# About hypoelliptic SDE equations with boundary conditions

Pierre Etoré, José Rafael León, Clémentine Prieur

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Pierre Etoré, José Rafael León, Clémentine Prieur. About hypoelliptic SDE equations with boundary conditions. 2025. hal-04984931

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# About hypoelliptic SDE equations with boundary conditions

Pierre Etoré¹, José R. León², and Clémentine Prieur³

¹Univ. Grenoble Alpes, CNRS, INRIA, Grenoble INP, LJK, F-38000 Grenoble, France, pierre.etore@univ-grenoble-alpes.fr

²Univ. de La República. IMERL. Montevideo, Uruguay, rlramos@fing.edu.uy

³Univ. Grenoble Alpes, CNRS, INRIA, Grenoble INP, LJK, F-38000 Grenoble, France, clementine.prieur@univ-grenoble-alpes.fr

## Abstract

In this paper, we are interested in Langevin dynamics on a bounded domain, with specular reflection at the boundary. We provide an explicit construction of the associated diffusion process and identify its infinitesimal generator. We then study probabilistic properties, namely irreducibility, aperiodicity and, adding a fluctuation-dissipation assumption, exponential ergodicity.

## 1 Introduction

The present research is deeply inspired by the seminal paper [7], which models the movement of a herd of elephants using an elliptic diffusion on the plane with Neumann boundary conditions over a non-empty bounded domain $O$ in $\mathbb{R}^2$ (i.e., a bounded, connected open set) such that $\partial O$ is $C^2$. The authors estimate several characteristics of this diffusion, such as the home range and the drift governing the movement. However, this modeling approach implies that the trajectories of the animals correspond to those of an elliptic diffusion, which are non-differentiable at any point. Addressing this limitation motivated Brillinger et al. [3] to propose a hypoelliptic model. As the authors in [3] underline, motion in Newtonian dynamics may be described by a potential function $V(x, t)$. Then the equations of motion take the form

$$ \begin{cases} dx(t) = y(t)dt \\ dy(t) = -(cy(t) + \nabla_x V(x(t), t))dt \end{cases} $$

where $x(t) = (x_1(t), x_2(t))$ represents the particle's location at time $t$ and $y$ is the particle's velocity. If the movement is perturbed by a white noise $dW$, the resulting equations are given by

$$ \begin{cases} dX_t = Y_t dt \\ dY(t) = -(cY_t + \nabla_x V(X_t, t))dt + \sigma dW_t. \end{cases} \quad (1) $$

The challenge now arises from the fact that the particle's motion occurs within a bounded domain $O$ in $\mathbb{R}^2$. This requires implementing a model that incorporates total or partial reflection at the boundary, ensuring the dynamics remains confined within the region. From now on, we consider an autonomous system where $V(x, t)$ depends only on $x$.

Our problem also admits a partial differential equation (PDE) formulation. Specifically, the heat equation associated with this process is given by:

$$ \partial_t u = \frac{\sigma^2}{2} \Delta_y u + \langle y, \nabla_x u \rangle - \langle cy + \nabla_x V(x), \nabla_y u \rangle $$
with $\langle \cdot, \cdot \rangle$ the Euclidean inner product in $\mathbb{R}^2$.

To constrain the movement, we impose a boundary condition. Notably, this condition applies solely to the particle's position, with no restrictions placed on the derivative. Then the domain in which the particle moves is the cylinder $\mathcal{D} = \mathcal{O} \times \mathbb{R}^2$. Consequently, the function $u(t, x, y)$ must satisfy appropriate boundary conditions on the set $\partial\mathcal{O} \times \mathbb{R}^2$, where $\partial\mathcal{O}$ represents the boundary of the spatial region $\mathcal{O}$. We review below some examples of such conditions, as presented in [4], [1], [15].

It is necessary to introduce first some notation. Let $\mathbf{n}(x)$ denote the normal vector to $\partial\mathcal{O}$ in $x$, pointing outwards from $\mathcal{O}$. The boundary $\Gamma = \partial\mathcal{O} \times \mathbb{R}^2$ of the cylindrical position/velocity domain $\mathcal{D}$ is the union of three disjoint subsets:

$$ \Gamma_+ = \{ (x, y) \in \Gamma : \langle \mathbf{n}(x), y \rangle > 0 \} \quad (\text{outgoing velocities}) $$

$$ \Gamma_- = \{ (x, y) \in \Gamma : \langle \mathbf{n}(x), y \rangle < 0 \} \quad (\text{ingoing velocities}) $$

$$ \Gamma_0 = \{ (x, y) \in \Gamma : \langle \mathbf{n}(x), y \rangle = 0 \} \quad (\text{null velocities}) $$

which we refer to as the outgoing, grazing, and ingoing sets, respectively.

