sqrt{\Sigma_{11}(B_t) - \frac{\Sigma_{12}^2(B_t)}{\Sigma_{22}(B_t)}} & \frac{\Sigma_{12}(B_t)}{\sqrt{\Sigma_{22}(B_t)}} \\ 0 & \sqrt{\Sigma_{22}(B_t)} \end{pmatrix} $$

satisfies $\sigma(B_t) \sigma(B_t)^T = \Sigma(B_t)$.

Let $(W^1, W^2)$ be a standard two-dimensional Brownian motion. By [24], there exists a pathwise unique solution $(B_t)_{t \ge 0}$ of equation (2) starting from $b_0 \in \mathbb{R}$. Then, equation (1) admits a pathwise unique solution starting from $a_0 \in \mathbb{R}$ given by:

$$ A_t = a_0 + \int_0^t \sqrt{\Sigma_{11}(B_s) - \frac{\Sigma_{12}^2(B_s)}{\Sigma_{22}(B_s)}} dW_s^1 + \int_0^t \frac{\Sigma_{12}(B_s)}{\sqrt{\Sigma_{22}(B_s)}} dW_s^2 + \int_0^t \mu_1(B_s) ds + \frac{\Sigma_{12}^+ - \Sigma_{12}^-}{\Sigma_{22}^+ + \Sigma_{22}^-} L_t^0(B). $$

Using a generalisation of the Yamada Watanabe Engelbert theorem (see [23, Proposition 2.10]), uniqueness in law also holds for the process starting from $(a_0, b_0)$. By standard arguments, we deduce that the family of probability laws indexed by $(a_0, b_0) \in \mathbb{R}^2$ defines a Markov process.

**Shape of the generator.** Let us prove that the generator can be written in divergence form. To this end, consider the continuous piecewise affine regularisations of $\Sigma$ and $\mu$ defined by: $\Sigma^n(b) = \Sigma^+$ if $b \ge \frac{1}{n}$, $\Sigma^n(b) = \Sigma^-$ if $b \le \frac{1}{n}$, and $\Sigma^n$ affine on $[-\frac{1}{n}, \frac{1}{n}]$ (similarly for $\mu^n$).

Let $(W^1, W^2)$ be a two-dimensional Brownian motion. By standard arguments, the SDE

$$ \begin{cases} dA_t^n = \sqrt{\Sigma_{11}^n(B_t^n) - \frac{\Sigma_{12}^n(B_t^n)}{\Sigma_{22}^n(B_t^n)}} dW_t^1 + \frac{\Sigma_{12}^n(B_t^n)}{\sqrt{\Sigma_{22}^n(B_t^n)}} dW_t^2 + \left(\mu_1^n(B_t^n) + \frac{1}{2} \frac{d}{dy} \Sigma_{12}^n(B_t^n)\right) dt, \\ dB_t^n = \sqrt{\Sigma_{22}^n(B_t^n)} dW_t^2 + \left(\mu_2^n(B_t^n) + \frac{1}{2} \frac{d}{dy} \Sigma_{22}^n(B_t^n)\right) dt \end{cases} $$

defines a Markov process with generator $\mathcal{L}_n = \frac{1}{2} \nabla \cdot (\Sigma^n(y)\nabla) + \mu^n(y) \cdot \nabla$. By [38], this family of processes converges in law to the strong Markov process with generator given by (1.6). Moreover, using arguments similar to those in [26], one can show that $(A_t^n, B_t^n)$ converges in the $L^1$ norm to $(A_t, B_t)$ for all $t \ge 0$. By identifying the limits, we conclude that (1.6) is indeed the generator of $(X, Y)$.

**2.2. Preliminary lemmas.** To derive the desired functional equation, we first establish a few preliminary lemmas. We begin by stating Aronson-type estimates for the transition density, along with some elementary consequences. These inequalities hold since the process admits a generator in divergence form.

**Lemma 2.1** (Some estimates due to Aronson). There exists a constant $M > 0$ such that, for all $t > 0$ and $z_0 \in \mathbb{R}^2$,

$$ (2.2) \qquad \frac{1}{Mt} e^{-M|z-z_0|^2/t-Mt} \le p_t^{z_0}(z) \le \frac{M}{t} e^{-|z-z_0|^2/Mt+Mt}. $$

Let $g_t^{z_0}(z)$ be defined by

$$ g_t^{z_0}(z) := \int_0^t p_s^{z_0}(u, 0) ds. $$

Then,

$$ (2.3) \qquad g_t^{z_0}(z) \le \frac{M^2 e^{-\frac{|z-z_0|^2}{Mt}} + Mt}{|z-z_0|^2} t. $$

Finally, for all $z_0 = (a_0, b_0) \in \mathbb{R}^2$ and $x \in \mathbb{R}$,

$$ (2.4) \qquad \mathbb{E}_{(a_0, b_0)} [e^{xAt}] \le e^{x^2Mt+a_0x}. $$
Proof. Inequalities (2.2) are proved in [38, Theorem II.3.8]. □

We now relate the occupation measure on the axis {$y = 0$}, defined in (1.10), to the values of the Green's function $g^{z_0}$ restricted to {$y = 0$}. This connection is made precise in the following lemma.

**Lemma 2.2** (Link between $H(z_0, \cdot)$ and $g^{z_0}(u, 0)$). Let $z_0 = (a_0, b_0) \in \mathbb{R}^2$ with $b_0 \neq 0$. Let $f: \mathbb{R} \to [0, +\infty)$ be a continuous nonnegative function. Then:

$$ (2.5) \qquad \mathbb{E}_{z_0} \left[ \int_0^{+\infty} f(A_s) dL_s^0(B) \right] = \left( \frac{\Sigma_{22}^{-} + \Sigma_{22}^{+}}{2} \right) \int_{\mathbb{R}} g^{z_0}(u, 0) f(u) du. $$

In particular, combining this with the asymptotic behavior as $t \to +\infty$ and the definition (1.11), we obtain the identity

$$ (2.6) \qquad \varphi^{z_0}(x) = \left( \frac{\Sigma_{22}^{-} + \Sigma_{22}^{+}}{2} \right) \int_{\mathbb{R}} g^{z_0}(u, 0) e^{-xu} du. $$

Proof. The proof reduces to showing that for all $t \ge 0$ and for every continuous, compactly supported, nonnegative function $f$, we have:

$$ (2.7) \qquad \mathbb{E}_{z_0} \left[ \int_0^t f(A_s) dL_s^0(B) \right] = \left( \frac{\Sigma_{22}^{-} + \Sigma_{22}^{+}}{2} \right) \int_{\mathbb{R}} g_t^{z_0}(u, 0) f(u) du. $$

Let $f$ be such a function and let $\epsilon > 0$. Then:

$$ \mathbb{E} \left[ \frac{1}{2\epsilon} \int_0^t f(A_s) \mathbb{1}_{B_s \in [-\epsilon, \epsilon]} d\langle B \rangle_s \right] = \int_0^t \int_{\mathbb{R}} f(a) \cdot \frac{1}{2\epsilon} \left( \Sigma_{22}^{-} \int_{[-\epsilon, 0]} p_s^{z_0}(a, b) db + \Sigma_{22}^{+} \int_{[0, \epsilon]} p_s^{z_0}(a, b) db \right) da ds. $$

Using the boundedness of $f$ and the Aronson estimates, the right-hand side converges to

$$ \frac{\Sigma_{22}^{-} + \Sigma_{22}^{+}}{2} \int_{0}^{t} \int_{\mathbb{R}} f(a) p_{s}^{z_{0}}(a, 0) da ds = \frac{\Sigma_{22}^{-} + \Sigma_{22}^{+}}{2} \int_{\mathbb{R}} g_{t}^{z_{0}}(a, 0) f(a) da. $$

as $\epsilon \to 0$. Moreover, we recall the almost sure identity:

$$ (2.8) \qquad \forall t \ge 0, \quad L_t^0(B) = \lim_{\epsilon \to 0} \int_0^t \frac{1}{2\epsilon} \mathbb{1}_{B_s \in [-\epsilon, \epsilon]} d\langle B \rangle_s. $$

If $s \mapsto H_s$ is a measurable step function with compact support, then:

$$ (2.9) \qquad \frac{1}{2\epsilon} \int_0^t H_s \mathbb{1}_{B_s \in [-\epsilon, \epsilon]} d\langle B \rangle_s \xrightarrow[\epsilon \to 0]{L^1, a.s.} \int_0^t H_s dL_s^0(B). $$

Since $s \mapsto f(A_s)$ is the uniform limit of step functions, the convergence in (2.9) remains valid when $H_s$ is replaced with $f(A_s)$. □

For later technical reasons, it is convenient to establish the finiteness of some exponential moments of $L_{\infty}^0(B)$. To do so, we establish the following two lemmas. For any real-valued trajectory $b = (b_t)_{t \ge 0}$, we define:

$$ T(b) := \inf\{t \ge 0 \mid |b_t| = 1\}. $$

**Lemma 2.3** (Exponential moments, part 1). Let $T = T(B)$. Then, for all $\lambda \ge 0$ and $z_0 \in \mathbb{R}^2$, we have:

$$ \mathbb{E}_{z_0} [e^{\lambda L_T(B)}] < +\infty. $$

Proof. Let us recall that $B$ satisfies:

$$ dB_t = \Sigma_{22}(B_t) dW_t + \mu_2(B_t) dt + q_2 dL_t^0(B), $$

where $W$ is a Brownian motion. By Girsanov's theorem, we can reduce the proof to the case $\mu_2 = 0$. Using the strong Markov property and the fact that the Stieltjes measure $dL^0(B)$ is supported on $\{t \ge 0 \mid B_t = 0\}$, we may further reduce the proof to the case $b_0 = 0$. By the Itô–Tanaka formula, we have:

$$ (2.10) \qquad |B_t| = \int_0^t \operatorname{sgn}(B_s) (\sigma_2(B_s) dW_s + q_2 dL_s^0(B)) + L_t^0(B) = \int_0^t \operatorname{sgn}(B_s) \sigma_2(B_s) dW_s + L_t^0(B) =: M_t + L_t^0(B). $$
Then, using the properties of exponential martingales:

$$ \begin{aligned} \mathbb{E} [e^{\lambda L_{t \wedge T}^0 (B)}] &\le \mathbb{E} [e^{\lambda |B_{t \wedge T}| + \lambda M_t}] \le \mathbb{E} [e^{\lambda + \frac{\lambda^2}{2} \langle M \rangle_{t \wedge T}}] \\ &\le e^{\lambda} \mathbb{E} [e^{\frac{\lambda^2}{2} (t \wedge T)}] \le e^{\lambda} \mathbb{E} [e^{\frac{\lambda^2}{2} T}]. \end{aligned} $$

Define the time-change $C_t := \int_0^t \Sigma_{22}(B_s)^2 ds$ for $t \ge 0$. Then, for all $t \ge 0$, $\min(\Sigma_{22}^-, \Sigma_{22}^+)t \le C_t \le \max(\Sigma_{22}^-, \Sigma_{22}^+)t$. As a result,

$$ T(B) \le \max(\Sigma_{22}^-, \Sigma_{22}^+) \cdot T(B_{C_t}). $$

Moreover, by the Dubins-Schwarz theorem, the process $(M_{C_t})_{t \ge 0}$ is a Brownian motion. Using (2.10), we deduce that $(|B_{C_t}|)_{t \ge 0}$ is a reflected Brownian motion. Finally,

$$ \mathbb{E} [e^{\lambda L_T}] \le e^{\lambda} \mathbb{E} [e^{\frac{\lambda^2}{2} \max(\Sigma_{22}^-, \Sigma_{22}^+) \cdot T(B_{C_t})}] < +\infty, $$

by standard results on reflected Brownian motion.

<figure><img src="image_11.png" /></figure>

Let us recall that we are working under Assumption (1.7).

**Lemma 2.4** (Exponential moments, part 2). There exists $\lambda^* > 0$ such that for all $0 \le \lambda < \lambda^*$ and $z_0 \in \mathbb{R}^2$, we have:

$$ \mathbb{E}_{z_0} [e^{\lambda L_{\infty}^0 (B)}] < +\infty. $$

*Proof.* For the proof, we use the strong Markov property and consider the back-and-forth excursions of the process between the sets $\{y=0\}$ and $\{|y| \ge 1\}$. As in the proof of Lemma 2.3, we may assume that $b_0 = 0$. Define the following stopping times:

$$ \sigma_0 = 0, \quad T_n = \inf\{t \ge \sigma_{n-1} \mid |B_t| = 1\}, \quad \sigma_n = \inf\{t \ge T_n \mid B_t = 0\}, \quad n \ge 1. $$

Note that, almost surely, there exists a (random) rank $n$ such that $\sigma_k = +\infty$ for all $k \ge n$. Then, by the strong Markov property,

$$ \begin{aligned} \mathbb{E}_{z_0} [e^{\lambda L_{\infty}^0 (B)}] &= \sum_{n=1}^{+\infty} \mathbb{E}_{z_0} \left[ \mathbb{1}_{\sigma_{n-1} < +\infty}, \sigma_n = +\infty \prod_{k=1}^{n} e^{\lambda(L_{T_k}^0(B) - L_{\sigma_{k-1}}^0(B))} \right] \\ &= \sum_{n=1}^{+\infty} \mathbb{E}_0 \left[ \mathbb{1}_{\sigma_1 = +\infty} e^{\lambda L_{T_1}^0(B)} \right] \cdot \mathbb{E}_0 \left[ \mathbb{1}_{\sigma_1 < +\infty} e^{\lambda L_{T_1}^0(B)} \right]^{n-1}. \end{aligned} $$

Moreover, by Lemma 2.3 and the dominated convergence theorem, we have:

$$ \mathbb{E}_0 \left[ \mathbb{1}_{\sigma_1 < +\infty} e^{\lambda L_{T_1}^0 (B)} \right] \underset{\lambda \to 0^+}{\longrightarrow} \mathbb{P}_0(\sigma_1 < +\infty) < 1. $$

This shows that $\mathbb{E}_{z_0} [e^{\lambda L_{\infty}^0 (B)}]$ is finite for $\lambda > 0$ small enough.

<figure><img src="image_12.png" /></figure>

This lemma allows us to prove the final technical result before the proof of Proposition 1.4.

**Lemma 2.5** (Convergence of exponential moments of $A_t$). Let $z_0 \in \mathbb{R}^2$. Then, the first component $A$ of $Z$ satisfies, $\mathbb{P}_{z_0}$-almost surely,

$$ A_t \underset{t \to \infty}{\longrightarrow} +\infty. $$

Furthermore, there exists a constant $\eta > 0$ such that for all $0 < u < \eta$,

$$ \mathbb{E}_{z_0} \left[ e^{-u A_t} \right] \underset{t \to +\infty}{\longrightarrow} 0. $$

*Proof.* The first statement follows directly from the fact that the process no longer touches the axis $\{y=0\}$ after a (random) time $T$. Indeed, after this time $T$, $A$ evolves as a Brownian motion with drift $\mu_1^+ > 0$ or $\mu_1^- > 0$, and therefore $A_t \to +\infty$ almost surely. For the second statement, recall that

$$ dA_t = \Sigma_{11}(B_t) d\tilde{B}_t + \mu_1(B_t) dt + q_1 dL_t^0(B), $$
where $\tilde{B}$ is a Brownian motion (possibly correlated with $B$). Define the martingale $M_t := \int_0^t \Sigma_{11}(B_s) d\tilde{B}_s$, $t \ge 0$. Then, by the Cauchy-Schwarz inequality, we obtain for all $u \ge 0$:

$$ \begin{aligned} \mathbb{E} [e^{-uA_t}]^2 &\le \mathbb{E} [e^{-2uM_t}] \cdot \mathbb{E} \left[ e^{-2uq_1 L_t^0(B) - 2u \int_0^t \mu_1(B_s) ds} \right] \\ &\le \mathbb{E} \left[ e^{-2uq_1 L_\infty^0(B)} \right] \cdot e^{-2ut(\min(\mu_1^+, \mu_1^-) - u \max(\Sigma_{11}^+, \Sigma_{11}^-))}. \end{aligned} $$

If we choose $u > 0$ small enough so that $\mathbb{E} [e^{-2uq_1 L_\infty^0(B)}] < +\infty$ by Lemma 2.4 and such that $\min(\mu_1^+, \mu_1^-) - u \max(\Sigma_{11}^+, \Sigma_{11}^-) > 0$, then the right-hand term tends to zero as $t \to +\infty$, which completes the proof. □

2.3. **Short version of the proof of the functional equation.** We now present a brief argument to establish Proposition 1.4. The idea of the proof is to apply Itô's formula to

$$ f(a, b) = e^{ax+by} \mathbb{1}_{b \ge 0} + e^{ax+bz} \mathbb{1}_{b < 0}. $$

Note that $f \notin C^2(\mathbb{R}^2)$. In this section, we assume that all local martingales are true martingales, ignore convergence issues, and apply the standard Itô formula despite the lack of smoothness of $f$. We compute:

$$ (2.11) \qquad \nabla f(a, b) := \begin{pmatrix} xf(a, b) \\ ye^{ax+by} \mathbb{1}_{b \ge 0} + ze^{ax+bz} \mathbb{1}_{b < 0} \end{pmatrix} $$

and

$$ \nabla^2 f(a, b) = \begin{pmatrix} x^2 f(a, b) & xye^{ax+by} \mathbb{1}_{b \ge 0} + xze^{ax+bz} \mathbb{1}_{b < 0} \\ xye^{ax+by} \mathbb{1}_{b \ge 0} + xze^{ax+bz} \mathbb{1}_{b < 0} & y^2 e^{ax+by} \mathbb{1}_{b \ge 0} + z^2 e^{ax+bz} \mathbb{1}_{b < 0} + (y-z)e^{ax} \delta_{b=0} \end{pmatrix}. $$

Formally, Itô's formula yields

$$ (2.12) \qquad f(Z_t) = f(Z_0) + \int_0^t \nabla f(Z_s) dZ_s + \frac{1}{2} \sum_{i=1}^2 \sum_{j=1}^2 \int_0^t \partial_i \partial_j f(Z_s) d\langle Z^i, Z^j \rangle_s $$

(with notation $Z^1 = A$ and $Z^2 = B$). Since $dZ_t = \sigma(B_t)dB_t + \mu(B_t)dt + q dL_t^0(B)$, we get

$$ \mathbb{E} \left[ \int_0^t \nabla f(Z_s) dZ_s \right] = (x\mu_1^+ + y\mu_2^+) \mathbb{E} \left[ \int_0^t \mathbb{1}_{B_s \ge 0} e^{xA_s + yB_s} ds \right] + (x\mu_1^- + z\mu_2^-) \mathbb{E} \left[ \int_0^t \mathbb{1}_{B_s < 0} e^{xA_s + zB_s} ds \right] \\ + \left( q_1 x + \frac{q_2(y+z)}{2} \right) \mathbb{E} \left[ \int_0^t e^{xA_s} dL_s^0(B) \right]. $$

Now, let us consider the second derivative terms. Similarly, we have:

$$ \begin{aligned} \mathbb{E} \left[ \frac{1}{2} \sum_{i=1}^{2} \sum_{j=1}^{2} \int_{0}^{t} \partial_{i} \partial_{j} f(Z_{s}) d\langle Z^{i}, Z^{j} \rangle_{s} \right] = \frac{1}{2} (x^{2} \Sigma_{11}^{+} + 2xy \Sigma_{12}^{+} + y^{2} \Sigma_{22}^{+}) \mathbb{E} \left[ \int_{0}^{t} \mathbb{1}_{B_{s} \ge 0} e^{xA_{s} + yB_{s}} ds \right] \\ + \frac{1}{2} (x^{2} \Sigma_{11}^{-} + 2xz \Sigma_{12}^{-} + z^{2} \Sigma_{22}^{-}) \mathbb{E} \left[ \int_{0}^{t} \mathbb{1}_{B_{s} < 0} e^{xA_{s} + zB_{s}} ds \right] \\ + \left( \frac{y-z}{2} \right) \mathbb{E} \left[ \int_{0}^{t} e^{xA_{s}} \delta_{B_{s}=0} ds \right]. \end{aligned} $$

The last term can be interpreted as $(\frac{y-z}{2}) \mathbb{E} [\int_0^t e^{xA_s} dL_t^0(B)]$. Then, by rearranging the terms in the expected value of (2.12), we obtain (2.13)

$$ \gamma_-(x, z) \mathbb{E} \left[ \int_0^t \mathbb{1}_{B_s < 0} e^{xA_s + zB_s} ds \right] + \gamma_+(x, y) \mathbb{E} \left[ \int_0^t \mathbb{1}_{B_s > 0} e^{xA_s + yB_s} ds \right] + \gamma(x, y, z) \mathbb{E} \left[ \int_0^t e^{xA_s} dL_s^0(B) \right] \\ = \mathbb{E}[f(Z_t)] - (e^{xa_0 + yb_0} \mathbb{1}_{b_0 \ge 0} + e^{xa_0 + zb_0} \mathbb{1}_{b_0 < 0}). $$

Assuming that $x, y$, and $z$ are chosen such that $\mathbb{E}[f(Z_t)] \to 0$ as $t \to +\infty$ and that the Laplace transforms converge, we obtain the desired equation.
2.4. Detailed version of the proof of the functional equation. In this section, we provide a detailed justification for the reasoning presented in Section 2.3 when $b_0 \neq 0$ (see Remark 2.6 for the case $b_0 = 0$). Note that an Itô–Tanaka formula exists in dimension 2 for non-smooth functions [12]. However, it is not straightforward to use, so instead we opted for an approximation using smooth functions. We then approximate the preceding function $f$ by smooth functions $f_\epsilon$ defined by

$$ (2.14) \qquad f_{\epsilon}(a, b) = e^{ax+bz} + \psi(b/\epsilon)(e^{ax+by} - e^{ax+bz}), $$

where $\psi$ is an increasing $C^\infty$ function satisfying $\psi(u) := \begin{cases} 0 & \text{if } u \le -1, \\ 1 & \text{if } u \ge 1, \\ \frac{1}{2} & \text{if } u = 0. \end{cases}$ Note that $f_\epsilon(a, b) = e^{ax+by}$ if $b \ge \epsilon$ and $f_\epsilon(a, b) = e^{ax+bz}$ if $b \le -\epsilon$. The Itô formula can now be applied to the $C^\infty$ function $f_\epsilon$: for all $t \ge 0$, we have

$$ (2.15) \qquad f_{\epsilon}(Z_t) = f_{\epsilon}(Z_0) + \int_{0}^{t} \nabla f_{\epsilon}(Z_s) dZ_s + \frac{1}{2} \sum_{i=1}^{2} \sum_{j=1}^{2} \int_{0}^{t} \partial_{i} \partial_{j} f_{\epsilon}(Z_s) d\langle Z^{i}, Z^{j} \rangle_{s}. $$

Step 1: Study of integrals against $dZ_s$. Note that

$$ (2.16) \qquad \nabla f_{\epsilon}(a, b) := \begin{pmatrix} x f_{\epsilon}(a, b) \\ ze^{ax+bz} + \frac{1}{\epsilon}\psi'(b/\epsilon)(e^{ax+by} - e^{ax+bz}) + \psi(b/\epsilon)(ye^{ax+by} - ze^{ax+bz}) \end{pmatrix}. $$