Then, the specular reflection is given by:

$$ u(t, x, y) |_{(x,y) \in \Gamma_+} = u(t, x, y - 2\langle y, \mathbf{n}(x) \rangle \mathbf{n}(x)). \quad (2) $$

This condition can be generalized by considering a kernel $R$ to describe the reflection. More precisely, for any $f: \mathbb{R}^+ \times \bar{\mathcal{D}} \to \mathbb{R}$, we define

$$ \gamma_- f(t, x, y) = \int_{\Gamma_+} R(t, x, y, y') \gamma_+ f(t, x, y') dy' $$

with $\gamma_\pm f(t, \cdot, \cdot) = \mathbf{1}_{\Gamma_\pm}((x, y))f(t, x, y)$ and $\mathbf{1}_{\Gamma_\pm}((\cdot, \cdot))$ the indicator function for $\Gamma_\pm$. Let us mention two particular cases. For $R(t, x, y, y') = \delta_{-y}(y')$, an individual reaching the boundary of the domain simply turns back, with unchanged modulus of velocity. Specular reflection corresponds to $R(t, x, y, y') = \delta_{y-2\langle y, \mathbf{n}(x) \rangle \mathbf{n}(x)}(y')$. Below, we present a figure that schematizes the process:

<figure><img src="image_3.png" /><figcaption>Figure 1: Specular reflection of a particle at the boundary of the domain. Normal vector $\mathbf{n}$ is pointing outwards.</figcaption></figure>

Interesting questions to address are:

* How can System (1) be modified to incorporate boundary conditions?

* What are the assumptions to ensure existence and uniqueness of a stationary probability measure of the underlying process, as far as ergodicity?

* Is it possible to estimate from a single trajectory the home range, the boundary of the domain, the drift and the invariant measure?
In the present paper, we address the two first questions, leaving the third one and the application to animal movements for future work. Let $(\Omega, \mathcal{F}, (\mathcal{F})_{t \ge 0}, \mathbb{P})$ a filtered probability space. We consider the more general stochastic hypoelliptic damped Hamiltonian process $(X_t, Y_t)_{t \ge 0}$ solution of:

$$ \begin{cases} dX_t = Y_t dt \\ dY_t = -(c(X_t, Y_t)Y_t + \nabla_x V(X_t))dt + \Sigma(X_t, Y_t)dW_t \end{cases} \quad (3) $$

with $(W_t, t \ge 0)$ a standard $d$-dimensional Brownian motion on $(\Omega, \mathcal{F}, (\mathcal{F})_{t \ge 0}, \mathbb{P})$. Following [10] we make the following assumptions on the potential $V$, the diffusion term $\Sigma$ and the damping force $c$:

(Av1) $V: \mathbb{R}^d \to \mathbb{R}$ is $C^1$ and lower bounded on $\mathbb{R}^d$.

(Ac1) $c: \mathbb{R}^{2d} \to \mathbb{R}^{d \times d}$ is continuous. In addition, there exist $\eta > 0$ and $L > 0$ such that $\forall y \in \mathbb{R}^d, |x| \ge L$,

$$ \frac{1}{2} [c(x, y) + c^T(x, y)] \ge \eta I_{\mathbb{R}^d}, $$

and for all $N > 0$,

$$ \sup_{|x| \le N, y \in \mathbb{R}^d} \|c(x, y)\|_{\text{HS}} < \infty, $$

where $\| \cdot \|_{\text{HS}}$ is the Hilbert-Schmidt norm of a matrix and $c^T$ is the transpose of $c$.

(AΣ1) $\Sigma: \mathbb{R}^{2d} \to \mathbb{R}^{d \times d}$ is symmetric, infinitely differentiable and for some $\sigma > 0$: $0 < \Sigma(x, y) \le \sigma I_d$ over $\mathbb{R}^{2d}$.

We consider specular reflection at the boundary. Exponential ergodicity will be proven under the more specific framework:

$$ \begin{cases} dX_t = Y_t dt \\ dY_t = - (\tilde{\Sigma}^2(X_t) I_d Y_t + \nabla_x V(X_t)) dt + (2\beta^{-1})^{1/2} \tilde{\Sigma}(X_t) dW_t. \end{cases} \quad (4) $$

Hypothesis (AΣ1) is replaced by:

(AΣ) $\Sigma: \mathbb{R}^d \to \mathbb{R}$ is a $C^1$ function, uniformly Lipschitz over $\mathbb{R}^d$, and such that for some $\Sigma_0 > 0$ and $\Sigma_\infty > 0$,

$$ \forall x \in \mathbb{R}^d, \Sigma_0 \le \Sigma(x) \le \Sigma_\infty. $$

The relation $\tilde{\Sigma}^2(t) = 2c(X_t)/\beta$ is called the fluctuation-dissipation relation (see, e.g., [12]) since it relates the magnitude of the dissipative term and the magnitude of the random term. This precise balance between the drift term which removes energy in average and the stochastic term ensures that the canonical measure is preserved by the dynamics on $\mathbb{R}^{2d}$. Under this condition we are able to exhibit a Lyapunov function $\Psi$ for the diffusion process on a bounded domain with specular reflection at the boundary, thus a spectral gap in the set $b_\Psi \mathcal{B}(\bar{D})$ of measurable functions $f$ on $\bar{D}$ such that $\sup_{x \in \bar{D}} |f(x)|/\Psi(x) < +\infty$. The invariant probability measure coincides then to the canonical measure renormalized on $\bar{D}$. Note that these results (exponential convergence to a unique invariant probability measure) could be generalized to other settings as soon as we are able to explicit a Lyapunov function which is compatible with boundary conditions.

Related results were obtained recently in [11] and [10], whose authors are interested by existence and unicity of a quasi-stationary distribution of the Langevin diffusion process on a bounded domain with absorption at the boundary. Our study relies strongly on these works. We also refer to [15] for the analysis of the diffusion process on a bounded domain of a manifold and with different boundary conditions, in
the framework where the damping force and the diffusion term are constant. The analysis in this last reference strongly relies on PDE techniques, whereas our proofs adopt a probabilistic point of view.

The paper is structured as follows. In Section 2 we explicit the construction of the reflected Langevin diffusion process and identify its infinitesimal generator. Then in Section 3, we study the properties of the process introduced in Section 2. In Section 3.1 we prove irreductibility and aperiodicity (Section 3.1). In Section 3.2 we give the explicit construction of a Lyapunov function in the case of dimension $d = 1$ and under fluctuation-dissipation condition. We then extend this construction to general dimensions in Section 3.3. We finally deduce in Section 3.4 exponential convergence to equilibrium.

## 2 Construction of confined Langevin diffusion process submitted to specular reflection

In this section we work under the assumptions (Av1), (Ac1) and $(\Lambda\Sigma1)$ on the coefficients $V$, $c$ and $\Sigma$.

As announced in the introduction we will study a confined version of (3) where the process $(X, Y)$ stays in the state space $D_1 = \Gamma_- \cup D$. This is related to specular reflection as described in the previous section. We define

$$J_t = -2 \sum_{s \le t} \langle Y_{s-}, \mathbf{n}(X_{s-}) \rangle \mathbf{n}(X_{s-}) \mathbf{1}_{\Gamma_+}(X_{s-}, Y_{s-}) \quad (5)$$

and consider the new system

$$\begin{cases} dX_t = Y_t dt \\ dY_t = \Sigma(X_t, Y_t) dW_t - (c(X_t, Y_t)Y_t + \nabla_x V(X_t)) dt + dJ_t. \end{cases} \quad (6)$$

Let us explain heuristically the behaviour of the system described by (6), and why the term specular reflection is used. When $(X, Y)$ reaches $\Gamma_-$ the ingoing velocity pushes the position $X$ inside $O$ and $(X, Y)$ stays automatically in $D$. When $(X, Y)$ approaches $\Gamma_+$ the jump term $J_t$ in (6) comes into play and changes immediately the outgoing velocity $Y$ into an ingoing velocity $Y - 2 < Y, \mathbf{n}(X) > \mathbf{n}(X)$: the position $X$ is pushed back inside $O$ and $(X, Y)$ stays in $D$.