Then,

$$ (2.17) \qquad \int_0^t \nabla f_\epsilon(Z_s) \mathbb{1}_{B_s \ge 0} dZ_s = x \int_0^t \mathbb{1}_{B_s \ge 0} f_\epsilon(Z_s) dA_s \\ + \int_0^t \mathbb{1}_{B_s \ge 0} \left( ze^{xA_s+zB_s} + \frac{1}{\epsilon}\psi'(B_s/\epsilon)(e^{xA_s+yB_s} - e^{xA_s+zB_s}) + \psi(B_s/\epsilon)(ye^{xA_s+yB_s} - ze^{xA_s+zB_s}) \right) dB_s, $$

where $\mathbb{1}_{B_s \ge 0} dA_s = (\Sigma_{11}^+ dB_s^1 + \Sigma_{12}^+ dB_s^2 + \mu_1^+ ds) + q_1 dL_t^0(B)$ and $\mathbb{1}_{B_s \ge 0} dB_s = (\Sigma_{12}^+ dB_s^1 + \Sigma_{22}^+ dB_s^2 + \mu_2^+ ds) + q_2 dL_t^0(B)$. Then, by Lemma 2.4, the Brownian integrals are martingales (not just local martingales), so their expectations vanish. Therefore,

$$ (2.18) \qquad \mathbb{E}_{z_0} \left[ \int_0^t \nabla f_\epsilon(Z_s) \mathbb{1}_{B_s \ge 0} dZ_s \right] = x \mu_1^+ \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{B_s \ge 0} f_\epsilon(Z_s) ds \right] + y \mu_2^+ \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{B_s \ge \epsilon} e^{xA_s+yB_s} ds \right] \\ + \mu_2^+ \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{0 \le B_s < \epsilon} \left( ze^{xA_s+zB_s} + \frac{1}{\epsilon}\psi'(B_s/\epsilon)(e^{xA_s+yB_s} - e^{xA_s+zB_s}) + \psi(B_s/\epsilon)(ye^{xA_s+yB_s} - ze^{xA_s+zB_s}) \right) ds \right] \\ + \left( q_1 x + \frac{q_2(y+z)}{2} \right) \mathbb{E}_{z_0} \left[ \int_0^t e^{xA_s} dL_t^0(B) \right]. $$

By the dominated convergence theorem and inequality (2.4),

$$ \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{0 \le B_s < \epsilon} \left( ze^{xA_s+zB_s} + \psi(B_s/\epsilon)(ye^{xA_s+yB_s} - ze^{xA_s+zB_s}) \right) ds \right] \xrightarrow{\epsilon \to 0} 0, $$

and

$$ \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{B_s \ge \epsilon} e^{xA_s+yB_s} ds \right] \xrightarrow{\epsilon \to 0} \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{B_s \ge 0} e^{xA_s+yB_s} ds \right]. $$

Moreover, by the definition of $g_t$, we have:

$$ \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{0 \le B_s < \epsilon} \frac{1}{\epsilon} \psi'(B_s/\epsilon)(e^{xA_s+yB_s} - e^{xA_s+zB_s}) ds \right] = \int_{\mathbb{R}} \int_0^\epsilon \frac{1}{\epsilon} \psi'(b/\epsilon)(e^{ax+by} - e^{ax+bz}) g_t(a, b) db da. $$

Since $b_0 \neq 0$, estimates (2.3) yield the following limit for all $a \in \mathbb{R}$ as $\epsilon \to 0$:

$$ (2.19) \qquad \int_0^\epsilon \frac{1}{\epsilon} \psi'(b/\epsilon)(e^{ax+by} - e^{ax+bz}) g_t(a, b) db = \int_0^1 \psi'(u)(e^{ax+\epsilon u y} - e^{ax+\epsilon u z}) g_t(a, \epsilon u) du $$

$$ (2.20) \qquad \xrightarrow{\epsilon \to 0} (e^{ax} - e^{ax}) g_t(a, 0) = 0. $$
(Note that $\frac{1}{\epsilon}\psi'(\cdot/\epsilon) \underset{\epsilon\to 0}{\longrightarrow} \delta_0$ in the sense of distributions.) Integrating in $a \in \mathbb{R}$ and using (2.3) again, we get

$$ (2.21) \qquad \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{0 \le B_s < \epsilon} \frac{1}{\epsilon} \psi'(B_s/\epsilon) (e^{xA_s+yB_s} - e^{xA_s+zB_s}) ds \right] \underset{\epsilon \to 0}{\longrightarrow} 0. $$

By a similar analysis on $\int_0^t \nabla f_\epsilon(Z_s) \mathbb{1}_{B_s<0} dZ_s$, the following limit holds:

$$ (2.22) \qquad \mathbb{E}_{z_0} \left[ \int_0^t \nabla f_\epsilon(Z_s) dZ_s \right] \underset{\epsilon \to 0}{\longrightarrow} (x\mu_1^+ + y\mu_2^+) \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{B_s \ge 0} e^{xA_s+yB_s} ds \right] + (x\mu_1^- + z\mu_2^-) \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{B_s < 0} e^{xA_s+zB_s} ds \right] \\ \qquad \qquad \qquad \qquad \qquad \qquad + \left( q_1 x + \frac{q_2(y+z)}{2} \right) \mathbb{E}_{z_0} \left[ \int_0^t e^{xA_s} dL_t^0(B) \right]. $$

### Step 2: Study of integrals against $d\langle Z^i, Z^j \rangle_s$.

Since $\partial_1 f_\epsilon(a, b) = x f_\epsilon(a, b)$, we can similarly show that

$$ (2.23) \qquad \mathbb{E}_{z_0} \left[ \int_0^t \partial_1 \partial_1 f_\epsilon(Z_s) d\langle A, A \rangle_s \right] \underset{\epsilon \to 0}{\longrightarrow} \Sigma_{11}^+ x^2 \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{B_s \ge 0} e^{xA_s+yB_s} ds \right] + \Sigma_{11}^- x^2 \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{B_s < 0} e^{xA_s+zB_s} ds \right] $$

and

$$ (2.24) \qquad \mathbb{E}_{z_0} \left[ \int_0^t \partial_1 \partial_2 f_\epsilon(Z_s) d\langle A, B \rangle_s \right] \underset{\epsilon \to 0}{\longrightarrow} \Sigma_{12}^+ xy \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{B_s \ge 0} e^{xA_s+yB_s} ds \right] + \Sigma_{12}^- xz \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{B_s < 0} e^{xA_s+zB_s} ds \right]. $$

Now, let us study the last term $i = j = 2$. We have $d\langle B, B \rangle_t = \Sigma_{22}^+ \mathbb{1}_{B_t \ge 0} dt + \Sigma_{22}^- \mathbb{1}_{B_t < 0} dt$, and

$$ \partial_2 \partial_2 f_\epsilon(a, b) = z^2 e^{ax+bz} + \frac{1}{\epsilon^2} \psi''(b/\epsilon) (e^{ax+by} - e^{ax+bz}) \\ + \frac{2}{\epsilon} \psi'(b/\epsilon) (ye^{ax+by} - ze^{ax+bz}) + \psi(b/\epsilon)(y^2e^{ax+by} - z^2e^{ax+bz}). $$

By the dominated convergence theorem and inequality (2.4), we obtain similarly:

$$ \mathbb{E}_{z_0} \left[ \int_0^t \left( z^2 e^{A_t x + B_t z} + \psi(b/\epsilon)(y^2 e^{A_t x + bB_t} - z^2 e^{A_t x + B_t z}) \right) d\langle B, B \rangle_t \right] \underset{\epsilon \to 0}{\longrightarrow} \Sigma_{22}^+ y^2 \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{B_s \ge 0} e^{xA_s+yB_s} ds \right] \\ + \Sigma_{22}^- z^2 \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{B_s < 0} e^{xA_s+zB_s} ds \right]. $$

By the same arguments as for (2.21), we have

$$ (2.25) \qquad \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{-\epsilon \le B_s < \epsilon} \frac{2}{\epsilon} \psi'(B_s/\epsilon) (ye^{xA_s+yB_s} - ze^{xA_s+zB_s}) d\langle B, B \rangle_s \right] \underset{\epsilon \to 0}{\longrightarrow} (\Sigma_{22}^- + \Sigma_{22}^+) (y-z) \int_{\mathbb{R}} e^{ax} g_t(a, 0) da. $$

Additionally,

$$ \begin{align*} \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{0 \le B_s < \epsilon} (1/\epsilon^2) \psi''(B_s/\epsilon) (e^{xA_s+yB_s} - e^{xA_s+zB_s}) ds \right] &= \int_{\mathbb{R}} \int_0^\epsilon (1/\epsilon^2) \psi''(b/\epsilon) (e^{xa+yb} - e^{xa+zb}) g_t(a, b) db da \\ &= \int_{\mathbb{R}} \int_0^1 \psi''(b) \frac{e^{ax+bey} - e^{ax+bez}}{\epsilon} g_t(a, \epsilon b) db da \\ &\longrightarrow (y-z) \int_{\mathbb{R}} e^{ax} g_t(a, 0) da \int_0^1 b \psi''(b) db \\ &= (y-z)(\psi(0) - \psi(1)) \int_{\mathbb{R}} e^{ax} g_t(a, 0) da = \frac{(z-y)}{2} \int_{\mathbb{R}} e^{ax} g_t(a, 0) da \end{align*} $$
Here, the limit is once again justified by the estimates in (2.3). Combining this with Lemma 2.2, we have shown that
(2.26)

$$
\frac{1}{2} \sum_{i=1}^{2} \sum_{j=1}^{2} \mathbb{E}_{z_0} \left[ \int_{0}^{t} \partial_{i} \partial_{j} f_{\epsilon}(Z_s) d\langle Z^i, Z^j \rangle_s \right] \underset{\epsilon \to 0}{\longrightarrow} \frac{1}{2} (\Sigma_{11}^{+} x^2 + 2\Sigma_{12}^{+} xy + \Sigma_{22}^{+} y^2) \mathbb{E}_{z_0} \left[ \int_{0}^{t} \mathbb{1}_{B_s \ge 0} e^{xA_s + yB_s} ds \right] \\
+ \frac{1}{2} (\Sigma_{11}^{-} x^2 + 2\Sigma_{12}^{-} xy + \Sigma_{22}^{-} y^2) \mathbb{E} \left[ \int_{0}^{t} \mathbb{1}_{B_s < 0} e^{xA_s + zB_s} ds \right] + (y-z) \mathbb{E}_{z_0} \left[ \int_{0}^{t} e^{xA_s} dL_s^0(B) \right].
$$

Step 3: Sum up and take the limit as $t \to +\infty$.

Combining (2.15), (2.22), and (2.26), we obtain:

$$
(2.27) \quad \mathbb{E}_{z_0}[f(Z_t)] = \gamma_-(x, z)\mathbb{E}_{z_0}\left[\int_0^t \mathbb{1}_{B_s<0}e^{xA_s+zB_s}ds\right] + \gamma_+(x, y)\mathbb{E}_{z_0}\left[\int_0^t \mathbb{1}_{B_s>0}e^{xA_s+yB_s}ds\right] \\
\qquad + \left(q_1x + \frac{y(1+q_2)+z(q_2-1)}{2}\right)\mathbb{E}_{z_0}\left[\int_0^t e^{xA_t}dL_s^0(B)\right] + e^{xa_0+yb_0}\mathbb{1}_{b_0>0} + zb_0\mathbb{1}_{b_0<0}.
$$

Moreover, note that $f(Z_t) \le e^{xA_t}$ since $y < 0$ and $z > 0$. By Lemma 2.5, $\mathbb{E}_{z_0}[f(Z_t)] \underset{t \to +\infty}{\longrightarrow} 0$ if $-\eta < x < 0$.

Note that $x < 0$, $y < 0$, $z > 0$ can be taken arbitrarily close to 0 such that the conditions $\gamma_+(x, y) < 0$, $\gamma_-(x, z) < 0$, $y - z < 0$, and $q_1x + \frac{y(1+q_2)+z(q_2-1)}{2} < 0$ are satisfied. For such $x, y, z$, the quantity

$$
\gamma_-(x, z) \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{B_s < 0} e^{xA_s + zB_s} ds \right] + \gamma_+(x, y) \mathbb{E}_{z_0} \left[ \int_0^t \mathbb{1}_{B_s > 0} e^{xA_s + yB_s} ds \right] \\
+ \left( q_1 x + \frac{y(1+q_2) + z(q_2-1)}{2} \right) \mathbb{E}_{z_0} \left[ \int_0^t e^{xA_t} dL_s^0(B) \right]
$$

converges to

$$
\gamma_-(x, z)\varphi_-^{z_0}(x, z) + \gamma_+(x, y)\varphi_+^{z_0}(x, y) + \left(q_1x + \frac{y(1+q_2)+z(q_2-1)}{2}\right)\varphi^{z_0}(x) \in [-\infty, 0)
$$

as $t \to +\infty$ (if some terms diverge, they are equal to $-\infty$ so the sum remains well defined). By (2.27), the previous sum is finite when $x \in (-\eta, 0)$, so $\varphi_-^{z_0}(x, z)$, $\varphi_+^{z_0}(x, y)$, and $\varphi^{z_0}(x)$ are finite for $x \in (-\eta, 0)$, $y < 0$ and $z > 0$ arbitrarily close to 0. Now that we know the Laplace transforms converge for such $x, y, z$, the asymptotics of (2.27) as $t \to +\infty$ yield the desired functional equation.

Remark 2.6 (Choice of $b_0 \neq 0$). The choice $b_0 \neq 0$ for the initial point $(a_0, b_0)$ is made for essentially technical reasons. Indeed, it ensures that $g_t$ remains finite on the strip $\mathbb{R} \times [-\epsilon, \epsilon]$ for $\epsilon > 0$ sufficiently small; this avoids unnecessary difficulties when taking the limit $\epsilon \to 0$ in (2.20) and (2.25). The functional equation also holds for $b_0 = 0$, which can be proved by taking the limit of (1.13) as $z_0 \to (a_0, 0)$.

3. PROOF OF THEOREM 1 AND 2: LAPLACE TRANSFORMS AND ASYMPTOTICS ON THE AXIS

Since the proof of Theorem 2 is quite independent of the proofs of the other theorems, it has been
placed in a separate section. Let us recall the expressions for $Y^{\pm}(x)$ and $Z^{\pm}(x)$ given in (1.14) and
(1.15).

**Lemma 3.1.** There exists $\epsilon > 0$ such that for all $u \in [-\epsilon, 0]$ and $v \in \mathbb{R}$,

$$
\Re(Y^{-}(u + iv)) < 0 \quad (\text{resp.} \quad \Re(Z^{+}(u + iv)) > 0).
$$

Proof. By elementary considerations (see [14], correcting an error without any consequence), we have
(3.1)

$$
\Re(Y^{\pm}(u+iv)) = \frac{1}{\Sigma_{22}^{+}} \left( -\Sigma_{12}^{+}u - \mu_{2}^{\pm} \pm \frac{\sqrt{\det(\Sigma^{+})}}{\sqrt{2}} \sqrt{(u - x_{min}^{+})(x_{max}^{+} - u) + v^2 + |(u + iv - x_{min}^{+})(x_{max}^{+} - u - iv)|} \right).
$$

Then, $\Re(Y^{-}(u + iv)) \le \Re(Y^{-}(u))$ for all $v \in \mathbb{R}$. The inequality $Y^{-}(0) = -\frac{2\mu_{2}^{+}}{\Sigma_{22}^{+}} < 0$ yields the first inequality. The second one is obtained symmetrically since $\mu_{2}^{-} < 0$. $\square$
We are now able to prove Theorem 1.

*Proof of Theorem 1.* For $x \in (-\epsilon, 0)$, we have $Y^{-}(x) < 0$ and $Z^{+}(x) > 0$. We can then substitute $(x, Y^{-}(x), Z^{+}(x))$ into the functional equation (1.13) to derive the equality (1.17) for $x \in (-\epsilon, 0)$. Since $\varphi$ is a Laplace transform, equation (1.17) holds in the entire domain of convergence of $\varphi$ by holomorphicity. Since the expression is meromorphic on $\mathbb{C} \setminus ((-\infty, \tilde{x}_b] \cup [x_b, +\infty))$, it remains only to show that the equation $\gamma(x, Y^{-}(x), Z^{+}(x)) = 0$ has no solution on $\mathbb{C} \setminus ((-\infty, \tilde{x}_b] \cup (x_b, +\infty))$. Using expressions (1.5), (1.14) and (1.15), we can write $\gamma(x, Y^{-}(x), Z^{+}(x))$ as:

$$ (3.2) \qquad \begin{aligned} \gamma(x, Y^{-}(x), Z^{+}(x)) &= \frac{1}{\Sigma_{22}^{+} + \Sigma_{22}^{-}} ((\Sigma_{12}^{+} - \Sigma_{12}^{-})x + \Sigma_{22}^{+}Y^{-}(x) - \Sigma_{22}^{-}Z^{+}(x)) \\ &= \frac{1}{\Sigma_{22}^{+} + \Sigma_{22}^{-}} \left( -\mu_{2}^{+} + \mu_{2}^{-} - \sqrt{(\Sigma_{12}^{+2} - \Sigma_{11}^{+}\Sigma_{22}^{+})x^{2} + 2(\mu_{2}^{+}\Sigma_{12}^{+} - \mu_{1}^{+}\Sigma_{22}^{+})x + (\mu_{2}^{+})^{2}} \right. \\ & \qquad \left. - \sqrt{(\Sigma_{12}^{-2} - \Sigma_{11}^{-}\Sigma_{22}^{-})x^{2} + 2(\mu_{2}^{-}\Sigma_{12}^{-} - \mu_{1}^{-}\Sigma_{22}^{-})x + (\mu_{2}^{-})^{2}} \right). \end{aligned} $$

Furthermore, by Assumption (1.7), $-\mu_2^+ + \mu_2^- < 0$ and by the same arguments as for (3.1),

$$ \Re \left( \sqrt{(\Sigma_{12}^{\pm 2} - \Sigma_{11}^{\pm}\Sigma_{22}^{\pm})x^2 + 2(\mu_2^{\pm}\Sigma_{12}^{\pm} - \mu_1^{\pm}\Sigma_{22}^{\pm})x + (\mu_2^{\pm})^2} \right) \geq 0. $$

Considering the real part, we see that $\gamma(x, Y^{-}(x), Z^{+}(x))$ does not vanish on $\mathbb{C} \setminus ((-\infty, \tilde{x}_b] \cup [x_b, +\infty))$, and the first result follows.

Equation (1.13), combined with the explicit expression of $\varphi^{z_0}$ (see Theorem 1), provides an explicit formula for $\varphi_+^{z_0}(x, y)$. Indeed, by substituting $z = Z^{+}(x) > 0$ (with $x < 0$ small enough) and $y < 0$ into (1.13), we obtain

$$ (3.3) \qquad \varphi_{+}^{z_0}(x, y) = \frac{-\gamma(x, y, Z^{+}(x))\varphi_{+}^{z_0}(x) - e^{xa_0 + yb_0 \mathbf{1}_{b_0 \ge 0} + zb_0 \mathbf{1}_{b_0 < 0}}}{\gamma_{+}(x, y)}. $$

Using the holomorphic properties of the Laplace transforms, (3.3) holds for all $(x, y)$ in the domain of convergence of $\varphi_+^{z_0}$. $\square$

We now state a final preparatory lemma prior to the proof of Theorem 2.

**Lemma 3.2** (Behaviour of $\varphi^{z_0}$ at $x_b$). Let $C_0$ and $f_0(z_0)$ be defined as in Theorem 2. Then,

$$ \varphi^{z_0}(x) = \varphi^{z_0}(x_b) - 2\sqrt{\pi} \left( \frac{\Sigma_{22}^{+} + \Sigma_{22}^{-}}{2} \right) C_0 f_0(z_0) \sqrt{x - x_b} + o(\sqrt{x - x_b}) $$

as $x \to x_b$, where $f_0(z_0)$ is defined in Theorem 2.

*Proof.* Without loss of generality, we assume in this proof that $x_b = x_{max}^+ < x_{max}^-$. We define $C_y$ as $C_y = \frac{\sqrt{\det(\Sigma^+(x_b - x_{min}^+))}}{\Sigma_{22}^{+}}$ so that $Y^{-}(x) = Y^{-}(x_b) - C_y \sqrt{x_b - x} + o(\sqrt{x_b - x})$.

Now, let us perform straightforward computations using the explicit expression of $\varphi^{z_0}$ given by (1.17). We have:

$$ \begin{aligned} \frac{1}{\gamma(x, Y^{-}(x), Z^{+}(x))} &= \frac{1}{\gamma(x_b, Y^{-}(x_b), Z^{+}(x_b)) - C_y \left(\frac{1+q_2}{2}\right) \sqrt{x_b - x}(1 + o(1))} \\ &= \frac{1}{\gamma(x_b, Y^{-}(x_b), Z^{+}(x_b))} \times \frac{1}{1 - \frac{C_y \left(\frac{1+q_2}{2}\right)}{\gamma(x_b, Y^{-}(x_b), Z^{+}(x_b))} \sqrt{x_b - x}(1 + o(1))} \\ &= \frac{1}{\gamma(x_b, Y^{-}(x_b), Z^{+}(x_b))} \left( 1 + \frac{C_y \left(\frac{1+q_2}{2}\right)}{\gamma(x_b, Y^{-}(x_b), Z^{+}(x_b))} \sqrt{x_b - x}(1 + o(1)) \right). \end{aligned} $$

Furthermore,

$$ e^{Y^{-}(x)b_0 \mathbf{1}_{b_0 > 0}} = e^{(Y^{-}(x_b) - \sqrt{x_b - x}(C_y + o(1)))b_0 \mathbf{1}_{b_0 > 0}} = e^{Y^{-}(x_b)b_0 \mathbf{1}_{b_0 > 0}} (1 - \sqrt{x_b - x}(C_y + o(1)))b_0 \mathbf{1}_{b_0 > 0}. $$
Multiplying the two quantities together, we obtain:

$$ \varphi^{z_0}(x) - \varphi^{z_0}(x_b) = \frac{C_y}{\gamma(x_b, Y^-(x_b), Z^+(x_b))} \times \left( b_0 \mathbf{1}_{b_0>0} - \frac{\frac{1+q_2}{2}}{\gamma(x_b, Y^-(x_b), Z^+(x_b))} \right) e^{a_0 x_b + Y^-(x_b) b_0 \mathbf{1}_{b_0>0} + Z^+(x_b) b_0 \mathbf{1}_{b_0<0}} \sqrt{x-x_b} + o(\sqrt{|x-x_b|}), $$

which is the desired formula. □

Since $\varphi^{z_0}$ is the Laplace transform of $\frac{\Sigma_{22}^+ + \Sigma_{22}^-}{2} g(x, 0)$ (see Lemma 2.2), we can now derive the asymptotics of $g^{z_0}(r, 0)$ as $r \to +\infty$ (i.e., Theorem 2) using the Tauberian Theorem A.1.