Note that because of the definition of the jump term $J$, the part $\Gamma_-$ of the boundary $\Gamma$ can be visited by $(X, Y)$, while $\Gamma_+$ cannot. Finally note that $\Gamma_0$ is of zero Lebesgue measure.

Note that (6) may be rewritten in the form

$$dZ_t = \Sigma(Z_t)dW_t + B(Z_t)dt + dJ_t \quad (7)$$

with $Z = (X, Y)$,

$$\Sigma(z = (x, y)) = \begin{pmatrix} 0 \\ \sigma(x, y) \end{pmatrix}, \quad B(z = (x, y)) = \begin{pmatrix} y \\ -c(x, y)y - \nabla_x V(x) \end{pmatrix}, \quad J_t = \begin{pmatrix} 0 \\ J_t \end{pmatrix}$$

We use this form frequently in the sequel for the sake of conciseness.

We first aim at showing that (6) (eq. (7)) has a weak solution and enjoys uniqueness in the sense of probability law. This can be performed in the following manner.

Let us consider for any $(s, z) \in \mathbb{R}_+ \times \mathbb{R}^{2d}$ the process

$$\bar{Z}_{s,z}(t) = z + \int_{s}^{t} \Sigma(\bar{Z}_{s,z}(u))dW_{u} + \int_{s}^{t} B(\bar{Z}_{s,z}(u))du, \quad t \ge s, \quad (8)$$

that is the solution of the non confined system (3) starting from $z$ at time $s$.
From Lemma 1.1 in [17] we know that there exists a unique weak solution $\bar{Z}_{0,z}$ to (8) for any $z \in \mathbb{R}^{2d}$, which is non explosive. Let $z \in \mathbb{R}^{2d}$ fixed. We set

$$ \tau_1(\bar{Z}_{0,z}) = \inf\{t \ge 0 : \bar{Z}_{0,z}(t-) \in \Gamma_+\}. $$

We know that $\tau_1$ is a stopping time (see, for instance, [15] page 113 for $\Sigma$ and $c$ constant). Moreover, the case general can be studied using the stopping time $\tau_R$ defined in [17] (pages 209-210).

Once $\bar{Z}_{0,z}(t-)$ has reached $\Gamma_+$ we construct $\bar{Z}_{0,z}(t)$, $t = \tau_1(\bar{Z}_{0,z})$, in the following manner

$$ \begin{aligned} \bar{X}_{0,z}(t) &= \bar{X}_{0,z}(t-) \\ \bar{Y}_{0,z}(t) &= \bar{Y}_{0,z}(t-) - 2 < \bar{Y}_{0,z}(t-), \mathbf{n}(\bar{X}_{0,z}(t-)) > \mathbf{n}(\bar{X}_{0,z}(t-)) \end{aligned} $$

Then we consider $\bar{Z}_{\tau_1(\bar{Z}_{0,z}), \bar{Z}_{0,z}(\tau_1(\bar{Z}_{0,z}))}(t)$, $t \ge \tau_1(\bar{Z}_{0,z})$, and

$$ \tau_2(\bar{Z}_{\tau_1(\bar{Z}_{0,z}), \bar{Z}_{0,z}(\tau_1(\bar{Z}_{0,z}))}(t)) = \inf\{t \ge 0 : \bar{Z}_{\tau_1(\bar{Z}_{0,z}), \bar{Z}_{0,z}(\tau_1(\bar{Z}_{0,z}))}(t-) \in \Gamma_+\}. $$

We thus construct $\bar{Z}_{\tau_1(\bar{Z}_{0,z}), \bar{Z}_{0,z}(\tau_1(\bar{Z}_{0,z}))}(t)$, $t = \tau_2(\bar{Z}_{\tau_1(\bar{Z}_{0,z}), \bar{Z}_{0,z}(\tau_1(\bar{Z}_{0,z}))}(t))$, in the following manner

$$ \begin{aligned} \bar{X}_{\tau_1(\bar{Z}_{0,z}), \bar{Z}_{0,z}(\tau_1(\bar{Z}_{0,z}))}(t) &= \bar{X}_{\tau_1(\bar{Z}_{0,z}), \bar{Z}_{0,z}(\tau_1(\bar{Z}_{0,z}))}(t-) \\ \bar{Y}_{\tau_1(\bar{Z}_{0,z}), \bar{Z}_{0,z}(\tau_1(\bar{Z}_{0,z}))}(t) &= \bar{Y}_{\tau_1(\bar{Z}_{0,z}), \bar{Z}_{0,z}(\tau_1(\bar{Z}_{0,z}))}(t-) \\ & \quad -2 < \bar{Y}_{\tau_1(\bar{Z}_{0,z}), \bar{Z}_{0,z}(\tau_1(\bar{Z}_{0,z}))}(t-), \mathbf{n}(\bar{X}_{\tau_1(\bar{Z}_{0,z}), \bar{Z}_{0,z}(\tau_1(\bar{Z}_{0,z}))}(t-)) > \mathbf{n}(\bar{X}_{\tau_1(\bar{Z}_{0,z}), \bar{Z}_{0,z}(\tau_1(\bar{Z}_{0,z}))}(t-)) \end{aligned} $$

We reiterate this procedure and denote in short $\tau_i$, $i \ge 0$, the successively encountered stopping times (with $\tau_0 = 0$). We therefore define a process $(Z_t)_{t \ge 0}$ which coincides on each time interval $[\tau_i, \tau_{i+1})$ with the process $\bar{Z}$ starting at time $\tau_i$ used in the construction. Note that by construction the $X$-component of $Z$ is time continuous while the $Y$-component experiences jumps governed by a term of type (5).

For the next theorem, we need to define $\bar{\mathcal{F}}$ as the $\sigma$-algebra generated by all the $\bar{Z}$ processes defined above.

**Theorem 1.** The afore defined process $Z$ is a Markov process that provides a weak solution of (6) (eq. (7)). Besides, (6) enjoys uniqueness in law.

*Proof.* Let us verify that $Z$ is a Markov process. For this, assume that $s < t$, $\mathcal{F}_s = \sigma(Z(u) : u \le s)$. Let us remark that $\mathcal{F}_s \subset \bar{\mathcal{F}}_s$ for all $s$.

We now compute

$$ \mathbb{E}[Z_t | \bar{\mathcal{F}}_s] = \sum_j \mathbb{E}[Z_t \mathbf{1}_{(\tau_j \le t < \tau_{j+1})} | \bar{\mathcal{F}}_s] = \sum_j \mathbb{E}[\bar{Z}_{\tau_j, t} \mathbf{1}_{(\tau_j \le t < \tau_{j+1})} | \bar{\mathcal{F}}_s] $$

where we have denoted $\bar{Z}_{\tau_j}$, the successive processes $\bar{Z}$ used in the construction of $Z$.

To compute the above expectations, we need to consider the following two sets: $\{\tau_j \le s < t\}$ and $\{s < \tau_j \le t\}$.

$$ \mathbb{E}[\bar{Z}_{\tau_j, t} \mathbf{1}_{(\tau_j \le t < \tau_{j+1})} | \bar{\mathcal{F}}_s] = \mathbb{E}[\bar{Z}_{\tau_j, t} \mathbf{1}_{(\tau_j \le s < t < \tau_{j+1})} | \bar{\mathcal{F}}_s] + \mathbb{E}[\bar{Z}_{\tau_j, t} \mathbf{1}_{(s < \tau_j \le t < \tau_{j+1})} | \bar{\mathcal{F}}_s] $$

For the first expectation, we have

$$ \mathbb{E}[\bar{Z}_{\tau_j, t} \mathbf{1}_{(\tau_j \le s < t < \tau_{j+1})} | \bar{\mathcal{F}}_s] = \mathbb{E}[\bar{Z}_t \mathbf{1}_{(\tau_j \le s < t < \tau_{j+1})} | \bar{Z}_s] = \mathbb{E}[Z_t \mathbf{1}_{(\tau_j \le s < t < \tau_{j+1})} | Z_s]. $$