*Proof of Theorem 2.* By symmetry, we only consider the asymptotics as $r \to +\infty$. First, suppose that $a_0 = 0$. Note that the first and third conditions of Theorem A.1 are satisfied by Theorem 1 and Lemma 3.2. Let us verify the second condition of Theorem A.1.

From (3.1), there exist $\delta > 0$ and constants $M, c > 0$ such that $\Re(Y^-(u+iv)) \le M - cv$ for all $u, v \in \mathbb{R}$ satisfying $|u| \le \frac{1}{\tan(\delta)}|v|$. Then, $e^{b_0 \Re(Y^-(x)) \mathbf{1}_{b_0>0}}$ is bounded for $x \in G_\delta^+(x_b)$. The same reasoning applies to $e^{b_0 \Re(Z^+(x)) \mathbf{1}_{b_0<0}}$.

Furthermore, from (3.1), possibly after choosing $\delta$ close enough to $\pi/2$, we have

$$ |\gamma(x, Y^{-}(x), Z^{+}(x))| \underset{x \in G_{\delta}^{+}(x_{b})}{\xrightarrow{|x| \to +\infty}} +\infty. $$

We therefore obtain the second condition of Theorem A.1 from the explicit expression of $\varphi^{z_0}$ given in (1.18). Then, by Theorem A.1, the following asymptotic holds:

$$ (3.4) \qquad g^{0,b_0}(r, 0) \underset{r \to +\infty}{\sim} \frac{-2\sqrt{\pi}}{\Gamma(-1/2)} C_0 f_0(z_0) \frac{e^{-rx_b}}{r^{3/2}} = C_0 f_0(z_0) \frac{e^{-rx_b}}{r^{3/2}}. $$

Now suppose that $a_0 \ne 0$. Note that if $Z \sim \mathbb{P}_{(0,b_0)}$, then $Z + (a_0, 0) \sim \mathbb{P}_{(a_0,b_0)}$. This follows from the pathwise uniqueness of the SDE (1.3). Therefore, $g^{(a_0,b_0)}(x, 0) = g^{(0,b_0)}(x - a_0, 0)$, which yields (3.4) using the identity $e^{a_0 x_b} f_0(0, b_0) = f_0(a_0, b_0)$ (see (1.25) and (1.26)). □

### 4. PROOFS OF THEOREM 3 TO 6: ASYMPTOTICS ALONG ALL DIRECTIONS

First, in Section 4.1, we establish a representation of the Green's functions $g^{z_0}$ as simple integrals using an inverse Laplace theorem and some complex analysis. These integrals are of saddle-point type. Therefore, in Section 4.2, we define paths of steepest descent in a neighbourhood of the saddle points $x(\alpha)$ corresponding to different directions $\alpha$. We should now shift the integration contours up to the saddle point, taking into account the singularities of the integrand. This procedure depends on the position of the saddle points $x(\alpha)$ relative to the branching points $\tilde{x}_b$ and $x_b$ of the integrand. In Section 4.3, we prove Theorems 3 and 4. In Section 4.4, we prove Theorem 5. Finally, we prove Theorem 6 in Section 4.5. In each section, we define a different path deformation for the integral representation of $g^{z_0}$. For each case, we identify the dominant contribution of the integral and determine which parts are negligible.

Without loss of generality, our proofs only consider angles $\alpha \in [0, \pi]$, since the remaining cases are symmetric.

#### 4.1. Laplace inverse and reduction to simple integrals.
Using the two-dimensional Laplace inversion theorem, we can express $g^{z_0}(a, b)$ in terms of its Laplace transform (see [8, Theorems 24.3 and 24.4] and [3]).

**Lemma 4.1 (Inverse Laplace theorem).** Let $z_0 \in \mathbb{R}^2$ and $(a, b) \in \mathbb{R}^2$ such that $b > 0$ and $(a, b) \ne z_0$. Then, for $\epsilon_1 > 0, \epsilon_2 > 0$ sufficiently small,

$$ (4.1) \qquad g^{z_0}(a, b) = \frac{1}{(2i\pi)^2} \int_{-\epsilon_1 - i\infty}^{-\epsilon_1 + i\infty} \int_{-\epsilon_2 - i\infty}^{-\epsilon_2 + i\infty} \varphi_{+}^{z_0}(x, y) e^{-ax-by} dy dx $$

where the convergence is understood in the sense of principal value.
Proof. This follows from the two-dimensional Laplace inversion theorem (see [8, Theorems 24.3 and 24.4] and [3]), provided that the double integral converges in the sense of principal value. This convergence is ensured by the explicit expression (1.18) and by standard computations based on integration by parts. $\square$

By similar considerations as in Section 1.3, we define branches $X^{\pm}(y)$ by $\gamma_+(X^{\pm}(y), y) = 0$. These branches have branching points at $y_{min}$ and $y_{max}$, defined by

$$y_{min} = \frac{\mu_1^+ \Sigma_{12}^+ - \mu_2^+ \Sigma_{11}^+ - \sqrt{D_2}}{\det(\Sigma^+)}; \quad y_{max} = \frac{\mu_1^+ \Sigma_{12}^+ + \mu_2^+ \Sigma_{11}^+ - \sqrt{D_2}}{\det(\Sigma^+)}$$

$$\text{where } D_2 = (\mu_1^+ \Sigma_{12}^+ - \mu_2^+ \Sigma_{11}^+)^2 + \mu_1^{+2} \det(\Sigma^+).$$

**Lemma 4.2** (Reduction to simple integrals). Let $z_0 \in \mathbb{R}^2$ and $(a, b) \in \mathbb{R}^2$ such that $b > 0$ and $(a, b) \neq z_0$. Then, for $\epsilon > 0$ small enough, $g^{z_0}(a, b)$ can be written as $g^{z_0}(a, b) = I_1^{z_0}(a, b) + I_2^{z_0}(a, b)$, where

$$(4.2) \qquad I_1^{z_0}(a, b) = \frac{1}{2i\pi} \int_{-\epsilon-i\infty}^{-\epsilon+i\infty} \varphi^{z_0}(x) \frac{\gamma(x, Y^+(x), Z^+(x))}{\partial_y \gamma_+(x, Y^+(x))} e^{-ax-bY^+(x)} dx$$

and

$$(4.3) \qquad I_2^{z_0}(a, b) = \frac{1}{2i\pi} \int_{-\epsilon-i\infty}^{-\epsilon+i\infty} \frac{e^{(a_0-a)x} \left( e^{(b_0-b)Y^+(x)} 1_{b_0>0} + e^{b_0Z^+(x)-bY^+(x)} 1_{b_0<0} \right)}{\partial_y \gamma_+(x, Y^+(x))} dx \quad \text{if } b > b_0.$$

Furthermore, the following expressions hold for $I_2^{z_0}(a, b)$ if $b_0 > 0$:

$$(4.4) \qquad I_2^{z_0}(a, b) = \frac{1}{2i\pi} \int_{-\epsilon-i\infty}^{-\epsilon+i\infty} \frac{e^{(a_0-a)X^+(y)+(b_0-b)y}}{\partial_x \gamma_+(X^+(y), y)} dy \quad \text{if } a > a_0$$

$$(4.5) \qquad I_2^{z_0}(a, b) = \frac{-1}{2i\pi} \int_{-\epsilon-i\infty}^{-\epsilon+i\infty} \frac{e^{(a_0-a)X^-(y)+(b_0-b)y}}{\partial_x \gamma_+(X^-(y), y)} dy \quad \text{if } a < a_0.$$

Proof. The proof is similar to that of [14, Lemma 4.1]. We choose $\epsilon_1 = \epsilon_2 = \epsilon$ such that $Y^-(\epsilon) < -\epsilon$ and $Y^+(\epsilon) > 0$ (this is possible since $Y^-(0) = \frac{-2\mu_2^+}{\Sigma_{22}^+} < 0$, $Y^+(0) = 0$, and $(Y^+)'(0) = \frac{-\mu_1^+}{\mu_2^+} < 0$). Then, for all $v \in \mathbb{R}$, we have $\Re(Y^-(\epsilon + iv)) \le \Re(Y^-(\epsilon)) < -\epsilon$ and $\Re(Y^+(\epsilon + iv)) \ge Y^+(\epsilon) > 0$ (see expression (3.1)).

Using (1.18) and (4.1), we derive the equality $g^{z_0}(a, b) = I_1^{z_0}(a, b) + I_2^{z_0}(a, b)$, where

$$I_1^{z_0}(a, b) = \frac{1}{(2i\pi)^2} \int_{-\epsilon-i\infty}^{-\epsilon+i\infty} \varphi^{z_0}(x) \left( \int_{-\epsilon-i\infty}^{-\epsilon+i\infty} \frac{-\gamma(x, y, Z^+(x))}{\gamma_+(x, y)} e^{-ax-by} dy \right) dx$$

and

$$I_2^{z_0}(a, b) = \frac{1}{(2i\pi)^2} \int_{-\epsilon-i\infty}^{-\epsilon+i\infty} \int_{-\epsilon-i\infty}^{-\epsilon+i\infty} \frac{-e^{xa_0+yb_0} 1_{b_0 \ge 0} + zb_0 1_{b_0 < 0}}{\gamma_+(x, y)} e^{-ax-by} dy dx.$$

It remains to show that $I_1^{z_0}(a, b)$ and $I_2^{z_0}(a, b)$ admit the desired integral representations. Let us focus on $I_1$. Fix $x \in [-\epsilon + i\mathbb{R}$. For $R > 0$, consider the path $\Gamma_R = \{-\epsilon + Re^{i\theta} | \theta \in [0, \pi]\}$. Then, for sufficiently large $R > 0$, the point $Y^+(x)$ lies inside the closed contour formed by $\Gamma_R \cup [-\epsilon - iR, -\epsilon + iR]$, see Figure 9 (recall that $\Re(Y^+(x)) \ge \Re(Y^+(\epsilon)) > -\epsilon$). By the residue theorem,

$$\frac{1}{2i\pi} \left( \int_{-\epsilon-iR}^{-\epsilon+iR} + \int_{\Gamma_R} \right) \frac{-\gamma(x, y, Z^+(x))}{\gamma_+(x, y)} e^{-ax-by} dy = \frac{\gamma(x, Y^+(x), Z^+(x))}{\partial_y \gamma_+(x, Y^+(x))} e^{-ax-bY^+(x)}.$$

Furthermore,

$$(4.6) \qquad \int_{\Gamma_R} \frac{\gamma(x, y, Z^+(x))}{\gamma_+(x, y)} e^{-ax-by} dy = e^{-ax} \int_0^\pi \frac{\gamma(x, -\epsilon + Re^{i\theta}, Z^+(x))}{\gamma_+(x, -\epsilon + Re^{i\theta})} e^{-b\epsilon - bRe^{i\theta}} d\theta.$$

Moreover, $\sup_{\theta \in [0, \pi]} \left| \frac{\gamma(x, -\epsilon + Re^{i\theta}, Z^+(x))}{\gamma_+(x, -\epsilon + Re^{i\theta})} \right| = O(1/R)$ as $R \to +\infty$, and $e^{-bRe^{i\theta}} \to 0$ as $R \to +\infty$, since $-b \cos(\theta) < 0$ for all $\theta \in (0, \pi)$. Therefore, (4.6) converges to 0 as $R \to +\infty$, which yields representation (4.2) for $I_1^{z_0}(a, b)$. The proof of the representations of $I_2^{z_0}(a, b)$ is analogous. $\square$
<figure><img src="image_13.png" /><figcaption>FIGURE 9. Changing path for representations (4.2) and (4.3) of $I_1^{z_0}(a, b)$ and $I_2^{z_0}(a, b)$.</figcaption></figure>

## 4.2. Paths of the saddle point method.
As the previous integrals are of the saddle-point type, we now introduce the associated paths. We recall the definition of $(x(\alpha), y(\alpha))$ for $\alpha \in [0, \pi]$ given by (1.19). Let us define

$$ (4.7) \qquad F(x, \alpha) = -\cos(\alpha) x - \sin(\alpha) Y^{+}(x) + \cos(\alpha) x(\alpha) + \sin(\alpha) y(\alpha). $$

By construction, for all $\alpha \in (0, \pi)$, we have $F(x(\alpha), \alpha) = 0$ and $F'_x(x(\alpha), \alpha) = 0$. It follows that $(Y^+)'(x(\alpha)) = -\frac{\cos(\alpha)}{\sin(\alpha)}$, and

$$ x'(\alpha) F''_x(x(\alpha), \alpha) + F''_{x,\alpha}(x(\alpha), \alpha) = 0. $$

Then,

$$ (4.8) \qquad x'(\alpha) F''_x(x(\alpha), \alpha) = -(\sin(\alpha) - (Y^+)'(x(\alpha)) \cos(\alpha)) = \frac{-1}{\sin(\alpha)}. $$

From computations carried out in [14, Section 5], we also have

$$ (4.9) \qquad F''_x(x(\alpha), \alpha) = \frac{\Sigma_{11}^+ \sin^2(\alpha) - 2\Sigma_{12}^+ \sin(\alpha) \cos(\alpha) + \Sigma_{22}^+ \cos^2(\alpha)}{\partial_y \gamma_+(x(\alpha), y(\alpha)) \sin(\alpha)} > 0, \quad \alpha \in (0, \pi). $$

Case $\alpha_0 \in (0, \pi)$. Let us fix $\alpha_0 \in (0, \pi)$. By the parameterised Morse Lemma [14, Lemma A.1], there exists a neighbourhood of $(0, \alpha_0) \in \mathbb{C} \times \mathbb{R}$, given by

$$ (4.10) \qquad \Omega(0, \alpha_0) = \{(\omega, \alpha) \in \mathbb{C} \times [0, \pi] \mid |\omega| \le L, |\alpha - \alpha_0| \le \eta\} $$

for some $L, \eta > 0$, and a function $x: \Omega(0, \alpha_0) \to \mathbb{C}$ of class $\mathcal{C}^\infty$ (and holomorphic in the first variable) such that

$$ x(0, \alpha) = x(\alpha) \quad \text{for all } \alpha \text{ with } |\alpha - \alpha_0| \le \eta, $$

and satisfying

$$ (4.11) \qquad F(x(\omega, \alpha), \alpha) = \omega^2, \quad \forall (\omega, \alpha) \in \Omega(0, \alpha_0). $$

Furthermore,

$$ (4.12) \qquad x'_{\omega}(0, \alpha) = \sqrt{\frac{2}{F''_{x}(x(\alpha), \alpha)}}. $$

Let $0 < \epsilon < K$. We define the contour of steepest descent as

$$ \Gamma_{x,\alpha} = \{x(it, \alpha) \mid t \in [-\epsilon, \epsilon]\}. $$
<figure><img src="image_14.png" /><figcaption>FIGURE 10. Changing path of Lemma 4.3</figcaption></figure>

Note that $F(x(it, \alpha), \alpha) = -t^2$. We denote by $x_{\alpha}^{+} = x(i\epsilon, \alpha)$ and $x_{\alpha}^{-} = x(-i\epsilon, \alpha)$ the endpoints of $\Gamma_{x,\alpha}$. It can be shown (see [14]) that for any $\alpha_0 \in (0, \pi)$, there exist $\eta > 0$ and $\nu > 0$ such that

$$ (4.13) \qquad \mathrm{Im}(x_{\alpha}^{+}) > \nu, \quad \mathrm{Im}(x_{\alpha}^{-}) < -\nu \quad \text{for all } \alpha \text{ with } |\alpha - \alpha_0| < \eta. $$

Case $\alpha_0 \in \{0, \pi\}$. We define

$$ (4.14) \qquad G(y, \alpha) = -\cos(\alpha)X^{+}(y) - \sin(\alpha)y + \cos(\alpha)x(\alpha) + \sin(\alpha)y(\alpha). $$

As before, we have $G(y(\alpha), \alpha) = 0$, $G'_{y}(y(\alpha), \alpha) = 0$, and $G''_{y}(y(\alpha), \alpha) > 0$ for $\alpha \in [0, \pi/2) \cup (\pi/2, \pi]$. Then, if $\alpha_0 \in [0, \pi/2) \cup (\pi/2, \pi]$ (and in particular if $\alpha_0 = 0$ or $\alpha_0 = \pi$), we define symmetrically the function $y(\omega, \alpha)$ corresponding to $G(y, \alpha)$ via the parameterised Morse Lemma. We then set

$$ \Gamma_{y,\alpha} = \{y(it, \alpha) \mid t \in [-\epsilon, \epsilon]\}, $$

whose endpoints are $y_{\alpha}^{+} = y(i\epsilon, \alpha)$ and $y_{\alpha}^{-} = y(-i\epsilon, \alpha)$. The paths $\Gamma_{x,\alpha}$ and $\Gamma_{y,\alpha}$ are linked by the relations

$$ (4.15) \qquad \Gamma_{x,\alpha} = \overleftarrow{X}^{+}(\Gamma_{y,\alpha}), \quad \Gamma_{y,\alpha} = \overrightarrow{Y}^{+}(\Gamma_{x,\alpha}), \quad \alpha \in (0, \pi/2) \cup (\pi/2, \pi). $$

where the arrows indicate that the paths are traversed in the reversed direction.

**4.3. Lemmas for red-arc parts.** Now, let us consider the asymptotics of Green's functions $g^{z_0}$ in the directions corresponding to the red arc parts, excluding $\alpha_b$ and $\tilde{\alpha}_b$ (see Figure 5). Recall the notation introduced in (1.16). We define

$$ S_{x,\alpha}^{+} = \{x_{\alpha}^{+} + it \mid t \ge 0\}, \quad S_{x,\alpha}^{-} = \{x_{\alpha}^{-} - it \mid t \ge 0\}, $$

$$ S_{y,\alpha}^{+} = \{y_{\alpha}^{+} + it \mid t \ge 0\}, \quad S_{y,\alpha}^{-} = \{y_{\alpha}^{-} - it \mid t \ge 0\}. $$

The following lemma describes how to modify the contours of integration in (4.2) and (4.3) so that they pass through the saddle point along the steepest descent path.

**Lemma 4.3** (Changing path through the saddle point). Let $z_0 \in \mathbb{R}^2$, $a \in \mathbb{R}$, $b > 0$ such that $(a, b) \neq z_0$, and let $\alpha \in (0, \pi)$ be such that $\tilde{x}_b < x(\alpha) < x_b$. Then:

$$ (4.16) \qquad I_1^{z_0}(a, b) = \frac{1}{2i\pi} \int_{S_{x,\alpha}^{-} \cup \Gamma_{x,\alpha} \cup S_{x,\alpha}^{+}} \varphi^{z_0}(x) \frac{\gamma(x, Y^{+}(x), Z^{+}(x))}{\partial_y \gamma_{+}(x, Y^{+}(x))} e^{-ax-bY^{+}(x)} dx, $$
and
(4.17)

$$I_2^{z_0}(a, b) = \frac{1}{2i\pi} \int_{S_{x,\alpha}^- \cup \Gamma_{x,\alpha} \cup S_{x,\alpha}^+} \frac{e^{(a_0-a)x} \left( e^{(b_0-b)Y^+(x)} \mathbb{1}_{b_0>0} + e^{b_0Z^+(x)-bY^+(x)} \mathbb{1}_{b_0<0} \right)}{\partial_y \gamma_+(x, Y^+(x))} dx, \quad \text{if } b > b_0.$$

Furthermore, if $b_0 > 0$, then

$$(4.18) \qquad I_2^{z_0}(a, b) = \frac{1}{2i\pi} \int_{S_{y,\alpha}^- \cup \Gamma_{y,\alpha} \cup S_{y,\alpha}^+} \frac{e^{(a_0-a)X^+(y)+(b_0-b)y}}{\partial_x \gamma_+(X^+(y), y)} dy, \quad \text{if } a > a_0,$$

and

$$(4.19) \qquad I_2^{z_0}(a, b) = -\frac{1}{2i\pi} \int_{S_{y,\alpha}^- \cup \Gamma_{y,\alpha} \cup S_{y,\alpha}^+} \frac{e^{(a_0-a)X^-(y)+(b_0-b)y}}{\partial_x \gamma_+(X^-(y), y)} dy, \quad \text{if } a < a_0.$$

*Proof.* We prove (4.16); the other identities are analogous. The argument is similar to the one given in [14, Lemma 6.2]. Using the Residue Theorem (see Figure 10) and Lemma 4.2, it suffices to verify that

$$(4.20) \quad \sup_{u \in [X^+(y_{\max}) - \eta, x_{\max} + \eta]} \left| \frac{\varphi^{z_0}(u + iv) \gamma(u + iv, Y^+(u + iv), Z^+(u + iv))}{\partial_y \gamma_+(u + iv, Y^+(u + iv))} e^{-a(u+iv)-bY^+(u+iv)} \right| \to 0$$

as $|v| \to \infty$, for some $\eta > 0$. This decay follows directly from (1.17) and the estimate on the real part in (3.1). $\square$

The following two lemmas provide the asymptotic contribution of integrals along the steepest descent contours.

**Lemma 4.4** (Asymptotics along $\Gamma_{x,\alpha}$ in the red arc region with $\alpha_0 \neq 0, \pi$). Let $\alpha_0 \in (0, \pi)$ such that $\tilde{x}_b < x(\alpha_0) < x_b$. Let $r = \sqrt{a^2 + b^2}$ and define $\alpha = \alpha(a, b) \in (0, \pi)$ such that $\cos(\alpha) = \frac{a}{\sqrt{a^2+b^2}}$, $\sin(\alpha) = \frac{b}{\sqrt{a^2+b^2}}$. Then, the following asymptotic holds as $r \to +\infty$ and $\alpha \to \alpha_0$:
(4.21)

$$\frac{1}{2i\pi} \int_{\Gamma_{x,\alpha}} \left( \frac{\varphi^{z_0}(x)\gamma(x, Y^+(x), Z^+(x))e^{-ax-bY^+(x)} + e^{(a_0-a)x} \left( e^{(b_0-b)Y^+(x)} \mathbb{1}_{b_0>0} + e^{b_0Z^+(x)-bY^+(x)} \mathbb{1}_{b_0<0} \right)}{\partial_y \gamma_+(x, Y^+(x))} \right) dx \\ \underset{\alpha \to \alpha_0}{\sim} C^+(\alpha_0)h_{\alpha_0}(z_0) \frac{e^{-r(\cos(\alpha)x(\alpha)+\sin(\alpha)y(\alpha))}}{\sqrt{r}},$$

where $C^+(\alpha_0)$ and $h_{\alpha_0}(z_0)$ are defined in (1.29) and (1.30).

*Proof.* We have $F_x''(x(\alpha_0), \alpha_0) > 0$ (see (4.9)). Then, by the parameterised saddle point method (see [14, Lemma 8.1]), for all $n \ge 0$, the integral (4.21) admits the following asymptotic expansion as $\alpha = \alpha(a, b) \to \alpha_0$ and $r = \sqrt{a^2 + b^2} \to +\infty$:

$$(4.22) \qquad e^{-r(\cos(\alpha)x(\alpha)+\sin(\alpha)y(\alpha))} \sum_{k=0}^{n} \frac{c_k(\alpha)}{r^k \sqrt{r}} + o\left(\frac{1}{r^n \sqrt{r}}\right),$$

where the coefficients $c_0(\alpha), \dots, c_n(\alpha)$ are continuous functions of $\alpha$ and satisfy

$$c_0(\alpha) = \frac{h_{\alpha}(z_0) x'_{\omega}(0, \alpha)}{2\sqrt{\pi} \partial_y \gamma_+(x(\alpha), y(\alpha))} = C(\alpha)h_{\alpha}(z_0)$$

(see (4.12)). In particular, $c_0(\alpha_0) > 0$ by expression (1.30). Therefore, the dominant asymptotic term is given by the $k=0$ term in (4.22), which concludes the proof. $\square$

**Lemma 4.5** (Asymptotics along $\Gamma_{x,\alpha}$ and $\Gamma_{y,\alpha}$ in the red arc region at $\alpha_0 = 0, \pi$). Suppose $x_{\max}^+ < x_{\max}^-$ and let $\alpha_0 = 0$. If $r = \sqrt{a^2 + b^2}$ and $\alpha = \alpha(a, b) \in (0, \pi)$ is the angle such that $\cos(\alpha) = \frac{a}{\sqrt{a^2+b^2}}$ and
$\sin(\alpha) = \frac{b}{\sqrt{a^2+b^2}}$, then the following asymptotics hold:

$$ (4.23) \quad \frac{1}{2i\pi} \int_{\Gamma_{x,\alpha}} \varphi^{z_0}(x) \frac{\gamma(x, Y^+(x), Z^+(x))}{\partial_y \gamma_+(x, Y^+(x))} e^{-ax-bY^+(x)} dx + \frac{1}{2i\pi} \int_{\Gamma_{y,\alpha}} \frac{e^{(a_0-a)X^+(y)+(b_0-b)y}}{\partial_x \gamma_+(X^+(y), y)} dy \\ \sim_{r \to +\infty, \alpha \to 0, \alpha \ge 0} C_0 f_0(z_0) \left( \kappa\alpha + \frac{1}{r} \right) \frac{e^{-r(\cos(\alpha)x(\alpha)+\sin(\alpha)y(\alpha))}}{\sqrt{r}}, $$

where $\kappa > 0$ is an explicit constant (see (4.29)) and $f_0(z_0)$ is given by (1.25). Furthermore,

$$ \partial_{\alpha} (C(\alpha)h_{\alpha}(z_0))|_{\alpha=0} = \kappa C_0 f_0(z_0). $$

If $x_{min}^+ > x_{min}^-$, the symmetric result holds for $\alpha \to \pi$ with $\alpha \le \pi$.

*Proof.* To prove this, we consider the integral representation along the contour $\Gamma_{y,\alpha}$. By performing the change of variables $x = X^+(y)$, we obtain:

$$ (4.24) \quad \int_{\Gamma_{x,\alpha}} \varphi^{z_0}(x) \frac{\gamma(x, Y^+(x), Z^+(x))}{\partial_y \gamma_+(x, Y^+(x))} e^{-ax-bY^+(x)} dx = \int_{\Gamma_{y,\alpha}} \varphi^{z_0}(X^+(y)) \frac{\gamma(X^+(y), y, Z^+(X^+(y)))}{\partial_x \gamma_+(X^+(y), y)} e^{-aX^+(y)-by} dy. $$

Recall that $(x(\alpha), y(\alpha)) \xrightarrow{\alpha \to 0} (x_{max}^+, Y^\pm(x_{max}^+))$. Using (1.17), we have:

$$ \varphi^{z_0}(X^+(y)) = - \frac{e^{X^+(y)a_0 + Y^-(X^+(y))b_0 \mathbf{1}_{b_0 \ge 0} + Z^+(X^+(y))b_0 \mathbf{1}_{b_0 < 0}}{\gamma(X^+(y), Y^-(X^+(y)), Z^+(X^+(y)))}. $$

By definition, $X^+(y)$ is holomorphic in a neighborhood of $Y^\pm(x_{max}^+)$. Moreover, by the Viète identities, the expression

$$ Y^-(X^+(y)) = \frac{\Sigma_{11}^+(X^+(y))^2 + 2\mu_1^+ X^+(y)}{\Sigma_{22}^+ y} $$

is holomorphic in a neighborhood of $Y^\pm(x_{max}^+)$. Since we are in the case $x_{max}^+ < x_{max}^-$, the function $Z^+(X^+(y))$ is also holomorphic in a neighborhood of $Y^\pm(x_{max}^+)$. Thus, the integrand on the right-hand side of (4.24) is holomorphic near the saddle point.

Applying the parameterized saddle-point method to the contour $\Gamma_{y,\alpha}$, we deduce that for any $\alpha_0 \in [0, \pi/2)$, the expression in (4.23) admits the following asymptotic expansion as $\alpha = \alpha(a, b) \to \alpha_0$ and $r = \sqrt{a^2 + b^2} \to \infty$:

$$ (4.25) \quad \left( \tilde{c}_0(\alpha) + \frac{\tilde{c}_1(\alpha)}{r} + o\left(\frac{1}{r}\right) \right) \frac{e^{-r(\cos(\alpha)x(\alpha)+\sin(\alpha)y(\alpha))}}{\sqrt{r}}, $$

where the coefficients $\tilde{c}_k(\alpha)$ are continuous in $\alpha$. By uniqueness of the asymptotic expansion, we necessarily have $c_k(\alpha) = \tilde{c}_k(\alpha)$ for all $\alpha \in (0, \pi/2]$ and for $k = 0, 1$. Recall that $X^+(y(\alpha)) = x(\alpha)$ and $Y^+(x(\alpha)) = y(\alpha)$ for all $\alpha \in (0, \pi/2)$. In particular, we have:

$$ (4.26) \quad c_0(\alpha) = C^+(\alpha) \cdot \frac{e^{x(\alpha)a_0 + b_0y(\alpha)\mathbf{1}_{b_0>0} + Z^+(x(\alpha))b_0\mathbf{1}_{b_0<0}}}{\gamma(x(\alpha), Y^-(x(\alpha)), Z^+(x(\alpha)))} \\ \times (-\gamma(x(\alpha), y(\alpha), Z^+(x(\alpha)))e^{(Y^-(x(\alpha))-y(\alpha))b_0\mathbf{1}_{b_0\ge 0}} + \gamma(x(\alpha), Y^-(x(\alpha)), Z^+(x(\alpha)))) $$

Furthermore, we observe that

$$ \begin{aligned} y(\alpha) - Y^-(X^+(y(\alpha))) &= Y^+(X^+(y(\alpha))) - Y^-(X^+(y(\alpha))) \\ &= \frac{2}{\Sigma_{22}^+} \sqrt{\det(\Sigma_+) \cdot (x_{max}^+ - X^+(y(\alpha))) \cdot (X^+(y(\alpha)) - x_{min}^+)} \end{aligned} $$

We also recall that $X^+(Y^\pm(x_{max}^+)) = x_{max}^+$, and that

$$ (X^+)'(y)|_{y=Y^\pm(x_{max}^+)} = 0, \quad (X^+)''(y)|_{y=Y^\pm(x_{max}^+)} = - \frac{\Sigma_{22}^+}{\partial_x \gamma_+(x_{max}^+, Y^\pm(x_{max}^+))}. $$
It follows that

$$
(4.27) \qquad y(\alpha) - Y^{-}(X^{+}(y(\alpha))) \sim \sqrt{\frac{2 \det(\Sigma^{+})(x_{max}^{+} - x_{min}^{+})}{\Sigma_{22}^{+} \partial_{x} \gamma_{+}(x_{max}^{+}, Y^{\pm}(x_{max}^{+}))}} \cdot \alpha \sim \Pi \alpha,
$$

where we define

$$
\Pi = \sqrt{\frac{2 \det(\Sigma^+)(x_{max}^+ - x_{min}^+)}{\Sigma_{22}^+ \partial_x \gamma_+(x_{max}^+, Y^\pm(x_{max}^+))}} = 2\sqrt{\frac{\det(\Sigma^+)}{\Sigma_{11}^+ \Sigma_{22}^+}},
$$

since $\partial_x \gamma_+(x_{max}^+, Y^\pm(x_{max}^+)) = \frac{\Sigma_{11}^+(x_{max}^+ - x_{min}^+)}{2}$.

Injecting (4.27) into (4.26), and using the asymptotics

$$
C^{+}(\alpha) \xrightarrow{\alpha \to 0^{+}} \frac{1}{\sqrt{2\pi \Sigma_{22}^{+} \partial_{x} \gamma_{+}(x_{max}^{+}, Y^{\pm}(x_{max}^{+})}}} = \frac{1}{\sqrt{\pi \Sigma_{11}^{+} \Sigma_{22}^{+} (x_{max}^{+} - x_{min}^{+})}},
$$

we obtain, after simplification,

$$
(4.28) \qquad C(\alpha)h_{\alpha}(z_0) = c_0(\alpha) \underset{\alpha \to 0^+}{\sim} \frac{2}{\Sigma_{11}^+ \Sigma_{22}^+} \sqrt{\frac{\det(\Sigma^+)}{\pi(x_{max}^+ - x_{min}^+)}} f_0(z_0) \alpha = \kappa C_0 f_0(z_0) \alpha,
$$

with

$$
(4.29) \qquad \kappa = \frac{-(\Sigma_{22}^{+} + \Sigma_{22}^{-})\gamma(x_{max}^{+}, Y^{\pm}(x_{max}^{+}), Z^{+}(x_{max}^{+}))}{\Sigma_{11}^{+}(x_{max}^{+} - x_{min}^{+})}.
$$

Moreover, evaluating (4.25) at $\alpha = 0$ gives

$$
g(r, 0) \underset{r \to \infty}{\sim} c_1(0) \cdot \frac{e^{-rx_b}}{r^{3/2}}.
$$

Since $c_1(\alpha) = \tilde{c}_1(\alpha)$ is continuous at $\alpha = 0^+$, we have $c_1(\alpha) \xrightarrow{\alpha \to 0} C_0 f(z_0) > 0$, we conclude that the full expansion (4.23) holds as claimed. $\square$

The next lemma shows that in representations (4.16), (4.17), (4.18) and (4.19), the integrals along $S_{x,\alpha}^{\pm}$ and $S_{y,\alpha}^{\pm}$ are negligible compared to those along $\Gamma_{x,\alpha}$ and $\Gamma_{y,\alpha}$.

**Lemma 4.6** (Negligibility of the integrals along $S_{x,\alpha}^{\pm}$ and $S_{y,\alpha}^{\pm}$). Let $z_0 \in \mathbb{R}^2$. For $(a,b) \in \mathbb{R} \times (0, +\infty)$, let $\alpha(a,b) \in (0, \pi)$ be the angle such that $\cos(\alpha) = \frac{a}{\sqrt{a^2+b^2}}$ and $\sin(\alpha) = \frac{b}{\sqrt{a^2+b^2}}$.

* Suppose $\alpha_0 \in (0, \pi)$. Then, for $\eta > 0$ small enough, there exist $r_0 > 0$ and $D > 0$ such that for all $(a,b)$ satisfying $\sqrt{a^2 + b^2} > r_0$ and $|\alpha(a,b) - \alpha_0| < \eta$, we have:

$$
(4.30) \qquad \left| \int_{S_{x,\alpha}^{\pm}} \varphi^{z_0}(x) \frac{\gamma(x, Y^{+}(x), Z^{+}(x))}{\partial_y \gamma_{+}(x, Y^{+}(x))} e^{-ax-bY^{+}(x)} dx \right| \le \frac{D}{b} e^{-ax(\alpha)-by(\alpha)-\epsilon^2 \sqrt{a^2+b^2}}.
$$

If $b > b_0$, then the following estimate also holds:

$$
(4.31) \qquad \left| \int_{S_{x,\alpha}^{\pm}} \frac{e^{(a_0-a)x} \left( e^{(b_0-b)Y^+(x)} \mathbb{1}_{b_0>0} + e^{b_0 Z^+(x)-bY^+(x)} \mathbb{1}_{b_0<0} \right)}{\partial_y \gamma_+(x, Y^+(x))} dx \right| \le \frac{D}{b-b_0} e^{-ax(\alpha)-by(\alpha)-\epsilon^2 \sqrt{a^2+(b-b_0)^2}}.
$$

* Suppose $x_b = x_{\max}^+$ and let $\alpha_0 = 0$. Then, for $\eta > 0$ small enough, there exist $r_0 > 0$ and $D > 0$ such that for all $(a,b)$ satisfying $\sqrt{a^2 + b^2} > r_0$ and $0 < \alpha(a,b) \le \eta$, we have:

$$
(4.32) \qquad \left| \int_{S_{x,\alpha}^{+}} \varphi^{z_0}(x) \frac{\gamma(x, Y^{+}(x), Z^{+}(x))}{\partial_y \gamma_{+}(x, Y^{+}(x))} e^{-ax-bY^{+}(x)} dx \right| \leq D e^{-ax(\alpha)-by(\alpha)-\epsilon^2 \sqrt{a^2+b^2}},
$$
$$
(4.33) \qquad \left| \int_{S_{y,\alpha}^+} \frac{e^{(a_0-a)X^+(y)+(b_0-b)y}}{\partial_x \gamma_+(X^+(y),y)} dy \right| \le \frac{D}{a} e^{-ax(\alpha)-by(\alpha)-\epsilon^2 \sqrt{a^2+b^2}}.
$$

• Symmetric estimates hold for α₀ = π when ˜x_b = x_min.

The same estimates hold for $S_{x,\alpha}^{-}$ and $S_{y,\alpha}^{-}$.

Proof. First, let us consider the case $0 < \alpha_0 < \pi$. Using the notation (4.7), inequality (4.30) can be
written as

$$
(4.34) \quad \left| \int_{v>0} \frac{\varphi^{z_0}(x_{\alpha}^{+} + iv)\gamma(x_{\alpha}^{+} + iv, Y^{+}(x_{\alpha}^{+} + iv), Z^{-}(x_{\alpha}^{+} + iv))}{\partial_y \gamma(x_{\alpha}^{+} + iv, Y^{+}(x_{\alpha}^{+} + iv))} \exp(\sqrt{a^2+b^2}(F(x_{\alpha}^{+} + iv, \alpha) - F(x_{\alpha}^{+}, \alpha))) dv \right| \le \frac{D}{b}.
$$

Furthermore,

$$
(4.35) \quad \left| \exp\left(\sqrt{a^2+b^2}\left(F(x_\alpha^+ + iv, \alpha) - F(x_\alpha^+, \alpha)\right)\right) \right| = \exp\left(-b(\mathfrak{R}Y^+(x_\alpha^+ + iv) - \mathfrak{R}Y^+(x_\alpha^+))\right).
$$

By (1.14), the quantity $\mathfrak{R}Y^+(x_\alpha^+ + iv) - \mathfrak{R}Y^+(x_\alpha^+)$ vanishes at $v=0$ and grows linearly as $v \to +\infty$.
Therefore, there exist constants $c > 0$ and $\eta > 0$ such that for all $\alpha$ satisfying $|\alpha - \alpha_0| \le \eta$ and all $v \ge 0$,

$$
(4.36) \qquad \mathfrak{R}Y^+(x_\alpha^+ + iv) - \mathfrak{R}Y^+(x_\alpha^+) \geq cv.
$$

By (1.14) and (1.15), the quantity $\gamma(u + iv, Y^{\pm}(u + iv), Z^{\pm}(u + iv))$ grows linearly as $v \to +\infty$, uniformly in $u \in [-\tilde{x}_b - \epsilon, x_b + \epsilon]$ for a given $\epsilon > 0$. Similarly, $\partial_y\gamma_+(u + iv, Y^{\pm}(u + iv))$ tends to 0 as $v \to +\infty$, also uniformly in $u \in [-\tilde{x}_b - \epsilon, x_b + \epsilon]$. Then, by (1.17), we have

$$
(4.37) \quad \sup_{v \ge 0, |\alpha - \alpha_0| \le \eta} \left| \frac{\varphi^{z_0}(x_\alpha^+ + iv) \gamma(x_\alpha^+ + iv, Y^+(x_\alpha^+ + iv), Z^+(x_\alpha^+ + iv))}{\partial_y \gamma_+(x_\alpha^+ + iv, Y^+(x_\alpha^+ + iv))} \right| < +\infty.
$$

Then, for all (a, b) such that $|\alpha(a, b) - \alpha_0| \le \eta$, the left-hand side of (4.34) is bounded by

$$
C \int_{0}^{\infty} e^{-bcv} \, dv = \frac{C}{cb},
$$

where $C > 0$ is a constant independent of $(a, b)$. This proves (4.30). Inequalities (4.31) and (4.33) follow
by analogous symmetric arguments.

Now, let us consider the case $\alpha_0 = 0$ and prove (4.32). This inequality can be written as (4.34), but
with the right-hand side $\frac{D}{b}$ replaced by $D$. The exponential term

$$
\exp\left(-\sqrt{a^2 + b^2}\left(F(x_{\alpha}^{+} + iv, \alpha) - F(x_{\alpha}^{+}, \alpha)\right)\right)
$$

can be bounded by 1. By similar arguments, there exists $\eta > 0$ such that

$$
(4.38) \quad C' := \sup_{v \ge 0, 0 < \alpha \le \eta} \left| \frac{\gamma(x_{\alpha}^{+} + iv, Y^{+}(x_{\alpha}^{+} + iv), Z^{+}(x_{\alpha}^{+} + iv))}{\partial_y \gamma_{+}(x_{\alpha}^{+} + iv, Y^{+}(x_{\alpha}^{+} + iv))} \right| < \infty.
$$

If $b_0 \neq 0$, then by (3.1) and (1.17), we have

$$
(4.39) \qquad C'' := \sup_{0 < \alpha \le \eta} \int_0^{+\infty} \left| \varphi^{z_0}(x_{\alpha}^{+} + iv) \right| dv < +\infty,
$$

which gives the desired result with $D = C'C''$. If $b_0 = 0$, an integration by parts argument justifies the
same estimate, using the explicit expression (1.17) (see [14, Lemma 7.3] for similar estimates). $\square$

We now have the tools to prove Theorems 3 and 4.
*Proof of Theorem 3.* Let $\alpha_0 \in (0, \pi)$ be such that $\tilde{x}_b < x(\alpha_0) < x_b$, and let $z_0 \in \mathbb{R}^2$. By Lemmas 4.2 and 4.3, the function $g^{z_0}(a, b)$ can be written as $g^{z_0}(a, b) = I_1^{z_0}(a, b) + I_2^{z_0}(a, b)$, where $I_1^{z_0}(a, b)$ and $I_2^{z_0}(a, b)$ are given by (4.16) and (4.17). Using the asymptotic expansion given in (4.21) from Lemma 4.4, together with the bounds (4.30) and (4.31) from Lemma 4.6, we see that the contributions of the integrals along the segments $S_{x,\alpha}^{\pm}$ are negligible compared to those along $\Gamma_{x,\alpha}$. Therefore, the asymptotics of $g^{z_0}(a, b)$ are given by (4.21). $\square$

*Proof of Theorem 4.* Without loss of generality, we consider the case $\alpha_0 = 0$. We use the representations of $I_1^{z_0}(a, b)$ and $I_2^{z_0}(a, b)$ given by (4.16) and (4.18). Then, applying the asymptotic expansion (4.23) from Lemma 4.5 and the estimates (4.32) and (4.33) from Lemma 4.6, we conclude the proof in the same way as for Theorem 3. $\square$

### 4.4. Lemmas for directions $\alpha_b$ and $\tilde{\alpha}_b$
Without loss of generality, we consider the asymptotic behaviour of $g^{z_0}$ in the direction $\alpha_b$. Since the integrands in (4.2) and (4.3) are no longer holomorphic at points $x > x_b$, we must introduce a different deformation of the integration path; see Figure 11b. Note that the integrands of (4.2) and (4.3) admit limits at the top and bottom edges of the branch cut $[x_b, +\infty)$. To define the new contour, we introduce the following notation.

*Notation 4.7.* Let $M > x_b$, and let $f$ be a continuous function defined in a neighbourhood of $[x_b, M]$ within $\{x \in \mathbb{C} : \Im(x) \ge 0\}$. We define

$$ \int_{\Omega_{+,M}} f(x) dx := \lim_{\eta \to 0} \int_{\Omega_{+,M}^{\eta}} f(x) dx, $$

where $\Omega_{+,M}^{\eta}$ denotes the straight segment connecting $M + i\eta$ to $x_b + i\eta$, oriented as in Figure 11a. Integration along $\Omega_{-,M}$ is defined analogously, following the orientation shown in Figure 11a.

If $\alpha < \alpha_b$ (i.e., $x(\alpha) > x_b$), we set $\Omega_{\pm,\alpha} = \Omega_{\pm,x(\alpha)}$. If $\alpha > \alpha_b$ (i.e., $x(\alpha) < x_b$), we adopt the convention $\Omega_{+,\alpha} = \Omega_{-,\alpha} = \{x(\alpha)\}$, i.e., the constant path reduced to the point $x(\alpha)$. We also define

$$ \Gamma_{x,\alpha}^{-} = \{x(it, \alpha) \mid t \in [-\epsilon, 0]\}, \quad \Gamma_{x,\alpha}^{+} = \{x(it, \alpha) \mid t \in (0, \epsilon]\}. $$

We can thus consider integrals of the relevant quantities along $\Omega_{\pm,\alpha}$. Let us now define the function $\Phi$ as

$$ (4.40) \qquad \Phi(x) = \varphi^{z_0}(x) \frac{\gamma(x, Y^{+}(x), Z^{+}(x))}{\partial_y \gamma_{+}(x, Y^{+}(x))} + \frac{e^{a_0 x + b_0 Y^{+}(x) \mathbb{1}_{b_0 > 0} + b_0 Z^{+}(x) \mathbb{1}_{b_0 < 0}}{\partial_y \gamma_{+}(x, Y^{+}(x))} $$

$$ (4.41) \qquad = \frac{e^{a_0 x + Z^{+}(x)b_0 \mathbb{1}_{b_0 < 0}}}{\partial_y \gamma_{+}(x, Y^{+}(x))} \left( -\frac{\gamma(x, Y^{+}(x), Z^{+}(x))}{\gamma(x, Y^{-}(x), Z^{+}(x))} e^{b_0 Y^{-}(x) \mathbb{1}_{b_0 > 0}} + e^{b_0 Y^{+}(x) \mathbb{1}_{b_0 > 0}} \right). $$

Observe that $\Phi$ is discontinuous along the cut $[x_b, +\infty)$. The following lemma provides the appropriate contour deformation when $\alpha_0 = \alpha_b$.