For the second expectation, let's first consider the case when $\tau_{j-1} \le s < \tau_j$. We obtain

$$ \begin{aligned} \mathbb{E}[\bar{Z}_{\tau_j, t} \mathbf{1}_{(\tau_{j-1} \le s < \tau_j)} \mathbf{1}_{(\tau_j \le t < \tau_{j+1})} | \bar{\mathcal{F}}_s] &= \mathbb{E}[\bar{Z}_{\tau_j, J(Z(s) + \mathcal{L}(s, \tau_j^-) + t)} \mathbf{1}_{(\tau_{j-1} \le s < \tau_j)} \mathbf{1}_{(\tau_j \le t < \tau_{j+1})}] \\ &= \mathbb{E}[Z_{\tau_j, J(Z(\tau_j^-)) + t} \mathbf{1}_{(\tau_{j-1} \le s < \tau_j)} \mathbf{1}_{(\tau_j \le t < \tau_{j+1})} | Z_s], \end{aligned} $$
where

$$ \mathcal{L}(s, \tau_j^-) := \int_s^{\tau_j^-} b(\bar{Z}_u) du + \begin{bmatrix} 0 \\ (W_{\tau_j^-} - W_s) \end{bmatrix}. $$

Let's assume now that $\tau_{j-2} \le s < \tau_{j-1}$. To compute the term

$$ \mathbb{E}[\bar{Z}_{\tau_j, J(Z(\tau_j^-)) + t} \mathbf{1}_{(\tau_{j-2} \le s < \tau_{j-1})} \mathbf{1}_{(\tau_j \le t < \tau_{j+1})} | \bar{\mathcal{F}}_s], \quad (9) $$

it is important to note that, due to the new restrictions imposed on the variable $s$, we have

$$ \bar{Z}_{\tau_j, J(Z(\tau_j^-)) + t} = J(J(Z_s + \mathcal{L}(s, \tau_{j-1}^-)) + \mathcal{L}(\tau_{j-1}, \tau_j^-)) + \mathcal{L}(\tau_j, t). $$

Thus, we obtain:

$$ \begin{align*} (9) &= \mathbb{E}[J(J(\bar{Z}_s + \mathcal{L}(s, \tau_{j-1}^-)) + \mathcal{L}(\tau_{j-1}, \tau_j^-)) + \mathcal{L}(\tau_j, t) \mathbf{1}_{(\tau_{j-2} \le s < \tau_{j-1})} \mathbf{1}_{(\tau_j \le t < \tau_{j+1})} | \bar{\mathcal{F}}_s] \\ &= \mathbb{E}[J(J(\bar{Z}_s + \mathcal{L}(s, \tau_{j-1}^-)) + \mathcal{L}(\tau_{j-1}, \tau_j^-)) + \mathcal{L}(\tau_j, t) \mathbf{1}_{(\tau_{j-2} \le s < \tau_{j-1})} \mathbf{1}_{(\tau_j \le t < \tau_{j+1})}] \\ &= \mathbb{E}[\bar{Z}_{\tau_j, J(\bar{Z}(\tau_j^-)) + t} \mathbf{1}_{(\tau_{j-2} \le s < \tau_{j-1})} \mathbf{1}_{(\tau_j \le t < \tau_{j+1})} | \bar{Z}_s] \\ &= \mathbb{E}[Z_{\tau_j, J(Z(\tau_j^-)) + t} \mathbf{1}_{(\tau_{j-2} \le s < \tau_{j-1})} \mathbf{1}_{(\tau_j \le t < \tau_{j+1})} | Z_s]. \end{align*} $$

By recursively applying the same reasoning, we can obtain the result for other cases.

Finally, returning to our original computation and utilizing the above results, we obtain the Markov property:

$$ \mathbb{E}[Z(t)|\mathcal{F}_s] = \sum_j \mathbb{E}[\bar{Z}_{\tau_j, t} \mathbf{1}_{(\tau_j \le t < \tau_{j+1})} | \bar{\mathcal{F}}_s] = \sum_j \mathbb{E}[Z_{\tau_j, t} \mathbf{1}_{(\tau_j \le t < \tau_{j+1})} | Z_s] = \mathbb{E}[Z(t)|Z_s] \text{ for } s < t. $$

□

We now aim at identifying the generator of the process $Z$ involved in Theorem 1. We recall that we say that a measurable function $f: \mathcal{D}_1 \to \mathbb{R}$ belongs to the extended domain $\mathcal{D}_e(\mathcal{L})$ of the generator $\mathcal{L}$ of $Z$ if there is a measurable function $g: \mathcal{D}_1 \to \mathbb{R}$ s.t. $\int_0^t |g(Z_s)|ds < \infty$ for all $t \ge 0$ and the process

$$ f(Z_t) - f(Z_0) - \int_0^t g(Z_s)ds, \quad t \ge 0 $$

is a $\mathbb{P}^z$-local martingale (possibly càd-làg) for all $z \in \mathcal{D}_1$. This defines $\mathcal{L}$ on $\mathcal{D}_e(\mathcal{L})$ by the relation $\mathcal{L}f = g$.