**Lemma 4.8** (Change of contour for direction $\alpha_b$). Assume that $x_{max}^{+} < x_{max}^{-}$, i.e., $\alpha_b \in (0, \pi)$. Let $z_0 \in \mathbb{R}^2$, $a \in \mathbb{R}$, and $b > b_0$ such that $(a, b) \neq z_0$. Then, for $0 < \alpha < \alpha_b$ (i.e., $x_{max}^{+} < x(\alpha)$), we have

$$ (4.42) \qquad I_1^{z_0}(a, b) + I_2^{z_0}(a, b) = \frac{1}{2i\pi} \int_{S_{x,\alpha}^{-} \cup \Gamma_{x,\alpha}^{-} \cup \Omega_{-,\alpha} \cup \Omega_{+,\alpha} \cup \Gamma_{x,\alpha}^{+} \cup S_{x,\alpha}^{+}} \Phi(x) e^{-ax - bY^{+}(x)} dx. $$

*Proof.* We apply the Residue theorem to the representations (4.2) and (4.3) of $I_1^{z_0}(a, b)$ and $I_2^{z_0}(a, b)$, using the contour described in Figure 11b. The asymptotic as $M \to +\infty$ is justified by (4.20), and the limit as $\eta \to 0$ follows from the dominated convergence theorem. This yields the desired identity (4.42). $\square$

For the asymptotic behaviour of Green's functions as $\alpha \to \alpha_b$, both contours $\Omega_{\pm,\alpha}$ and $\Gamma_{x,\alpha}^{\pm}$ contribute to the leading term. Since the function $\Phi$ is not holomorphic at $x_b$, the standard saddle-point expansion does not apply. However, we present an adapted version of the method.

Recall the notation (4.10). From the expressions (1.14) and (1.17), there exists a neighbourhood $V$ of $x_b$, containing the disc $\mathbb{D}(0, L)$ for some $L > 0$, and a holomorphic function $\psi$ on $V$ such that

$$ \Phi(x) = \psi(\sqrt{x_b - x}), \quad \text{for all } x \in V \setminus [x_b, +\infty). $$
<figure><img src="image_15.png" /><figcaption>FIGURE 11. Illustration of the integration paths $\Omega_{+,\alpha}^\eta$ and the contour deformation for direction $\alpha_b$.</figcaption></figure>

By separating the even and odd parts in the power series expansion of $\psi$ around $x_b$, we may write

$$ (4.43) \qquad \Phi(x) = \Phi_1(x) + \sqrt{x_b - x} \Phi_2(x) $$

for $x \in V \setminus [x_b, +\infty)$, where $\Phi_1$ and $\Phi_2$ are holomorphic in a neighbourhood of $x_b$. This decomposition allows us to establish the following lemma.

**Lemma 4.9** (Contribution of $\Gamma_{x,\alpha}^{\pm}$ in direction $\alpha_b$). Let $z_0 \in \mathbb{R}^2$. Suppose $x_{\max}^{+} < x_{\max}^{-}$. Then, the following asymptotic expansion holds:

$$ (4.44) \qquad \frac{1}{2i\pi} \int_{\Gamma_{x,\alpha}^{-} \cup \Gamma_{x,\alpha}^{+}} \Phi(x) e^{-r(\cos(\alpha)x + \sin(\alpha)Y^{+}(x))} dx \underset{\substack{r \to +\infty \\ \alpha \to \alpha_b}}{\sim} C^{+}(\alpha_b) h_{\alpha_b}(z_0) \frac{e^{-r(\cos(\alpha)x(\alpha) + \sin(\alpha)y(\alpha))}}{\sqrt{r}} $$

where $C^{+}(\alpha_b)$ and $h_{\alpha_b}(z_0)$ are given by (1.29) and (1.30).

*Proof.* Applying the saddle-point method to the holomorphic function $\Phi_1$, as developed in [14, Lemma 8.1], we obtain:

$$ (4.45) \qquad \frac{1}{2i\pi} \int_{\Gamma_{x,\alpha}} \Phi_1(x) e^{-r(\cos(\alpha)x + \sin(\alpha)Y^+(x))} dx \underset{\substack{r \to +\infty \\ \alpha \to \alpha_b}}{\sim} C^+(\alpha_b) h_{\alpha_b}(z_0) \frac{e^{-r(\cos(\alpha)x(\alpha) + \sin(\alpha)y(\alpha))}}{\sqrt{r}}, $$

since $\Phi(x_b) = \Phi_1(x_b)$. Moreover, for all $(t, \alpha) \in \Omega(0, \alpha_b)$, we have

$$ |\sqrt{x_b - x(it, \alpha)}| \le \sqrt{|x(0, \alpha_b) - x(0, \alpha)| + |x(0, \alpha) - x(it, \alpha)|} \le C (\sqrt{|\alpha_b - \alpha|} + \sqrt{|t|}), $$
for some constant $C > 0$ independent of $t$ and $\alpha$. Since $\Phi_2$ is holomorphic, it is bounded on every compact sets. Therefore, there exist constants $M, M' > 0$ such that, for all $\alpha$ with $|\alpha - \alpha_b| < \eta$, we have:

$$ (4.46) \qquad \left| \int_{\Gamma_{x,\alpha}^{+}} \sqrt{x_b-x} \Phi_2(x) e^{-r(\cos(\alpha)x+\sin(\alpha)Y^{+}(x))} dx \right| = e^{-r(\cos(\alpha)x(\alpha)+\sin(\alpha)y(\alpha))} \times \left| \int_{0}^{\epsilon} \sqrt{x_b-x(it,\alpha)} \Phi_2(x(it,\alpha)) e^{-rt^2} i x'_{\omega}(it,\alpha) dt \right| \\ \leq M e^{-r(\cos(\alpha)x(\alpha)+\sin(\alpha)y(\alpha))} \int_{0}^{\epsilon} (\sqrt{|\alpha_b-\alpha|} + \sqrt{t}) e^{-rt^2} dt \\ \underset{\alpha \to \alpha_b}{\sim} M' e^{-r(\cos(\alpha)x(\alpha)+\sin(\alpha)y(\alpha))} \left( \frac{\sqrt{|\alpha_b-\alpha|}}{\sqrt{r}} + \frac{1}{r} \right), $$

which is negligible compared to (4.45) as $r \to +\infty$ and $\alpha \to \alpha_b$. The same estimate holds for the integral over $\Gamma_{x,\alpha}^-$. This completes the proof of (4.44). $\square$

We split the proof of the contribution of the integrals along $\Omega_{\pm,\alpha}$ into two lemmas.

**Lemma 4.10** (Value of $\Phi_2$ at $x_b$). Recall the notation from (4.43). Suppose $x_{max}^- < x_{max}^+$. Then:

$$ \Phi_2(x_b) = -2 \frac{\sqrt{\det(\Sigma^-)(x_b - x_{min}^-)}}{\Sigma_{22}^- (\Sigma_{22}^+ + \Sigma_{22}^-)} f_0(z_0) $$

where $f_0(z_0)$ is given by (1.26).

*Proof.* The proof reduces to showing that

$$ (4.47) \qquad \Phi(x) = \Phi(x_b) - 2 \frac{\sqrt{\det(\Sigma^-)(x_b - x_{min}^-)}}{\Sigma_{22}^- (\Sigma_{22}^+ + \Sigma_{22}^-)} f_0(z_0) \sqrt{x_b - x} + o(\sqrt{x_b - x}) $$

as $x \to x_b$. To prove the above formula, we use the explicit expressions of $\Phi$ and $\varphi^{z_0}$ given by (4.40) and (1.17). Let us denote $C_z = \frac{\sqrt{\det(\Sigma^-)(x_b - x_{min}^-)}}{\Sigma_{22}^-}$ so that $Z^+(x) = Z^+(x_b) + C_z \sqrt{x_b - x} + o(\sqrt{x_b - x})$. We have:

$$ \begin{aligned} \frac{\gamma(x, Y^+(x), Z^+(x))}{\gamma(x, Y^-(x), Z^+(x))} &= 1 + \frac{\left(\frac{1+q_2}{2}\right) (Y^+(x) - Y^-(x))}{\gamma(x, Y^-(x), Z^+(x))} \\ &= 1 + \left(\frac{1+q_2}{2}\right) \frac{Y^+(x) - Y^-(x)}{\gamma(x_b, Y^-(x_b), Z^+(x_b)) + \frac{q_2-1}{2} C_z \sqrt{x_b - x} + o(\sqrt{x_b - x})} \\ &= 1 + \left(\frac{1+q_2}{2}\right) \frac{Y^+(x_b) - Y^-(x_b)}{\gamma(x_b, Y^-(x_b), Z^+(x_b))} \left(1 - \frac{\frac{q_2-1}{2} C_z \sqrt{x_b - x} (1+o(1))}{\gamma(x_b, Y^-(x_b), Z^+(x_b))}\right). \end{aligned} $$

Furthermore,

$$ \begin{aligned} e^{Z^+(x)b_01_{b_0<0}} &= e^{(Z^+(x_b)+\sqrt{x_b-x}(C_z+o(1)))b_01_{b_0<0}} \\ &= e^{Z^+(x_b)b_01_{b_0<0}}(1+\sqrt{x_b-x}(C_z+o(1)))b_01_{b_0<0}). \end{aligned} $$

Then, if $b_0 < 0$, (4.41) can be written as

$$ \partial_y \gamma_+(x, Y^+(x)) \Phi(x) = e^{x_b a_0 + Z^+(x_b) b_0} \left(1 + C_z \sqrt{x_b - x} (b_0 + o(1))\right) \\ \times - \left(\frac{1+q_2}{2}\right) \frac{Y^+(x_b) - Y^-(x_b)}{\gamma(x_b, Y^-(x_b), Z^+(x_b))} \left(1 - \frac{\frac{q_2-1}{2} C_z \sqrt{x_b - x} (1+o(1))}{\gamma(x_b, Y^-(x_b), Z^+(x_b))}\right) $$

which yields (4.47), using the identity $\partial_y \gamma_+(x_b, Y^+(x_b)) = \frac{\Sigma_{22}^+}{2}(Y^+(x_b) - Y^-(x_b))$. If $b_0 > 0$, the computations are analogous. $\square$
Before setting the asymptotics along $\Omega_{\pm,\alpha}$, we need to introduce some notation. We recall that $x(0, \alpha_b) = x_b$ and that $x'_\omega(0, \alpha_b) = \sqrt{\frac{2}{F_x''(x(\alpha_b),\alpha)}} \neq 0$ by (4.9). Then, by the implicit function theorem, there exists a $C^\infty$ function, denoted by $\tau$, and a neighbourhood $W$ of $(0, \alpha_b)$ in $\mathbb{R}_\omega \times \mathbb{R}_\alpha$ (contained in $\Omega(0, \alpha_b)$, see (4.10)) such that

$$ \forall (\omega, \alpha) \in W, \quad [x(\omega, \alpha) = x_b] \iff [\omega = \tau(\alpha)]. $$

In particular, $x(\tau(\alpha), \alpha) = x_b$, and $\tau'(\alpha_b) = \frac{-x'(\alpha_b)}{x'_\omega(0, \alpha_b)} > 0$. Then, for $\alpha < \alpha_b$, we have $\tau(\alpha) < 0$ and

$$ \tau(\alpha) = \frac{-x'(\alpha_b)}{x'_\omega(0, \alpha_b)}(\alpha - \alpha_b)(1 + o_{\alpha \to \alpha_b}(1)). $$

Note that, by (4.8) and (4.12), we have $|K_+| = \frac{-x'(\alpha_b)}{x'_\omega(0, \alpha_b)}$ with the notation (1.34). Since, by (4.11), $F(x_b, \alpha) = F(x(\tau(\alpha), \alpha), \alpha) = \tau^2(\alpha)$, we deduce from (4.7) the following identity:

$$ (4.48) \qquad \cos(\alpha)x(\alpha_b) + \sin(\alpha)y(\alpha_b) = \cos(\alpha)x(\alpha) + \sin(\alpha)y(\alpha) + \tau^2(\alpha). $$

**Lemma 4.11** (Contribution of $\Omega_{\pm,\alpha}$ in direction $\alpha_b$). Suppose that $x_{max}^+ < x_{max}^-$. Then,

* If $r(\alpha - \alpha_b)^2$ remains bounded, then the asymptotic contribution of the integrals along $\Omega_{\pm,\alpha}$ in (4.42) is negligible compared to (4.44) as $r \to +\infty$ and $\alpha \to \alpha_b$.

* If $r(\alpha - \alpha_b)^2 \to +\infty$ and $\alpha \le \alpha_b$, the following asymptotics hold:

$$ (4.49) \qquad \frac{1}{2i\pi} \int_{\Omega_{\pm,\alpha}} \Phi(x) e^{-r(\cos(\alpha)x + \sin(\alpha)Y^+(x))} dx \\ \sim_{\substack{r \to +\infty \\ \alpha \to \alpha_b, \alpha \le \alpha_b}} \frac{\sin^{3/2}(\alpha_b) \sqrt{\det(\Sigma^-)(x_b - x_{min}^-)}}{\sqrt{\pi \Sigma_{22}^- (\Sigma_{22}^+ + \Sigma_{22}^-)}} f_0(z_0) e^{-r(\cos(\alpha)x(\alpha) + \sin(\alpha)y(\alpha))} \frac{e^{r\tau(\alpha)^2}}{(r(\alpha_b - \alpha))^{3/2}}. $$

*Proof.* First, since $\Phi_1$ is holomorphic, we have

$$ \frac{1}{2i\pi} \int_{\Omega_{\bar{x},\alpha} \cup \Omega_{\bar{x},\alpha}^{+}} \Phi_1(x) e^{-r(\cos(\alpha)x + \sin(\alpha)Y^+(x))} dx = 0. $$

For $\Phi_2$, we note that

$$ \int_{\Omega_{\bar{x},\alpha}^{-}} \sqrt{x_b - x} \Phi_2(x) e^{-ax - bY^+(x)} dx = \int_{\Omega_{\bar{x},\alpha}^{+}} \sqrt{x_b - x} \Phi_2(x) e^{-ax - bY^+(x)} dx $$

by the definition of integration over $\Omega_{\pm,\alpha}$ and the convention for the complex square root. We parameterise the path $\Omega_{+,\alpha}$ as $\Omega_{+,\alpha} = \{x(t, \alpha) | t \in [\tau(\alpha), 0]\}$. Then,

$$ (4.50) \qquad \int_{\Omega_{\bar{x},\alpha}^{+}} \sqrt{x_b - x} \Phi_2(x) e^{-r(\cos(\alpha)x + \sin(\alpha)Y^+(x))} dx \\ = -ie^{-r(\cos(\alpha)x(\alpha) + \sin(\alpha)y(\alpha))} \int_{\tau(\alpha)}^{0} \sqrt{x(t, \alpha) - x_b} \Phi_2(x(t, \alpha)) e^{rt^2} x'_\omega(t, \alpha) dt. $$

Moreover, for any $t \in (\tau(\alpha), 0)$, there exists $\xi_{t,\alpha} \in (\tau(\alpha), t)$ such that

$$ x(t, \alpha) - x_b = x(t, \alpha) - x(\tau(\alpha), \alpha) = x'_\omega(\xi_{t,\alpha}, \alpha)(t - \tau(\alpha)). $$

Therefore,

$$ (4.51) \qquad \left| \int_{\tau(\alpha)}^{0} \sqrt{x(t, \alpha) - x_b} \Phi_2(x(t, \alpha)) e^{rt^2} x'_\omega(t, \alpha) dt - \Phi_2(x(0, \alpha)) x'_\omega(0, \alpha)^{3/2} \int_{\tau(\alpha)}^{0} \sqrt{t - \tau(\alpha)} e^{rt^2} dt \right| \\ \le \epsilon(\alpha) \int_{\tau(\alpha)}^{0} \sqrt{t - \tau(\alpha)} e^{rt^2} dt $$
where

$$
\begin{align*}
\epsilon(\alpha) &:= \sup_{t \in [\tau(\alpha), 0]} \left| \sqrt{x'_{\omega}(\xi_{t,\alpha}, \alpha)} \Phi_2(x(t, \alpha)) x'_{\omega}(t, \alpha) - \sqrt{x'_{\omega}(0, \alpha)} \Phi_2(x(0, \alpha)) x'_{\omega}(0, \alpha) \right| \\
&\le \sup_{\substack{t \in [\tau(\alpha), 0] \\ \xi \in [\tau(\alpha), 0]}} \left| \sqrt{x'_{\omega}(\xi, \alpha)} \Phi_2(x(t, \alpha)) x'_{\omega}(t, \alpha) - \sqrt{x'_{\omega}(0, \alpha)} \Phi_2(x(0, \alpha)) x'_{\omega}(0, \alpha) \right| \underset{\alpha \to \alpha_b}{\to} 0.
\end{align*}
$$

Using the change of variables $t = \tau(\alpha)s$, we have

$$
(4.52) \qquad \int_{\tau(\alpha)}^{0} \sqrt{t - \tau(\alpha)} e^{rt^2} dt = |\tau(\alpha)|^{3/2} \int_{0}^{1} \sqrt{1 - s} e^{r\tau(\alpha)^2 s^2} ds.
$$

Assume first that $r\tau(\alpha)^2$ remains bounded. Then, (4.52) is also bounded. By (4.51), (4.50) is then bounded by $|\tau(\alpha)|^{3/2}e^{-r(\cos(\alpha)x(\alpha)+\sin(\alpha)y(\alpha))} \le M\sqrt{|\tau(\alpha)|} \frac{e^{-r(\cos(\alpha)x(\alpha)+\sin(\alpha)y(\alpha))}}{\sqrt{r}}$ for some constant $M > 0$, which is negligible compared to (4.44). This proves (ii).

Now, suppose that $r\tau(\alpha)^2 \to +\infty$. By Lemma B.1, the following asymptotics hold:

$$
|\tau(\alpha)|^{3/2} \int_0^1 \sqrt{1-s} e^{r\tau(\alpha)^2 s^2} ds \sim |\tau(\alpha)|^{3/2} \frac{\sqrt{\pi}}{4\sqrt{2}} \frac{e^{r\tau(\alpha)^2}}{(r\tau(\alpha)^2)^{3/2}} = \frac{\sqrt{\pi}}{4\sqrt{2}} \frac{e^{r\tau(\alpha)^2}}{(r|\tau(\alpha)|)^{3/2}}
$$

Hence,

$$
\begin{align*}
\frac{1}{2i\pi} \int_{\Omega_{x,\alpha}^{-} \cup \Omega_{x,\alpha}^{+}} \Phi(x) e^{-r(\cos(\alpha)x + \sin(\alpha)Y^{+}(x))} dx &\sim \frac{-i}{i\pi} \frac{\sqrt{\pi}}{4\sqrt{2}} \Phi_2(x_b) x'_{\omega}(0, \alpha_b)^{3/2} e^{-r(\cos(\alpha)x(\alpha) + \sin(\alpha)y(\alpha))} \frac{e^{r\tau(\alpha)^2}}{(r\tau(\alpha))^{3/2}} \\
&\sim \frac{-\sin^{3/2}(\alpha_b)}{2\sqrt{\pi}} \Phi_2(x_b) e^{-r(\cos(\alpha)x(\alpha) + \sin(\alpha)y(\alpha))} \frac{e^{r\tau(\alpha)^2}}{(r\sin(\alpha_b - \alpha))^{3/2}}
\end{align*}
$$

since

$$
\frac{x'_{\omega}(0, \alpha_b)^3}{2\sqrt{2}|x'(\alpha_b)|^{3/2}} = \frac{1}{|F''(x(\alpha_b), \alpha_b)x'(\alpha_b)|^{3/2}} = |\sin(\alpha_b)|^{3/2}
$$

(see (4.8) and (4.12)). The conclusion follows from Lemma 4.10. $\square$

Proof of Theorem 5. By Lemmas 4.2 and 4.8, the function $g^{z_0}$ can be written as $g^{z_0} = I_1^{z_0}(a, b) + I_2^{z_0}(a, b)$, with representation given by (4.42). According to Lemmas 4.6, 4.9, and 4.11, the contributions of the integrals along the paths $S_{x,\alpha}^{\pm}$ are negligible compared to those along $\Gamma_{x,\alpha}^{\pm}$ and $\Omega_{x,\alpha}^{\pm}$. Thus, we are left with a competition between the contributions from $\Gamma_{x,\alpha}^{\pm}$ and $\Omega_{x,\alpha}^{\pm}$.

(i) If $\alpha > \alpha_b$, then the integrals along $\Omega_{\pm,\alpha}$ vanish (since $\Omega_{\pm,\alpha} = \{x(\alpha)\}$), and the result (4.44) follows from Lemma 4.9.

(ii) The statement in point (ii) directly follows from Lemma 4.11.

(iii) – (iv) – (v) Now suppose that $\alpha < \alpha_b$ and $r(\alpha - \alpha_b)^2 \to +\infty$. Then we have: We have

$$
\begin{align*}
(4.53) \qquad \frac{e^{r\tau^2(\alpha)}}{r|\tau(\alpha)|^{3/2}} &= \frac{e^{r\tau^2(\alpha)}}{r|\tau(\alpha)|^2} \sqrt{|\tau(\alpha)|} \\
&= \exp\left(r\tau^2(\alpha) - \ln(r\tau^2(\alpha)) + \frac{1}{2}\ln(|\tau(\alpha)|)\right) \\
&= \exp\left(K_+^2 r(\alpha_b - \alpha)^2 (1 + o(1)) + \frac{1}{2}\ln(\alpha_b - \alpha) + \frac{1}{2}\ln(|K_+|)\right) \\
(4.54) \qquad &= \sqrt{|K_+|} \exp\left(K_+^2 r(\alpha_b - \alpha)^2 \left(\frac{\ln(\alpha_b - \alpha)}{2K_+^2 r(\alpha_b - \alpha)^2} + 1 + o(1)\right)\right).
\end{align*}
$$

If $\frac{\ln(\alpha_b - \alpha)}{2K_+^2 r(\alpha_b - \alpha)^2} \to l \in [-\infty, -1)$, then (4.53) tends to 0, and the asymptotics of $g(r \cos(\alpha), r \sin(\alpha))$ are governed by (4.44). Note that in this case, $\frac{K_+^2 r(\alpha_b - \alpha)^2}{r(\alpha_b - \alpha)^{3/2}} \to 0$. If $\frac{\ln(\alpha_b - \alpha)}{2K_+^2 r(\alpha_b - \alpha)^2} \to l \in (-1, 0]$, then (4.53) diverges to $+\infty$, and the asymptotics of $g(r \cos(\alpha), r \sin(\alpha))$ are given by (4.49), which coincides with (1.35) thanks to (4.48). In this case, $\frac{K_+^2 r(\alpha_b - \alpha)^2}{r(\alpha_b - \alpha)^{3/2}} \to +\infty$. It remains to
<figure><img src="image_16.png" /><figcaption>FIGURE 12. Path deformation in Lemma 4.12</figcaption></figure>

analyze the critical case where $\frac{\ln(\alpha_b - \alpha)}{2K_+^2 r(\alpha_b - \alpha)^2} \to -1$. In this situation, we have $r \sim \frac{-\ln(\alpha_b - \alpha)}{2K_+^2 (\alpha_b - \alpha)^2}$ so that

$$r\tau^2(\alpha) = rK_+^2(\alpha_b - \alpha)^2 + O(r(\alpha_b - \alpha)^3) = rK_+^2(\alpha_b - \alpha)^2 + o(1)$$

and consequently,

$$(4.55) \qquad \frac{e^{r\tau^2(\alpha)}}{r|\tau(\alpha)|^{3/2}} \sim \frac{e^{K_+^2 r(\alpha_b - \alpha)^2}}{r|K_+(\alpha - \alpha_b)|^{3/2}}.$$

Hence, both contributions (4.44) and (4.49) may be significant, and expression (4.55) provides the conclusion for points (iii) to (v), depending on the possible limiting behavior of $\frac{e^{K_+^2 r(\alpha_b - \alpha)^2}}{r|K_+(\alpha - \alpha_b)|^{3/2}}$. $\square$

**4.5. Lemmas for the non-red-arc zones.** For angles outside of the red zones, we use a different changing path again. For $\delta > 0$, we set

$$(4.56) \qquad R_{\delta}^{+} = \{x_b + \delta + iv \mid v > 0\}, \quad R_{\delta}^{-} = \{x_b + \delta - iv \mid v > 0\}.$$

The following lemma performs the appropriate path deformation; see Figure 12.

**Lemma 4.12** (Changing path for directions $0 \le \alpha < \alpha_b$). Suppose that $x_{max}^+ < x_{max}^-$ (i.e., $\alpha_b \in (0, \pi)$). Let $\delta > 0$ and let $z_0 \in \mathbb{R}^2$, $a \in \mathbb{R}$, and $b > 0$ such that $(a, b) \neq z_0$. Then, for $0 \le \alpha < \alpha_b$,

$$(4.57) \qquad I_1^{z_0}(a, b) = \frac{1}{2i\pi} \int_{R_{\delta}^{-} \cup \Omega_{-, x_b+\delta} \cup \Omega_{+, x_b+\delta} \cup R_{\delta}^{-}} \varphi^{z_0}(x) \frac{\gamma(x, Y^{+}(x), Z^{+}(x))}{\partial_y \gamma_{+}(x, Y^{+}(x))} e^{-ax-bY^{+}(x)} dx.$$

Furthermore, if $b_0 > 0$ and $a > a_0$, then (4.18) holds. Finally, if $b_0 \le 0$, then

$$(4.58) \qquad I_2^{z_0}(a, b) = \frac{1}{2i\pi} \int_{R_{\delta}^{-} \cup \Omega_{-, x_b+\delta} \cup \Omega_{+, x_b+\delta} \cup R_{\delta}^{+}} \frac{e^{(a_0-a)x+b_0Z^{+}(x)-bY^{+}(x)}}{\partial_y \gamma_{+}(x, Y^{+}(x))} dx.$$

*Proof.* The proof is identical to that of Lemma 4.8, except that the paths $\Gamma_{x,\alpha}^{\pm}$ are replaced by straight lines. $\square$

Once again, we split the analysis of the asymptotics. We recall the notation from (4.40). The following lemma provides the main asymptotic contribution.
**Lemma 4.13** (Contribution of $\Omega_{-,x_b+\delta}$). Let $z_0 \in \mathbb{R}^2$. Suppose that $x_{max}^+ < x_{max}^-$ (i.e., $\alpha_b \in (0, \pi)$), and let $\alpha_0 \in [0, \alpha_b)$. If $b_0 \le 0$, then
(4.59)

$$ \frac{1}{2i\pi} \int_{\Omega_{-,x_b+\delta} \cup \Omega_{+,x_b+\delta}} \Phi(x) e^{-r(\cos(\alpha)x + \sin(\alpha)Y^+(x))} dx \underset{\substack{r \to \infty \\ \alpha \to \alpha_0, \alpha \ge 0}}{\sim} C_{br} f_0(z_0) \frac{e^{-r(\cos(\alpha)x(\alpha_b) + \sin(\alpha)y(\alpha_b))}}{r^{3/2} \sin(\alpha_b - \alpha)^{3/2}} $$

where $C_{br} = |\sin(\alpha_b)|^{3/2}C_0$ (see (1.24)), and where $f_0(z_0)$ is defined in (1.26). If $b_0 > 0$, then

$$ (4.60) \qquad \frac{1}{2i\pi} \int_{\Omega_{-,x_b+\delta} \cup \Omega_{+,x_b+\delta}} \varphi^{z_0}(x) \frac{\gamma(x, Y^+(x), Z^+(x))}{\partial_y \gamma_+(x, Y^+(x))} e^{-r(\cos(\alpha)x + \sin(\alpha)Y^+(x))} dx \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad C_{br} f_0(z_0) \frac{e^{-r(\cos(\alpha)x(\alpha_b) + \sin(\alpha)y(\alpha_b))}}{r^{3/2} \sin(\alpha_b - \alpha)^{3/2}}. $$

*Proof.* First, assume that $b_0 \le 0$. Once again, by the convention on the complex square root,

$$ \frac{1}{2i\pi} \int_{\Omega_{-,x_b+\delta} \cup \Omega_{+,x_b+\delta}} \Phi(x) e^{-ax-bY^+(x)} dx = \frac{2}{2i\pi} \int_{\Omega_{+,x_b+\delta}} \sqrt{x_b-x} \Phi_2(x) e^{-r(\cos(\alpha)x + \sin(\alpha)Y^+(x))} dx \\ = \frac{e^{-r(\cos(\alpha)x(\alpha_b) + \sin(\alpha)y(\alpha_b))}}{i\pi} \int_{\Omega_{+,x_b+\delta}} \sqrt{x_b-x} \Phi_2(x) e^{-rU(x,\alpha)} dx $$

where

$$ (4.61) \qquad U(x, \alpha) = \cos(\alpha)(x - x(\alpha_b)) + \sin(\alpha)(Y^+(x) - y(\alpha_b)). $$

By the definition of $(x(\alpha_b), y(\alpha_b))$, we have $(Y^+)'(x_b) = -\frac{\cos(\alpha_b)}{\sin(\alpha_b)}$. Therefore,

$$ U'_{x}(x_b, \alpha_0) = \cos(\alpha_0) - \sin(\alpha_0) \frac{\cos(\alpha_b)}{\sin(\alpha_b)} = \frac{\sin(\alpha_b - \alpha_0)}{\sin(\alpha_b)} > 0. $$

Moreover, $U(x_b, \alpha_0) = 0$. Then, by the implicit function theorem, there exists a neighborhood $V \subset \mathbb{R} \times [0, 2\pi]$ of $(0, \alpha_0)$ given by

$$ (4.62) \qquad V = \{(t, \alpha), \quad |t| \le L, |\alpha - \alpha_0| \le \eta\} $$

and a map $z: V \to \mathbb{R}$ such that, for all $(t, \alpha) \in V$,

$$ (4.63) \qquad U(x, \alpha) = t \iff x = z(t, \alpha) $$

with $x_b = z(0, \alpha_0)$. This means that $U(z(t, \alpha), \alpha) = t$. In particular, we obtain $U(z(0, \alpha), \alpha) = 0$. Moreover, by the definition of $U(x, \alpha)$, we have $U(x_b, \alpha) = 0$. By the uniqueness stated in (4.63), we then deduce that $z(0, \alpha) = x_b$ for every $\alpha$ satisfying $|\alpha - \alpha_0| \le \eta$. Furthermore,

$$ (4.64) \qquad z'_{t}(0, \alpha_{0}) = \frac{1}{U'_{x}(x_{b}, \alpha_{0})} = \frac{\sin(\alpha_{b})}{\sin(\alpha_{b} - \alpha_{0})} > 0. $$

Again by the implicit function theorem, if $\delta > 0$ is sufficiently small, there exists a smooth function $t(\alpha)$ (possibly requiring a smaller $\eta$ in (4.62)), defined for $|\alpha - \alpha_0| \le \eta$, such that

$$ z(t(\alpha), \alpha) = x_b + \delta \quad \forall |\alpha - \alpha_0| \le \eta. $$

By choosing $L > 0$ and $\eta > 0$ sufficiently small in (4.62), we can assume that $z'_t(t, \alpha) > 0$ for all $(t, \alpha) \in V$. Since $z(0, \alpha) = x_b$, it follows that $t(\alpha) > 0$. In particular, $t(\alpha) \to t(\alpha_0) > 0$ as $\alpha \to \alpha_0$. We now parametrize the segment $\Omega_{+,x_b+\delta}$ by $\{z(t, \alpha) | t \in [0, t(\alpha)]\}$. Then,

$$ \int_{\Omega_{+,x_b+\delta}} \sqrt{x_b-x} \Phi_2(x) e^{-rU(x,\alpha)} dx = -i \int_0^{t(\alpha)} \sqrt{z(t,\alpha) - x_b} \Phi_2(z(t,\alpha)) z'_t(t,\alpha) e^{-rt} dt. $$

Then, by the Taylor-Lagrange inequality,

$$ |z(t, \alpha) - z(0, \alpha) - tz'_{t}(0, \alpha)| \le C_{1}t^{2}, \quad \forall (t, \alpha) \in V $$
where $C_1 = \frac{1}{2} \sup_{(t,\alpha) \in V} |z_t''(0, \alpha)|$. Since $|\sqrt{a} - 1| = \frac{|a-1|}{\sqrt{a+1}} \le |a-1|$ for $a \ge 0$, we obtain

$$ \left| \sqrt{\frac{z(t, \alpha) - z(0, \alpha)}{tz_t'(0, \alpha)}} - 1 \right| \le C_2 |t| $$

for some constant $C_2 > 0$ (since $V$ can be chosen so that $1/z_t'(0, \alpha)$ is bounded on $V$). Hence,

$$ \left| \sqrt{z(t, \alpha) - z(0, \alpha)} - \sqrt{tz_t'(0, \alpha)} \right| \le C_3 |t|^{3/2} \quad \forall (t, \alpha), 0 \le t \le L, \forall |\alpha - \alpha_0| \le \eta $$

for some constant $C_3 > 0$. Similarly, there exists a constant $C_4 > 0$ such that

$$ |\Phi_2(z(t, \alpha))z_t'(t, \alpha) - \Phi_2(x_b)z_t'(0, \alpha)| \le C_4|t| \quad \forall (t, \alpha), 0 \le t \le L, \forall |\alpha - \alpha_0| \le \eta. $$

Thus, there exists $C > 0$ such that

$$ \left| \sqrt{z(t, \alpha) - z(0, \alpha)} \Phi_2(z(t, \alpha)) z_t'(t, \alpha) - \Phi_2(x_b) z_t'(0, \alpha) \sqrt{tz_t'(0, \alpha)} \right| \le C |t|^{3/2} \quad \forall (t, \alpha), 0 \le t \le K, \forall |\alpha - \alpha_0| \le \eta. $$

It follows that

$$ -i \int_{0}^{t(\alpha)} \sqrt{z(t, \alpha) - x_b} \Phi_2(z(t, \alpha)) z_t'(t, \alpha) e^{-rt} dt = -i \Phi_2(x_b) |z_t'(0, \alpha)|^{3/2} \int_{0}^{t(\alpha)} \sqrt{t} e^{-rt} dt + R(r, \alpha) $$

where

$$ |R(r, \alpha)| \le C \int_{0}^{t(\alpha)} t^{3/2} e^{-rt} dt = r^{-5/2} \int_{0}^{rt(\alpha)} t^{3/2} e^{-t} dt = O(r^{-5/2}) $$

as $r \to +\infty$ and $\alpha \to \alpha_0$. Furthermore,

$$ \int_{0}^{t(\alpha)} \sqrt{t} e^{-rt} dt \underset{\alpha \to \alpha_0}{\sim} r^{-3/2} \int_{0}^{+\infty} \sqrt{t} e^{-t} dt = \frac{\sqrt{\pi}}{2r^{3/2}}. $$

Hence,

$$ \begin{array}{l} (4.65) \qquad \displaystyle \frac{1}{2i\pi} \int_{\Omega_{-}, x_b + \delta \cup \Omega_{+}, x_b + \delta} \Phi(x) e^{-r(\cos(\alpha)x + \sin(\alpha)Y^+(x))} dx \underset{\substack{r \to \infty \\ \alpha \to \alpha_0}}{\sim} \frac{-ie^{-r(\cos(\alpha)x(\alpha_b) + \sin(\alpha)y(\alpha_b))}}{i\pi} \Phi_2(x_b) |z_t'(0, \alpha_0)|^{3/2} \frac{\sqrt{\pi}}{2r^{3/2}} \\ (4.66) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \underset{\substack{r \to \infty \\ \alpha \to \alpha_0}}{\sim} C_{br} f_0(z_0) \frac{e^{-r(\cos(\alpha)x(\alpha_b) + \sin(\alpha)y(\alpha_b))}}{r^{3/2} \sin(\alpha_b - \alpha_0)^{3/2}}. \end{array} $$

by (4.64) and Lemma 4.10.

If $b_0 > 0$, the same argument applies, replacing $\Phi(x)$ with $\varphi^{z_0}(x) \frac{\gamma(x, Y^+(x), Z^+(x))}{\partial_y \gamma_+(x, Y^+(x))}$ which yields the result in (4.60). $\square$

To show that the asymptotics of the previous integrals yield the leading term in the asymptotics of the Green's function, we establish the following lemma.

**Lemma 4.14 (Negligibility of remaining integrals).** Suppose $x_{max}^+ < x_{max}^- (\text{i.e., } \alpha_b \in (0, \pi))$. Let $0 \le \alpha_0 < \alpha_b$. Then, for $\delta > 0$ small enough, there exist constants $D > 0$, $\epsilon' > 0$, $\eta > 0$, and $r_0 > 0$ such that for all $r \ge r_0$ and all $\alpha \ge 0$ satisfying $|\alpha - \alpha_0| < \eta$, the following estimates hold:

$$ (4.67) \qquad \left| \int_{R_{\delta}^{\pm}} \varphi^{z_0}(x) \frac{\gamma(x, Y^{+}(x), Z^{+}(x))}{\partial_y \gamma_{+}(x, Y^{+}(x))} e^{-r(\cos(\alpha)x + \sin(\alpha)Y^{+}(x))} dx \right| \le D e^{-r(\cos(\alpha)x(\alpha_b) + \sin(\alpha)y(\alpha_b) + \epsilon')} $$

and:

$$ \bullet \text{ If } b_0 > 0, \text{ then} \qquad (4.68) \qquad I_2(r \cos(\alpha), r \sin(\alpha)) \le D e^{-r(\cos(\alpha)x(\alpha_b) + \sin(\alpha)y(\alpha_b) + \epsilon')} $$

$$ \bullet \text{ If } b_0 \le 0, \text{ then} \qquad (4.69) \qquad \left| \int_{R_{\delta}^{\pm}} \frac{e^{(a_0-a)x+b_0Z^{+}(x)-bY^{+}(x)}}{\partial_y \gamma_{+}(x, Y^{+}(x))} e^{-r(\cos(\alpha)x + \sin(\alpha)Y^{+}(x))} dx \right| \le D e^{-r(\cos(\alpha)x(\alpha_b) + \sin(\alpha)y(\alpha_b) + \epsilon')} $$
**Proof.** We begin by proving (4.67). The argument is analogous to that used in the proof of Lemma 4.6. Suppose first that $\alpha_0 > 0$. Then (4.67) can be rewritten as (4.70)

$$ \left| \int_{0}^{+\infty} \varphi^{z_0}(x_b+\delta+iv) \frac{\gamma(x_b+\delta+iv, Y^{+}(x_b+\delta+iv), Z^{+}(x_b+\delta+iv))}{\partial_y \gamma_{+}(x_b+\delta+iv, Y^{+}(x_b+\delta+iv))} e^{-r(F(x_b,\alpha)-F(x_b+\delta+iv)-\epsilon')} dx \right| \le D. $$

We decompose the exponent as follows:

$$ F(x_b, \alpha) - F(x_b + \delta + iv) - \epsilon' = (F(x_b, \alpha) - F(x_b + \delta, \alpha) - \epsilon') + (F(x_b + \delta, \alpha) - F(x_b + \delta + iv)). $$

Now, observe that the function $x \mapsto F(x, \alpha_0)$ is strictly decreasing on $[-\pi, x(\alpha_0)]$ and strictly increasing on $[x(\alpha_0), x_{max}]$. Since $\alpha_0 < \alpha_b$, it follows that $x(\alpha_0) > x_b$. Therefore, for some $\delta > 0$ and $\epsilon' > 0$ small enough, there exists $\eta > 0$ such that $F(x_b, \alpha) - F(x_b + \delta, \alpha) - \epsilon' \ge 0$ for all $\alpha \ge 0$ such that $|\alpha - \alpha_0| < \eta$. Moreover, using the same reasoning as in the proof of Lemma 4.6, we find that there exists $c > 0$ such that for all $v \ge 0$,

$$ (4.71) \qquad F(x_b + \delta, \alpha) - \Re F(x_b + \delta + iv) \ge c \sin(\alpha)v. $$

Then, by applying (4.37), the left-hand side of (4.70) is bounded by $\frac{C}{r \sin(\alpha)} \le D$, for some $D > 0$ independent of $r$ and $\alpha \ge \alpha_0 - \eta > 0$ (for $\eta > 0$ sufficiently small). This proves (4.70) when $\alpha_0 > 0$.

If $\alpha_0 = 0$, (4.67) can be obtained similarly by integration by parts, see [14, Lemma 7.3] for similar considerations. The proof of (4.69) is symmetrical.

We now turn to the proof of (4.68). Suppose $b_0 \ge 0$ and consider the representation of $I_2^{z_0}(a, b)$ given by (4.18). Applying the saddle-point method to (4.18) (see Lemmas 4.4, 4.5, and 4.6), we obtain

$$ I_2^{z_0}(r \cos(\alpha), r \sin(\alpha)) = O\left( \frac{e^{-r(\cos(\alpha)x(\alpha)+\sin(\alpha)y(\alpha))}}{\sqrt{r}} \right) = o\left( e^{-r(\cos(\alpha)x(\alpha_b)+\sin(\alpha)y(\alpha_b)+\epsilon')} \right) $$

as $r \to +\infty$ and $\alpha \to \alpha_0$, for $\epsilon' > 0$ small enough by the definition of $(x(\alpha), y(\alpha))$. This proves (4.68) for some constant $D > 0$. $\square$

**Proof of Theorem 6.** If $b_0 \le 0$, we consider the representations of $I_1^{z_0}(a, b)$ and $I_2^{z_0}(a, b)$ given by (4.57) and (4.58), respectively, in Lemma 4.12. The contributions of these integrals along $\Omega_{\pm, x_b+\delta}$ are exactly given by (4.59) from Lemma 4.13. By Lemma 4.14, and more precisely by (4.67) and (4.69), the asymptotic contributions of the integrals along $R_{\delta}^{\pm}$ in the representations (4.57) and (4.58) are negligible compared to those along $\Omega_{\pm, x_b+\delta}$. It follows that the asymptotics of $g^{z_0}(r \cos(\alpha), r \sin(\alpha))$ are given by (4.59), as claimed.

If $b_0 > 0$, we consider the representation of $I_1^{z_0}(a, b)$ given by (4.57), and the inequality for $I_2^{z_0}(a, b)$ provided in (4.68). The contribution of the integral (4.57) along $\Omega_{\pm, x_b+\delta}$ is given by (4.60). By (4.67), the integrals along $R_{\delta}^{\pm}$ from the representation (4.57) are asymptotically negligible compared to (4.60). Moreover, by (4.68), the asymptotic contribution of $I_2^{z_0}(r \cos(\alpha), r \sin(\alpha))$ is also negligible compared to (4.60). Therefore, the asymptotics of $g^{z_0}(r \cos(\alpha), r \sin(\alpha))$ are given by (4.60), which completes the proof. $\square$

## 5. PROOF OF THEOREM 7: FULL AND MINIMAL MARTIN BOUNDARY

In this section, we prove the result concerning the Martin boundary using Theorems 2 to 6. In the general theory of Martin boundaries, Martin functions are usually excessive (see [20, Theorem 3]). Nevertheless, for Markov chains with a finite number of steps from each state, one can immediately verify that Martin functions are harmonic. They are also harmonic for (drifted) Brownian motion in $\mathbb{R}^d$. In the next lemmas, we show that the functions $f_0$, $f_\pi$, and $h_\alpha$, $\alpha \in \mathcal{M} \setminus \{0, \pi\}$, in our model are harmonic as well. Due to the fundamental role of $z_0$ in this section, we write $g(z_0, z)$ for $g^{z_0}(z)$.

**Lemma 5.1 (Harmonicity).** The functions $f_0$, $f_\pi$, and $h_\alpha$, $\alpha \in \mathcal{M} \setminus \{0, \pi\}$, are harmonic.

**Proof.** Let $U$ be an open set that is relatively compact in $\mathbb{R}^2$. By [20, Proposition 6.2], if $z \in \mathbb{R}^2$, then $g(\cdot, z)$ is harmonic on $\mathbb{R}^2 \setminus \{z\}$. Hence the Martin kernel $K(\cdot, z)$ defined by $K(z_0, z) = \frac{g(z_0, z)}{g(0, z)}$ is harmonic as well on $\mathbb{R}^2 \setminus \{z\}$. Therefore, if $z \notin U$ and $z_0 \in \mathbb{R}^2$ satisfy $z \ne z_0$, then

$$ (5.1) \qquad K(z_0, z) = \mathbb{E}_{z_0}[K(Z_{\tau_{Uc}}, z)]. $$
Let $\alpha_0 \in \mathcal{M}$. We consider $z = (r \cos(\alpha), r \sin(\alpha))$ with $\alpha \in \mathcal{M}$, and let $r \to \infty$ and $\alpha \to \alpha_0$. By Theorems 3 to 5, we have

$$K(z_0, (r \cos(\alpha), r \sin(\alpha))) = \frac{h_{\alpha_0}(z_0) + \varepsilon(z_0, r, \alpha)}{h_{\alpha_0}(0) + \varepsilon(0, r, \alpha)}$$

where $\varepsilon(z_0, r, \alpha) \xrightarrow[r \to +\infty, \alpha \to \alpha_0]{} 0$ (and where, for consistency of notation, we have set $h_0(z_0) = f_0(z_0)$ and $h_\pi(z_0) = f_\pi(z_0)$). If $\varepsilon(z_0, r, \alpha)$ converges to zero as $r \to +\infty$, $\alpha \to \alpha_0$ uniformly in $z_0 \in U$, the dominated convergence theorem applied to (5.1) yields

$$h_{\alpha_0}(z_0) = \mathbb{E}_{z_0}[h_{\alpha_0}(Z_{\tau_U \varepsilon})]$$

and completes the proof. The uniform convergence in $z_0 \in U$ can be established using arguments similar to those in [33, Lemmas 5.3 and Section 6]. The only directions specific to our model to which these arguments do not apply directly are $\alpha_b$ and $\tilde{\alpha}_b$ corresponding to branching points, cf. (1.16), (1.21), and (1.22). It remains to establish uniform convergence for these directions. Let us consider case (i) of Theorem 5. We prove that the equivalence (4.45) is uniform in $z_0 \in U$ and that the constant $M$ in (4.46) can be chosen independently of $z_0 \in U$. Recall the definitions (4.43) of $\Phi_1$ and $\Phi_2$, which we denote by $\Phi_1^{z_0}$ and $\Phi_2^{z_0}$. For (4.45), it suffices (see the proof of [14, Lemma 8.1]) to show that

$$(5.2) \qquad \sup_{\substack{z_0 \in U \\ x \in K}} |\Phi_1^{z_0}(x)| < +\infty$$

for some compact neighborhood $K \subset \mathbb{C}$ of $x_b$. For (4.46), it suffices to prove the analogue of (5.2) with $\Phi_2^{z_0}$ instead of $\Phi_1^{z_0}$. To do this, we use the explicit expression (4.41) of $\Phi_2^{z_0}(x)$. Note that $\Phi_2^{z_0}(x)$ can be written as $(\Psi_1(x, z_0) + \Psi_2(x, z_0)\sqrt{x_b - x})e^{f(x)\sqrt{x_b - x}b_01_{b_0<0}}$ where $\Psi_i(x, z_0)$, $i = 1, 2$ are locally bounded in $(x, z_0)$ and both $\Psi_i(x, z_0)$, $i = 1, 2$ and $f(x)$ are holomorphic in $x$ in a neighborhood of $x_b$. Expanding the exponential into a Taylor series gives

$$e^{f(x)\sqrt{x_b - x}b_01_{b_0<0}} = \underbrace{\sum_{k=0}^{+\infty} \frac{f(x)^{2k}}{(2k)!} (x_b - x)^k b_0^{2k} 1_{b_0<0}}_{=\Xi_1(x, b_0)} + \underbrace{\sqrt{x_b - x} \sum_{k=0}^{+\infty} \frac{f(x)^{2k+1}}{(2k+1)!} (x_b - x)^k b_0^{2k+1} 1_{b_0<0}}_{=\Xi_2(x, b_0)}$$

A straightforward argument shows that $\Xi_1(x, b_0)$ and $\Xi_2(x, b_0)$ are bounded when $x$ lies in a sufficiently small neighborhood of $x_b$ and $b_0$ ranges over a compact set. Since $\Phi_1(x, z_0) = \Psi_1(x, z_0)\Xi_1(x, b_0) + (x_b - x)\Psi_2(x, z_0)\Xi_2(x, b_0)$ and $\Phi_2(x, z_0) = \Psi_1(x, z_0)\Xi_2(x, b_0) + \Psi_2(x, z_0)\Xi_1(x, b_0)$, this yields the desired uniform bounds. $\square$

To prove Theorem 7, we state the following technical lemma.