We now reformulate a bit the "specular reflection condition" that has been evoked in the introduction in (2):

$$ \varphi(x, y)|_{\Gamma_+} = \varphi(x, y - 2 < y, \mathbf{n}(x) > \mathbf{n}(x))|_{\Gamma_-} \quad \forall (x, y) \in \Gamma_+, \quad (10) $$

this equation making sense for functions $\varphi$ having a trace on $\Gamma_{\pm}$. Note that in the statements below when we say $\varphi \in C^{1,2}(\mathcal{D}_1)$ we mean that $\varphi$ is the restriction to $\mathcal{D}_1$ of a function in $C^{1,2}(\mathbb{R}^{2d})$ (the set of continuous functions on $\mathbb{R}^d$ with continuous first order derivatives with respect to the $x$-variables and continuous derivatives up to order 2 in the $y$ variables). So by their nature those functions have a trace on $\Gamma_{\pm}$.

We aim at proving the following result.

**Proposition 1.** The domain $\mathcal{D}_e(\mathcal{L})$ contains the functions $\varphi$ in $C^{1,2}(\mathcal{D}_1)$ and that satisfy (10). For those functions one has

$$ \mathcal{L}\varphi(z) = \frac{1}{2} \sum_{i,j=1}^{d} [\sigma\sigma^T]_{ij}(z) \partial_{y_i y_j} \varphi(z) + \langle y, \nabla_x \varphi(z) \rangle - \langle c(x, y) y + \nabla_x V(x), \nabla_y \varphi(z) \rangle, \quad \forall z = (x, y) \in \mathcal{D}_1. $$
**Lemma 1.** Let $\varphi \in C^{1,2}(D_1)$ and that satisfy (10). We set

$$g(z) = \frac{1}{2} \sum_{i,j=1}^{d} [\sigma \sigma^T]_{ij}(z) \partial_{y_i y_j} \varphi(z) + \langle y, \nabla_x \varphi(z) \rangle - \langle c(x, y) y + \nabla_x V(x), \nabla_y \varphi(z) \rangle, \quad \forall z = (x, y) \in D_1. \tag{11}$$

We set $\tau_1 = \inf\{t \ge 0 : Z_{t-} \in \Gamma_+\}$. On time interval $[0, \tau_1]$ the process

$$\varphi(Z_t) - \varphi(Z_0) - \int_0^t g(Z_s) ds, \quad 0 \le t \le \tau_1,$$

can be identified to a local martingale.

*Proof.* Applying an Itô formula with jumps and using $d\langle Y^i, Y^j \rangle_t = \sum_{k=1}^d (\sigma_{ik}\sigma_{jk})(X_t, Y_t)dt = [\sigma\sigma^T]_{ij}(X_t, Y_t)dt$ we get

$$\begin{align*}
\varphi(Z_t) - \varphi(Z_0) - \int_0^t g(Z_s) ds &= - \int_0^t g(Z_s) ds + \int_0^t \langle \nabla_x \varphi(Z_s), Y_s \rangle ds \\
&\quad - \int_0^t \langle \nabla_y \varphi(Z_s), c(X_s, Y_s) Y_s + \nabla_x V(X_s) \rangle ds \\
&\quad + \int_0^t \langle \nabla_y \varphi(Z_s), \sigma(X_s, Y_s) dW_s \rangle \\
&\quad + \frac{1}{2} \sum_{i,j=1}^d \int_0^t \partial_{y_i y_j}(Z_s) d\langle Y^i, Y^j \rangle_s + \int_0^t [\varphi(Z_s) - \varphi(Z_{s-})] dJ_s \\
&= \int_0^t \langle \nabla_y \varphi(Z_s), \sigma(X_s, Y_s) dW_s \rangle + \int_0^t [\varphi(Z_s) - \varphi(Z_{s-})] dJ_s.
\end{align*}$$

For $t < \tau_1$ it is clear that the above quantity is equal to $\int_0^t \nabla_x \varphi(Z_s) \cdot \sigma(X_s, Y_s)(Z_s)dW_s$. We examine precisely what happens for $t = \tau_1$. We consider the quantity $\varphi(Z_{\tau_1}) - \varphi(Z_{\tau_1-}) = \varphi(X_{\tau_1}, Y_{\tau_1}) - \varphi(X_{\tau_1-}, Y_{\tau_1-})$.