**Lemma 5.2** (Asymptotic inequalities for $h_\alpha, f_0, f_\pi$). Let $\alpha, \beta \in \mathcal{M}$ with $\alpha \neq \beta$. Then:

* Suppose that $\beta \in \mathcal{M} \cap (0, \pi)$. Then there exist $\epsilon > 0$, $\eta > 0$, $r_0 > 0$, and $C > 0$ such that for all $\gamma \in \mathcal{M} \cap (\beta - \epsilon, \beta + \epsilon)$ and all $r \ge r_0$,

$$(5.3) \qquad h_\gamma(r \cos(\beta), r \sin(\beta)) \ge C^{-1} e^{r(x_\beta) \cos(\alpha) + y_\beta \sin(\beta) - \eta} \\ \qquad \text{and such that for all } r \ge r_0,$$

$$(5.4) \qquad h_\alpha(r \cos(\beta), r \sin(\beta)) \le C e^{r(x_\beta) \cos(\alpha) + y_\beta \sin(\beta) - 2\eta}.$$

* Suppose that $\beta \in \mathcal{M} \cap (\pi, 2\pi)$. Then there exist $\epsilon > 0$, $\eta > 0$, $r_0 > 0$, and $C > 0$ such that for all $\gamma \in \mathcal{M} \cap (\beta - \epsilon, \beta + \epsilon)$ and all $r \ge r_0$,

$$(5.5) \qquad h_\gamma(r \cos(\beta), r \sin(\beta)) \ge C^{-1} e^{r(x_\beta) \cos(\alpha) + z_\beta \sin(\beta) - \eta} \\ \qquad \text{and such that for all } r \ge r_0$$

$$(5.6) \qquad h_\alpha(r \cos(\beta), r \sin(\beta)) \le C e^{r(x_\beta) \cos(\alpha) + z_\beta \sin(\beta) - 2\eta}.$$

Inequalities (5.4) and (5.6) also hold if $\alpha$ is replaced by 0 (resp. $\pi$) and $h_\alpha$ by $f_0$ (resp. $f_\pi$).

*Proof.* This is a direct consequence of expressions (1.26) and (1.30). $\square$
Proof of Theorem 7. By Theorems 2 to 6 and Lemma 5.1, it remains to show that functions $h_\alpha$ and $f_0$, $f_\pi$ are minimal. The main idea of the proof is that $h_\beta(r \cos(\beta), r \sin(\beta))$ is asymptotically of greater order than $h_\alpha(r \cos(\beta), r \sin(\beta))$ as $r \to +\infty$ for any fixed $\beta$ such that $\beta \neq \alpha$.

Let $\alpha \in \mathcal{M} \setminus \{0, \pi\}$. We now rigorously prove that $h_\alpha$ is minimal. To do so, we show that if $c_0, c_\pi \ge 0$ and if $\mu$ is a Borel measure on $\mathcal{M} \setminus \{0, \pi\}$ such that

$$ (5.7) \qquad h_{\alpha} = \int_{\mathcal{M} \setminus \{0, \pi\}} h_{\alpha} d\mu(\alpha) + c_0 f_0 + c_{\pi} f_{\pi}, $$

then $c_0 = c_\pi = 0$ and $\mu = \delta_\alpha$ (we excluded 0 and $\pi$ in the integral term to avoid problems of notation). Let $\beta \in \mathcal{M} \cap (0, \pi)$ with $\alpha \neq \beta$, and let $\epsilon, \eta, C, r_0 > 0$ be the constants provided by Lemma 5.2. Then, using inequalities (5.3), (5.4), and the representation (5.7), for all $r \ge r_0$ we have

$$ \begin{aligned} Ce^{r(x(\beta) \cos(\alpha) + y(\beta) \sin(\beta) - 2\eta)} &\ge h_{\alpha}(r \cos(\beta), r \sin(\beta)) \\ &\ge \int_{(\beta - \epsilon, \beta + \epsilon) \cap \mathcal{M}} h_{\beta}(r \cos(\gamma), r \sin(\gamma)) d\mu(\gamma) \\ &\ge C^{-1} \mu((\beta - \epsilon, \beta + \epsilon) \cap \mathcal{M}) e^{r(x(\beta) \cos(\alpha) + y(\beta) \sin(\beta) - \eta)}. \end{aligned} $$

Hence, for $r \ge r_0$,

$$ C^2 e^{-\eta r} \ge \mu((\beta - \epsilon, \beta + \epsilon) \cap \mathcal{M}). $$

Taking the limit as $r \to +\infty$, we obtain $\mu((\beta - \epsilon, \beta + \epsilon) \cap \mathcal{M}) = 0$. By a standard property of Borel measures, it follows that $\mu(\mathcal{M} \cap (0, \pi) \setminus \{\alpha\}) = 0$. Using symmetric arguments along with inequalities (5.5) and (5.6), we similarly get $\mu(\mathcal{M} \cap (\pi, 2\pi) \setminus \{\alpha\}) = 0$. Hence, $\mu = c\delta_\alpha$, where $\delta_\alpha$ denotes the Dirac measure at $\alpha$. By (5.7), we get $h_\alpha = ch_\alpha + c_0f_0 + c_\pi f_\pi$. Since $h_\alpha, f_0$, and $f_\pi$ are clearly linearly independent, it follows that $c_0 = c_\pi = 0$ and $c = 1$. The proofs of the minimality of $f_0$ and $f_\pi$ are analogous. $\square$

Let us now prove Corollary 1.7.

*Proof of Corollary 1.7.* By standard arguments, the mapping

$$ h : (a_0, b_0) \mapsto \mathbb{P}_{(a_0, b_0)}(B_t \xrightarrow{t \to +\infty} +\infty) $$

is harmonic. Moreover, note that $h_{\alpha_{\mu+}}$ and $h_{\alpha_{\mu-}}$ are the only bounded harmonic functions among $f_0$, $f_\pi$, and $h_\alpha$ for $\alpha \in \mathcal{M} \setminus \{0, \pi\}$ (see (1.25), (1.26), (1.30), and (1.31)). Hence, applying the representation (1.44) to the bounded harmonic function $h$ yields $h = c_+h_{\alpha_{\mu+}} + c_-h_{\alpha_{\mu-}}$ for some nonnegative constants $c_+$ and $c_-$. By properties of standard Brownian motion (denoted here by $W$), we have

$$ h(a_0, b_0) \ge \mathbb{P} \left[ \left\{ b_0 + \sqrt{\Sigma_{22}} W_t + \mu_2^+ t \xrightarrow{t \to \infty} +\infty \right\} \cap \left\{ b_0 + \sqrt{\Sigma_{22}} W_t + \mu_+ t > 0 \quad \forall t \ge 0 \right\} \right] \xrightarrow{b_0 \to +\infty} 1 $$

by Hypothesis (1.7) on the drift. Since $h_{\alpha_{\mu-}}(a_0, b_0) \xrightarrow{b_0 \to +\infty} 0$ and $h_{\alpha_{\mu+}}(a_0, b_0) \xrightarrow{b_0 \to +\infty} 1$ (see (1.30) and (1.31)), we obtain $c_+ = 1$. Considering now the limit as $b_0 \to -\infty$, we deduce that $c_- = 0$, and the desired result follows. $\square$

# 6. GENERALIZATION TO SKEW DIFFUSIONS ACTING IN ANY OBLIQUE DIRECTION q

Up to this point, we have imposed the condition $q = q_0$ (see (1.5)) to have a divergence form generator. The goal of this section is to study the extension of our results for $q \in \mathbb{R} \times (-1, 1)$.

In Section 6.1, we state hypotheses 1 and 2 on $q$ under which this analysis can be fruitful. Section 6.2 describes new properties satisfied by the Laplace transforms. In particular, the Laplace transform $\varphi^{z_0}$ may now have a pole. In Section 6.3, we introduce a new angular region (arising from the pole of $\varphi^{z_0}$) and generalise Theorems 3–6 to this broader range of parameters $q$. Sections 6.4 and 6.5 are devoted to the study of the asymptotic behaviour of the Green’s functions $g^{z_0}$ within this new region. Finally, in Section 6.6, we identify the Martin boundary and the minimal Martin boundary for the generalised processes.
Let $\Sigma^+, \Sigma^-$ be covariance matrices, and let $\mu^+, \mu^-$ be $\mathbb{R}^2$ satisfy (1.7). Let $q \in \mathbb{R} \times (-1, 1)$. Using the same method as in Section 2.1, one can show that for every $(a_0, b_0) \in \mathbb{R}^2$, there exists a unique pathwise solution $Z = (A, B)$ to equation (1.3), driven by a Brownian motion $(W_t)_{t \ge 0}$.

**6.1. Hypotheses.** We now state two technical hypotheses on the parameter $q \in \mathbb{R} \times (-1, 1)$ under which our extended results hold.

**Hypothesis 1 (Markov property and density).** The process $Z$ defines a strong Markov process. Moreover, for any $z_0 \in \mathbb{R}^2$ and any $t > 0$, the law of $Z_t$ starting from $z_0$ admits a density $p_t^{z_0}(z)$ with respect to the Lebesgue measure on $\mathbb{R}^2$. Finally, the function $z \mapsto p_t^{z_0}(z)$ has limits at the top and bottom of the axis $\{y=0\}$.

**Hypothesis 2 (Aronson-like inequality).** There exists a constant $M > 0$ such that for all $t > 0$ and all $z_0 \in \mathbb{R}^2$,

$$ (6.1) \qquad p_t^{z_0}(z) \le \frac{M}{t} e^{-|z-z_0|^2/Mt+Mt} $$

Assuming Hypothesis 1, we may define the Green's functions by $g^{z_0}(z) = \int_0^{+\infty} p_t^{z_0}(z) dt$ for $z = (x, y) \in \{y \ne 0\}$ and $z_0 \in \mathbb{R}^2$. Under Hypothesis 2, inequality (2.4) holds, and the functions $g^{z_0}$ satisfy inequalities (2.3).

**6.2. Analogous results for Laplace transforms.** In this section, we investigate properties of the Laplace transforms for a general parameter $q$. The functional equation (1.11) and the continuation formula for $\varphi^{z_0}$ given in (1.17) remain valid and can be proved using the same arguments.

**Proposition 6.1 (Functional equation and continuation of $\varphi^{z_0}$).** There exists $\eta > 0$ such that for all $x \in (-\eta, 0)$, $y < 0$, and $z > 0$, the functions $\varphi_-^{z_0}(x, z)$, $\varphi_+^{z_0}(x, y)$, and $\varphi^{z_0}(x)$ are finite and satisfy (1.13). Furthermore, $\varphi^{z_0}$ admits a meromorphic continuation on $\mathbb{C} \setminus ((-\infty, \tilde{x}_b] \cup [x_b, +\infty))$ given by (1.17).

The main difference with the case $q = q_0$ is that $\varphi^{z_0}$ may now have a pole.

**Proposition 6.2 (Poles of $\varphi^{z_0}$ on $(\tilde{x}_b, x_b)$).** On $[\tilde{x}_b, x_b]$, the equation (E) : $\gamma(x, Y^{-}(x), Z^{+}(x)) = 0$ has at most one solution, denoted by $x^*$. If this equation has no solution, we adopt the convention $x^* = +\infty$. Then, on $(\tilde{x}_b, x_b)$, the extended function $\varphi^{z_0}$ has a pole if and only if (E) has a solution, and the pole is located at $x^*$ whenever it exists. In that case, $(x^*, Y^{-}(x^*), Z^{+}(x^*))$ solves the system

$$ (6.2) \qquad \begin{cases} \gamma(x, y, z) = 0 \\ \gamma_+(x, y) = 0 \\ \gamma_-(x, z) = 0 \end{cases} $$

and

$$ \mathrm{res}_{x=x^*} \varphi^{z_0}(x) = - \frac{e^{x^* a_0 + Y^{-}(x^*) b_0 \mathbb{1}_{b_0>0} + Z^{+}(x^*) b_0 \mathbb{1}_{b_0<0}}}{\frac{d}{dx} \gamma(x, Y^{-}(x), Z^{+}(x)) \Big|_{x=x^*}} $$

Furthermore, if $x^* > 0$ (resp. $x^* < 0$), then $x^*$ is the unique pole of $\varphi^{z_0}$ in the strip $\{\tilde{x}_b - \epsilon < \Re(x) < x^* + \epsilon\}$ (resp. $\{x^* - \epsilon < \Re(x) < x_b + \epsilon\}$) for sufficiently small $\epsilon > 0$.

**Proof.** From (1.17), it is clear that $x$ is a pole of $\varphi^{z_0}$ if and only if $(x, Y^{-}(x), Z^{+}(x))$ satisfies (6.2). Note that the function $\gamma(x, Y^{-}(x), Z^{+}(x))$ is convex, being the sum of convex functions. Moreover, its explicit expression shows that at least one of the inequalities $\gamma(x, Y^{-}(x), Z^{+}(x))|_{x=\tilde{x}_b} < 0$ or $\gamma(x, Y^{-}(x), Z^{+}(x))|_{x=x_b} < 0$ must hold. Hence, $\gamma(x, Y^{-}(x), Z^{+}(x))$ has at most one zero on $(\tilde{x}_b, x_b)$, and $\varphi^{z_0}$ has at most one pole in this interval. The last statement of the lemma follows from a straightforward analysis of $\gamma(x, Y^{-}(x), Z^{+}(x))$. $\square$

Note that $x^* \ne 0$ since $\gamma(0, Y^{-}(0), Z^{+}(0)) = (\frac{1+q_2}{2}) Y^{-}(0) + (\frac{q_2-1}{2}) Z^{+}(0) < 0$.
Remark 6.3 (Values of Green’s functions at the boundary). In the case $q = q_0$, the Green’s function is continuous on the axis, meaning that the upper and lower limits $g^{z_0}(u, 0^+)$ and $g^{z_0}(u, 0^-)$ are equal. When $q \neq q_0$, this is no longer the case. Using the initial value theorem for Laplace transforms together with (1.13), we obtain for all $u \in \mathbb{R}$ with $z_0 \neq (u, 0)$,

$$ (6.3) \qquad \Sigma_{22}^{+} \left( \frac{1-q_2}{2} \right) g^{z_0}(u, 0^+) = \Sigma_{22}^{-} \left( \frac{1+q_2}{2} \right) g^{z_0}(u, 0^-). $$

By the same argument as in the proof of Lemma 2.2, we also have

$$ (6.4) \qquad \varphi^{z_0}(x) = \int_{\mathbb{R}} \frac{\Sigma_{22}^{-} g^{z_0}(u, 0^{-}) + \Sigma_{22}^{+} g^{z_0}(u, 0^{+})}{2} e^{-xu} du $$

which can be interpreted as the Laplace transform of the Green's functions on the axis. In fact, if we adopt the convention $g^{z_0}(u, 0) = \frac{\Sigma_{22}^{z_0}(u, 0^{-}) + \Sigma_{22}^{z_0}(u, 0^{+})}{\Sigma_{22}^{-} + \Sigma_{22}^{+}}$, then $\varphi^{z_0}$ is the Laplace transform of $\frac{\Sigma_{22}^{-} + \Sigma_{22}^{+}}{2} g^{z_0}(u, 0)$ exactly as in the case $q = q_0$. Theorem 2 for $\alpha = \alpha_0 \in \{0, \pi\}$ could be extended to the case of any $q \in \mathbb{R} \times (-1, 1)$ in this sense.

### 6.3. Generalisation of Theorems 3 to 6 and introduction of a new region for angles.

6.3.1. Case without pole. If $x^* = +\infty$, then all the arguments remain unchanged, and Theorems 3 to 6 hold for general $q \in \mathbb{R} \times (-1, 1)$.

6.3.2. Notation if $\varphi^{z_0}$ has a pole. Now suppose that $\varphi^{z_0}$ has a pole $x^*$, and let us introduce a new region of angles for the asymptotic analysis. For simplicity, we assume that $x^* \neq \tilde{x}_b$ and $x^* \neq x_b$.

We recall the notation $\alpha_{\mu^+} = \arctan(\mu_2^+ / \mu_1^+) \in (0, \pi)$ and $\alpha_{\mu^-} = \arctan(\mu_2^- / \mu_1^-) + 2\pi \in (3\pi/2, 2\pi)$. Note that $x(\alpha) > 0$ for $\alpha \in (0, \alpha_{\mu^+}) \cup (\alpha_{\mu^-}, 2\pi)$, and that $x(\alpha) < 0$ for $\alpha \in (\alpha_{\mu^+}, \pi) \cup (\pi, \alpha_{\mu^-})$ (see Section 1.4.2). Let $\alpha_-^* \in (0, \pi)$ and $\alpha_+^* \in (\pi, 2\pi)$ be the angles defined by

$$ (6.5) \qquad (x(\alpha_+^*), y(\alpha_+^*)) = (x^*, Y^+(x^*)), \quad (x(\alpha_-^*), y(\alpha_-^*)) = (x^*, Z^-(x^*)). $$

(see Figure 13). From the above remarks on the sign of $x(\alpha)$, we observe that:

* If $x^* \in (0, x_b)$, then $\alpha_+^* \in (0, \alpha_{\mu^+})$ and $\alpha_-^* \in (\alpha_{\mu^-}, 2\pi)$.

* If $x^* \in (\tilde{x}_b, 0)$, then $\alpha_+^* \in (\alpha_{\mu^+}, \pi)$ and $\alpha_-^* \in (\pi, \alpha_{\mu^-})$.

This defines a new (pink) region of angles; see Figure 14 for the case $x^* \in (0, x_b)$.

6.3.3. Generalisation of Theorems 3 to 6 when $\varphi^{z_0}$ has a pole. Outside the pink region, the same method as in the case $q = q_0$ applies. In these areas, the contour deformations do not involve the pole of $\varphi^{z_0}$, so the asymptotics in Theorems 3 to 6 remain valid in the corresponding non-pink regions (see Figure 14 for the case $x^* > 0$ and $\tilde{x}_b = x_{min}$).

6.4. Study of asymptotics (strictly) inside the dotted pink region. Without loss of generality, we focus on the case where $x^* \in (0, x_b)$. In this section, we study the asymptotics of the Green's functions $g^{z_0}$ in the directions $\alpha_0 \in (\alpha_-^*, 2\pi) \cup [0, \alpha_+^*)$.

Note that Lemmas 4.1, 4.2, and the constructions in Section 4.2 still hold, namely that $g^{z_0} = I_1^{z_0} + I_2^{z_0}$.

**Lemma 6.4** (Changing path for directions $\alpha_0 \in [0, \alpha_+^*)$). Suppose $x^* \in (0, x_b)$. Let $x^* < x_c < x_b$. Then, for $a \in \mathbb{R}$, $b > 0$, and $z_0 \in \mathbb{R}^2$ with $(a, b) \neq z_0$, we have:

$$ (6.6) \qquad I_1^{z_0}(a, b) = \operatorname{res}_{x=x^*} \varphi^{z_0}(x) \frac{\gamma(x^*, Y^+(x^*), Z^+(x^*))}{\partial_y \gamma_+(x^*, Y^+(x^*))} e^{-ax^*-bY^+(x^*)} \\ \qquad + \frac{1}{2i\pi} \int_{x_c+i\mathbb{R}} \frac{\varphi^{z_0}(x) \gamma(x, Y^+(x), Z^+(x)) e^{-ax-bY^+(x)}}{\partial_y \gamma_+(x, Y^+(x))} dx $$

where $\gamma(x^*, Y^+(x^*), Z^+(x^*)) \neq 0$. Furthermore,

$$ (6.7) \qquad I_2^{z_0}(a, b) = \frac{1}{2i\pi} \int_{x_c+i\mathbb{R}} \frac{e^{(a_0-a)x} \left( e^{(b_0-b)Y^+(x)} 1_{b_0>0} + e^{b_0Z^+(x)-bY^+(x)} 1_{b_0<0} \right)}{\partial_y \gamma_+(x, Y^+(x))} dx \quad \text{if } b > b_0 $$
<figure><img src="image_17.png" /><figcaption>FIGURE 13. Definition of $\alpha_{\pm}^*$ from the data $x^*$. In this case, $x^* \in (\tilde{x}_b, 0)$.</figcaption></figure>

and

$$ (6.8) \qquad I_2^{z_0}(a, b) = \frac{1}{2i\pi} \int_{Y^+(x_c)+i\mathbb{R}} \frac{e^{(a_0-a)X^+(y)+(b_0-b)y}}{\partial_x\gamma_+(X^+(y), y)} dy \quad \text{if } a > a_0. $$

*Proof.* The deformation of the integration path is justified using the same arguments as in Lemma 4.12. Equation $\gamma(x^*, Y^+(x^*), Z^+(x^*)) \neq 0$ follows from the fact that it is not possible to have both $\gamma(x, Y^+(x), Z^+(x)) = 0$ and $\gamma(x, Y^-(x), Z^+(x)) = 0$ simultaneously, except possibly at $x = x_{min}^+$ and $x = x_{max}^+$. $\square$

The term $I_1^{z_0}$ is decomposed into the sum of two terms, namely the residue at the pole and a remaining integral term. In the next lemma, we show that this integral term and also $I_2(a, b)$ are negligible in the asymptotics compared with the residue at the pole.

**Lemma 6.5** (Negligibility of integrals compared to the pole). Suppose $x^* \in (0, x_b)$. Let $z_0 \in \mathbb{R}^2$, $\alpha_0 \in [0, \alpha_+^*)$, and let $x_c$ be such that $x^* < x_c < \min(x(\alpha_0), x_b)$. Let $r = \sqrt{a^2 + b^2}$ and let $\alpha(a, b) \in (0, \pi)$ denote the angle satisfying $\cos(\alpha) = \frac{a}{\sqrt{a^2+b^2}}$ and $\sin(\alpha) = \frac{b}{\sqrt{a^2+b^2}}$. Then, for $\eta > 0$ small enough, there exist constants $r_0 > 0$ and $D > 0$ such that for all $(a, b)$ satisfying $r > r_0$, $\alpha(a, b) > 0$, and $|\alpha(a, b) - \alpha_0| < \eta$, we have:

$$ \left| \frac{1}{2i\pi} \int_{x_c+i\mathbb{R}} \frac{\varphi^{z_0}(x)\gamma(x, Y^+(x), Z^+(x))e^{-ax-bY^+(x)}}{\partial_y\gamma_+(x, Y^+(x))} dx \right| + |I_2(a, b)| \leq D e^{-r(\cos(\alpha)x_c+\sin(\alpha)Y^+(x_c))} $$

*Proof.* The proof is identical to that of Lemma 4.14. $\square$

We now prove the corresponding theorem for the dotted pink region (see Figure 14).
<figure><img src="image_18.png" /><figcaption>FIGURE 14. Summary of asymptotics for the extended results. In this case, $\tilde{x}_b = x_{min}^+$ and $x^* \in (0, x_b)$. We omit $\alpha_b$ for readability.</figcaption></figure>

**Theorem 8** (Asymptotics strictly inside the dotted pink region). Suppose that $x^* \in (0, x_b)$. Let $z_0 \in \mathbb{R}^2$ and $\alpha_0 \in [0, \alpha_+^*)$. Then:

$$ (6.9) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\alpha \to \alpha_0, \alpha > 0}{\sim} C_*^+ f_*(z_0) e^{-r(\cos(\alpha)x^* + \sin(\alpha)Y^+(x^*))} $$

where

$$ (6.10) \qquad f_*(a_0, b_0) = \begin{cases} e^{x^* a_0 + Y^-(x^*) b_0} & \text{if } b_0 \ge 0 \\ e^{x^* a_0 + Z^+(x^*) b_0} & \text{if } b_0 < 0. \end{cases} $$

and

$$ (6.11) \qquad C_*^+ = \frac{\gamma(x^*, Y^+(x^*), Z^+(x^*))}{\partial_y \gamma_+(x^*, Y^+(x^*)) \frac{d}{dx} \gamma(x, Y^-(x), Z^+(x)) \Big|_{x=x^*}}. $$

The symmetric result holds for angles $\alpha_0 \in (\alpha_-, 2\pi]$.

*Proof.* Choose $x^* < x_c < \min(x(\alpha_0), x_b)$ and consider the representation $g^{z_0} = I_1^{z_0} + I_2^{z_0}$ from Lemma 6.4. By the choice of $x_c$, we have $\cos(\alpha_0)x^* + \sin(\alpha_0)Y^+(x^*) < \cos(\alpha_0)x_c + \sin(\alpha_0)Y^+(x_c)$. Then, by Lemma 6.5, both the integral term in (6.6) and the integral in (6.7) are negligible compared with the residue as $\alpha \to \alpha_0$ and $r \to +\infty$. The result follows immediately. $\square$

6.5. **Asymptotics in directions $\alpha_\pm^*$.** We now consider the asymptotics in the directions $\alpha_\pm^*$ when $x^* \neq +\infty$. For simplicity, we assume that $x^* \in (0, x_b)$. This “pole-type” asymptotics has been studied in detail in [14, Section 10], and the same arguments apply here. Therefore, we state the following theorem without providing a proof. Let erf denote the function defined by $\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x \exp(-s^2) ds$.

**Theorem 9** (Asymptotics as $\alpha \to \alpha_\pm^*$). Suppose that $x^* \in (0, x_b)$. Then, the following asymptotics hold:
<figure><img src="image_19.png" /><figcaption>FIGURE 15. Type of asymptotics according to directions: saddle point type, pole or branching point.</figcaption></figure>

(i) If $\alpha > \alpha_+^*$ and $r(\alpha - \alpha_+^*)^2 \to \infty$, then

$$ (6.12) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\substack{r \to +\infty \\ \alpha \to \alpha_+^*}}{\sim} C_*' f_*(z_0) \frac{e^{-r(\cos(\alpha)x(\alpha)+\sin(\alpha)y(\alpha))}}{\sqrt{r(x(\alpha)-x^*)}} $$

where

$$ C_*' = -C^+(\alpha_+^*) \frac{\gamma(x^*, Y^+(x^*), Z^+(x^*))}{\left. \frac{d}{dx} \gamma(x, Y^-(x), Z^+(x)) \right|_{x=x^*}} > 0 $$

and where $C^+(\alpha_+^*)$ is given by (1.29).

(ii) If $\alpha > \alpha_+^*$ and $r(\alpha - \alpha_+^*)^2 \to c > 0$, then

$$ (6.13) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\substack{r \to +\infty \\ \alpha \to \alpha_+^*}}{\sim} \frac{1}{2} C_*^+ \left( 1 - \operatorname{erf}\left(\sqrt{c}A(\alpha_+^*)\right) \right) f_*(z_0) e^{-r(\cos(\alpha)x^*+\sin(\alpha)y^*)}. $$

where $C_*^+$ is given by (6.11), $A(\alpha_+^*) = \frac{-x'(\alpha_+^*)}{x'_\omega(0, \alpha_+^*)}$ and where $x'_\omega(0, \alpha_+^*)$ is defined in (4.12).

(iii) If $r(\alpha - \alpha_+^*)^2 \to 0$, then

$$ (6.14) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\substack{r \to +\infty \\ \alpha \to \alpha_+^*}}{\sim} \frac{1}{2} C_*^+ f_*(z_0) e^{-r(\cos(\alpha)x^*+\sin(\alpha)y^*)}. $$

(iv) If $\alpha < \alpha_+^*$ and $r(\alpha - \alpha_+^*)^2 \to c > 0$, then

$$ (6.15) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\substack{r \to +\infty \\ \alpha \to \alpha_+^*}}{\sim} \frac{1}{2} C_*^+ \left( 1 + \operatorname{erf}\left(\sqrt{c}A(\alpha_+^*)\right) \right) f_*(z_0) e^{-r(\cos(\alpha)x^*+\sin(\alpha)y^*)}. $$

(v) If $\alpha < \alpha_+^*$ and $r(\alpha - \alpha_+^*)^2 \to \infty$, then

$$ (6.16) \qquad g^{z_0}(r \cos(\alpha), r \sin(\alpha)) \underset{\substack{r \to +\infty \\ \alpha \to \alpha_+^*}}{\sim} C_*^+ f_*(z_0) e^{-r(\cos(\alpha)x^*+\sin(\alpha)y^*)}. $$

The symmetric asymptotics hold as $\alpha \to \alpha_-^*$.

6.6. **Martin boundary, general case.** The study of the asymptotics carried out in Sections 6.4 and 6.5, together with the results established in Sections 6.3.3 and 6.3.1, leads to the following theorem on the Martin boundary.

**Theorem 10** (Martin Boundary and harmonic functions for general $q$). If $x^* = +\infty$, then Theorem 7 applies. Assume now that $x^* \in (0, x_b)$. Then, the Martin boundary $\Gamma$ is given by

$$ (6.17) \qquad \Gamma = \left[ \{e^{i\alpha}\}_{\alpha \in [\alpha_+, \alpha_-]} \cap \mathcal{M} \cup \{ue^{i\tilde{\alpha}_b}\}_{u \in [1, 2]} \right] / \mathcal{R}_1 \sim \mathbb{S}^1 $$
where $\mathcal{R}_1$ is the equivalence relation given by $e^{i\alpha_{+}^{*}} \mathcal{R}_1 e^{i\alpha_{+}^{*}}$, $e^{i\pi} \mathcal{R}_1 2e^{i\alpha_{b}}$, and $[x\mathcal{R}_1y \iff x = y]$ for the other points (see Figure 16). Furthermore, the minimal Martin boundary, denoted by $\Gamma_{min}$, is given by

$$ (6.18) \qquad \Gamma_{min} = [\{e^{i\alpha}\}_{\alpha \in [\alpha_{+}^{*}, \alpha_{-}^{*}] \cap \mathcal{M}}] / \mathcal{R}_2 \sim [0, 1] $$

where $\mathcal{R}_2$ is the equivalence relation given by $e^{i\alpha_{-}^{*}} \mathcal{R}_2 e^{i\alpha_{+}^{*}}$ and $[x\mathcal{R}_2y \iff x = y]$ for the other points (see again Figure 16). Moreover, for each positive harmonic function $h$ (with respect to the process), there exists a unique Radon measure $\nu$ on $[\alpha_{+}^{*}, \alpha_{-}^{*}) \cap \mathcal{M}$ such that

$$ (6.19) \qquad \forall z \in \mathbb{R}^2, \quad h(z) = \int_{[\alpha_{+}^{*}, \alpha_{-}^{*}) \cap \mathcal{M}} \frac{h_{\alpha}(z)}{h_{\alpha}(0)} d\nu(\alpha) $$

where the conventions $\frac{h_{\alpha_{+}^{*}}(z_0)}{h_{\alpha_{+}^{*}}(0)} = \frac{f_{*}(z_0)}{f_{*}(0)}$ and $\frac{h_{\pi}(z_0)}{h_{\pi}(0)} = \frac{f_{\pi}(z_0)}{f_{\pi}(0)}$ are used, since

$$ \lim_{\substack{\alpha \to \alpha_{+}^{*} \\ \alpha \in \mathcal{M}}} \frac{h_{\alpha}(z_0)}{h_{\alpha}(0)} = \lim_{\substack{\alpha \to \alpha_{-}^{*} \\ \alpha \in \mathcal{M}}} \frac{h_{\alpha}(z_0)}{h_{\alpha}(0)} = \frac{f_{*}(z_0)}{f_{*}(0)} \quad \text{and} \quad \frac{h_{\alpha}(z_0)}{h_{\alpha}(0)} \underset{\alpha \to \pi}{\longrightarrow} \frac{f_{\pi}(z_0)}{f_{\pi}(0)}. $$

Finally, the case $x^* \in (\tilde{x}_b, 0)$ is symmetric.

<figure><img src="image_20.png" /><figcaption>FIGURE 16. Martin Boundary and Minimal Martin Boundary for the general process in the case $x^* \in (0, x_b)$ and $x_{min}^- < x_{min}^+$.</figcaption></figure>
## APPENDIX A. TWO-SIDED TAUBERIAN THEOREM

Tauberian theorems are known for Laplace transforms on $\mathbb{R}^+$, giving a link between the asymptotics of a function and its Laplace transform (cf [8, Theorem 37.1] or [4, Lemma C.2]). In fact, the proof can be directly adapted to two-sided Laplace transforms without any technical issue.

**Lemma A.1.** Let $f: \mathbb{R} \to \mathbb{R}^+$ be a continuous function and $\varphi(z) := \int_{\mathbb{R}} e^{xz} f(x)dx$. Suppose that there are $a < 0 < b$ such that $\varphi$ converges absolutely in $a < \Re(z) < b$. For $\delta > 0$ and $s \in \mathbb{C}$, let $G_{\delta}^{+}(s) = \{z \in \mathbb{C}, \Re(z) > 0, z \neq s, |\arg(z-s)| > \delta\}$ (where the complex argument is taken in $(-\pi, \pi]$). Suppose that there exists $0 \le \delta < \pi/2$ such that

* Function $\varphi$ has a holomorphic continuation on $G_{\delta}^{+}(b)$.

* The following asymptotics hold:

$$ \varphi(z) \underset{z \to b}{\underset{|z| \to +\infty}{\longrightarrow}} 0. $$

* There exists $\lambda \in \mathbb{R}$ and constants $c \in \mathbb{C}$, $A \neq 0$ such that

$$ (A.1) \qquad (\varphi(z) - c)(b - z)^{\lambda} \underset{z \to b}{\underset{z \in G_{\delta}^{+}(b)}{\longrightarrow}} A. $$

Then:

$$ f(x) \underset{x \to +\infty}{\sim} \frac{A}{\Gamma(\lambda)} e^{-bx} x^{\lambda-1} $$

Obviously, the symmetric lemma for the asymptotics of $f$ in $-\infty$ holds.

## APPENDIX B. AN INTEGRAL ASYMPTOTIC LEMMA

**Lemma B.1.** The following asymptotics hold as $q \to +\infty$

$$ \int_0^1 \sqrt{1 - s} e^{qs^2} ds \underset{q \to +\infty}{\sim} \frac{\sqrt{\pi}}{4\sqrt{2}} \frac{e^q}{q^{3/2}}. $$

*Proof.* It suffices to establish that, for some $\epsilon > 0$, the integral from over $[1-\epsilon, 1]$ has the same asymptotic. Since the asymptotic of the integral over $[0, 1-\epsilon]$ is bounded by $O(\exp(q(1-\epsilon)^2))$, as $q \to \infty$. Let us look at

$$ J(q, \epsilon) = \exp(-q) \int_{1-\epsilon}^{1} \sqrt{1-s} \exp(qs^2) ds $$

for some $\epsilon > 0$ small enough. After the change of variables $t = q - qs^2$, it becomes

$$ J(q, \epsilon) = \frac{1}{2q} \int_{0}^{q(1-(1-\epsilon)^2)} \frac{\sqrt{1-\sqrt{1-t/q}}}{\sqrt{1-t/q}} \exp(-t) dt. $$

For any $\epsilon > 0$ small enough there exists a constant $C > 0$ such that

$$ \left| \frac{\sqrt{1 - \sqrt{1 - s}}}{\sqrt{1 - s}} - \sqrt{\frac{s}{2}} \right| \le C s \sqrt{\frac{s}{2}}, \quad \forall s \in [0, 1 - (1 - \epsilon)^2]. $$

Then

$$ J(q, \epsilon) = \frac{1}{2q} \int_{0}^{q(1-(1-\epsilon)^2)} \sqrt{\frac{t}{2q}} \exp(-t) dt + R(q, \epsilon) $$

where

$$ |R(q, \epsilon)| \le \frac{1}{2q} \int_{0}^{\infty} \sqrt{\frac{t}{2q}} \frac{t}{q} \exp(-t) dt = O(q^{-5/2}), \quad q \to \infty $$

is of smaller order than the first term. It remains to compute

$$ \int_{0}^{q(1-(1-\epsilon)^2)} \sqrt{t} \exp(-t) dt \underset{q \to \infty}{\longrightarrow} \frac{\sqrt{\pi}}{2}, \quad q \to \infty. $$

□
**Acknowledgments.** S.F. and M.P. thank Antoine Lejay for very instructive discussions on stochastic processes in discontinuous layered media and porous barriers. I.K. and M.P. are grateful to Lorenzo Zambotti for generously sharing his expertise on the links between PDEs and stochastic processes, and in particular for his useful suggestions regarding the analysis of transition densities. I.K. is indebted to Irina Ignatiouk-Robert and Kilian Raschel for enriching discussions on discrete random walks analogous to the diffusion processes studied in the present paper.
REFERENCES

[1] O. Aryasova, I. Pavlyukevich, and A. Pilipenko, Homogenization of a multivariate diffusion with semipermeable interfaces, *J. Theoret. Probab.* **37** (2024), 1787–1823.

[2] R. Atar and A. Budhiraja, On the multi-dimensional skew Brownian motion, *Stochastic Process. Appl.* **125** (2015), no. 5, 1911–1925.

[3] Brychkov, I. Multidimensional Integral Transformations. (Gordon,1992)

[4] Dai, J. & Miyazawa, M. Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution. *Stochastic Systems*. **1**, 146 – 208 (2011)

[5] DeBlassie, R. The Adjoint Process of Killed Reflected Brownian Motion in a Cone and Applications. *The Annals Of Probability*. **27** (1999,10), http://dx.doi.org/10.1214/aop/1022874812

[6] DeBlassie, D. The Growth of the Martin Kernel in a Horn-shaped Domain. *Indiana University Mathematics Journal*. **57**, 3115-3129 (2008), http://www.jstor.org/stable/24903089

[7] DeBlassie, D. The Martin Kernel for Unbounded Domains. *Potential Analysis*. **32**, 389-404 (2009,9), http://dx.doi.org/10.1007/s11118-009-9156-2

[8] G. Doetsch. *Introduction to the Theory and Application of the Laplace Transformation*. Springer Berlin Heidelberg, Berlin, Heidelberg, 1974.

[9] P. A. Ernst and S. Franceschi. Asymptotic behavior of the occupancy density for obliquely reflected Brownian motion in a half-plane and Martin boundary. *The Annals of Applied Probability*, 31(6):2991 – 3016, 2021.

[10] Étoré, P. (2006). On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients. *Electronic Communications in Probability*, **11**, Paper no. 9, 249–275.

[11] G. Fayolle, R. Iasnogorodski, and V. Malyshev. *Random Walks in the Quarter-Plane: Algebraic Methods, Boundary Value Problems, Applications to Queueing Systems and Analytic Combinatorics*. 2nd ed., Springer Publishing Company, Incorporated, 2017.

[12] Feng, C. & Zhao, H. A Generalized Ito's Formula in Two-Dimensions and Stochastic Lebesgue-Stieltjes Integrals. *Electronic Journal Of Probability*, **12**, 1568 - 1599 (2007), https://doi.org/10.1214/EJP.v12-468

[13] E. R. Fernholz, T. Ichiba, and I. Karatzas. Two Brownian particles with rank-based characteristics and skew-elastic collisions. *Stochastic Processes and their Applications*, **123**(8):2999–3026, 2013.

[14] Franceschi, S., Kourkova, I. & Petit, M. Asymptotics for the Green's functions of a transient reflected Brownian motion in a wedge. *Queueing Systems*. **108**, 321-382 (2024,8), http://dx.doi.org/10.1007/s11134-024-09925-y

[15] J. M. Harrison and L. A. Shepp, On skew Brownian motion, *Ann. Probab.* **9** (1981), no. 2, 309–313.

[16] I. Ignatiouk-Roubert, I. Kurkova, K. Raschel, Reflected random walks and unstable Martin boundary, *Annales de l'Institut Henri Poincaré* **60** (2024), 549-587

[17] Ignatyuk, I., Malyshev, V. & Scherbakov, V. Boundary effects in large deviation problems. *Russian Mathematical Surveys*. **49**, 41 (1994,4), https://dx.doi.org/10.1070/RM1994v049n02ABEH002204

[18] Kobayashi, M., Sakuma, Y. & Miyazawa, M. Join the shortest queue among k parallel queues: tail asymptotics of its stationary distribution. *Queueing Systems*. **74**, 303-332 (2013,3), http://dx.doi.org/10.1007/s11134-013-9353-y

[19] Kobayashi, M., Miyazawa, M. & Sakuma, Y. Heavy traffic limit of stationary distribution of the multi-level single server queue. (2025), https://arxiv.org/abs/2505.03504

[20] Kunita, H. & Watanabe, T. Markov processes and martin boundaries part I. *Illinois Journal Of Mathematics*. **9**, 485 - 526 (1965)

[21] I. Kurkova, V. Malyshev, Martin boundary and Ellipric Curves, *Markov Processes and Relatdes Fields* **4** (1998) 203–272

[22] Kurkova, I. & Suhov, Y. Malyshev's theory and JS-queues. Asymptotics of stationary probabilities. *The Annals Of Applied Probability*. **13**, 1313 – 1354 (2003), https://doi.org/10.1214/aoap/1069786501

[23] Kurtz, T. The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities. *Electronic Journal Of Probability*. **12**, 951 – 965 (2007), https://doi.org/10.1214/EJP.v12-431

[24] Le Gall, J. One - dimensional stochastic differential equations involving the local times of the unknown process. *Stochastic Analysis And Applications*. pp. 51-82 (1984)

[25] Lejay, A. On the constructions of the skew Brownian motion. *Probability Surveys*. **3**, 413 – 466 (2006), https://doi.org/10.1214/154957807000000013

[26] Lejay, A. Simulation of a stochastic process in a discontinuous layered medium. *Electronic Communications In Probability*. **16**, 764 – 774 (2011), https://doi.org/10.1214/ECP.v16-1686

[27] L. Lenôtre. *Study and simulation of skew diffusion processes*. PhD thesis, Université de Rennes, 2015. https://tel.archives-ouvertes.fr/tel-01247310v2.

[28] Miyazawa, M. Light tail asymptotics in multidimensional reflecting processes for queueing networks. *TOP*. **19** pp. 233-299 (2011,12)

[29] Miyazawa, M. Tail Decay Rates in Double QBD Processes and Related Reflected Random Walks. *Mathematics Of Operations Research*. **34**, 547-575 (2009,8), http://dx.doi.org/10.1287/moor.1090.0375

[30] Miyazawa, M. Two Sided DQBD Process and Solutions to the Tail Decay Rate Problem and Their Applications to the Generalized Join Shortest Queue. (2009,1)
[31] Miyazawa, M. A superharmonic vector for a nonnegative matrix with QBD block structure and its application to a Markov-modulated two-dimensional reflecting process. Queueing Syst 81, 1-48 (2015). https://doi.org/10.1007/s11134-015-9454-x

[32] Miyazawa, M. Multi-level reflecting Brownian motion on the half line and its stationary distribution. (2024), https://arxiv.org/abs/2405.16764

[33] Petit, M. Martin boundary of a degenerate Reflected Brownian Motion in a wedge. (2024), https://arxiv.org/abs/2411.02156

[34] N. I. Portenko, Diffusion processes with a generalized drift coefficient, Theory Probab. Appl. 24 (1979), no. 1, 62-78.

[35] N. I. Portenko, Stochastic differential equations with generalized drift vector, Theory Probab. Appl. 24 (1979), no. 2, 338-353.

[36] M. Portenko and B. Kopytko, On constructing a multidimensional diffusion process with a membrane located on a given hyperplane and acting in an oblique direction, Sci. Res. Inst. Math. Comput. Sci. 11 (2012), no. 3, 137-147.

[37] S. E. Shreve. Reflected Brownian motion in the “bang-bang” control of Brownian drift. *SIAM Journal on Control and Optimization*, 19(4):469–478, 1981. doi:10.1137/0319027.

[38] Stroock, D. Diffusion semigroups corresponding to uniformly elliptic divergence form operators. *Séminaire De Probabilités XXII*. pp. 316-347 (1988), https://doi.org/10.1007/BFb0084145

[39] R. J. Williams. Semimartingale reflecting Brownian motions in the orthant. *Stochastic Networks*, vol. 13, 1995.

[40] Zhao, Y. The kernel method tail asymptotics analytic approach for stationary probabilities of two-dimensional queueing systems. *Queueing Systems*. **100** (2022,1)

[41] L. Zaitseva, On the Markov property of strong solutions to SDE with generalized coefficients, *Theory Stoch. Process.* **11** (2005), no. 27, 140–146.

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