As the paths of $X$ are continuous we have $X_{\tau_1} = X_{\tau_1-}$. In addition

$$\begin{align*}
Y_{\tau_1} &= Y_{\tau_1-} + J_{\tau_1} - J_{\tau_1-} \\
&= Y_{\tau_1-} - 2 \sum_{s \le \tau_1} \langle Y_{s-}, \mathbf{n}(X_{s-}) \rangle \mathbf{n}(X_{s-}) \mathbf{1}_{\Gamma_+}(X_{s-}, Y_{s-}) \\
&\quad + 2 \sum_{s \le \tau_1-} \langle Y_{s-}, \mathbf{n}(X_{s-}) \rangle \mathbf{n}(X_{s-}) \mathbf{1}_{\Gamma_+}(X_{s-}, Y_{s-}) \\
&= Y_{\tau_1-} - 2 \langle Y_{\tau_1-}, \mathbf{n}(X_{\tau_1-}) \rangle \mathbf{n}(X_{\tau_1-})
\end{align*}$$

Thus, using $(X_{\tau_1-}, Y_{\tau_1-}) \in \Gamma_+$ and (10), we get

$$\varphi(X_{\tau_1}, Y_{\tau_1}) - \varphi(X_{\tau_1-}, Y_{\tau_1-}) = \varphi(X_{\tau_1-}, Y_{\tau_1-} - 2 \langle Y_{\tau_1-}, \mathbf{n}(X_{\tau_1-}) \rangle \mathbf{n}(X_{\tau_1-})) - \varphi(X_{\tau_1-}, Y_{\tau_1-}) = 0.$$

Thus $\int_0^{\tau_1} [\varphi(Z_s) - \varphi(Z_{s-})]dJ_s = 0$. This completes the proof. $\square$

*Proof of Proposition 1.* Let $\varphi \in C^{1,2}(D_1)$ and that satisfy (10). Let us set $g$ as in (11).

If now we define $\tau_i = \inf\{t \ge \tau_{i-1} : Z_{t-} \in \Gamma_+\}, i \ge 2$, we could mimic the proof of Lemma 1 on each time interval $[\tau_{i-1}, \tau_i]$ and finally prove that for any $\varphi$ in $C^{1,2}(D_1)$ and that satisfy (10), one has that

$$\varphi(Z_t) - \varphi(Z_0) - \int_0^t g(Z_s) ds, \quad t \ge 0$$

is a local martingale (note that of course by continuity of $g$ one has $\int_0^t |g|(Z_s)ds < \infty$ for all $t \ge 0$, so that the above involved integrals are well defined). This completes the proof. $\square$
# 3 Main properties of reflected Langevin diffusion process

In this section, we make Assumptions (Av1), (Ac1) and (AΣ) on $V$, $c$ and $\Sigma$. We also assume specular reflection at the boundary. We aim at studying the properties of the associated Markov process with initial state $z_0 \in \mathcal{D}$. The state space of this process is the set $\mathcal{D}_1 := \mathcal{D} \cup \Gamma_-$ where $\Gamma_-$ has been defined in Section 1. In Section 3.1 we prove that the confined process is aperiodic and $\varphi_0$-irreducible in the sense of [14], with $\varphi_0$ a maximal irreducibility measure. Then in Section 3.2, we exhibit a Lyapunov function $\Psi \ge 1$ with a drift condition for the extended generator $\mathcal{L}$ of the confined process, under the assumption of fluctuation-dissipation and in dimension $d = 1$. Thus result is extended to any dimension $d \ge 1$ ins Section 3.3. Then, by applying [9, Theorem 5.2], we get that the aperiodic and $\Psi$-irreducible confined process is $\Psi$-uniformly ergodic (see Section 3.4).

## 3.1 $\varphi_0$-irreducibility and aperiodicity of the confined process

This section is devoted to the proof of Proposition 2 stated below.

**Proposition 2.** Under Assumptions (Av1), (Ac1) and (AΣ), the confined Langevin diffusion process with specular reflexion is $\varphi_0$-irreducible and aperiodic.

*Proof.* We first prove $\varphi$-irreducibility of the reflected confined process $(Z_t, t \ge 0)$. That is, we want to prove that there exists some $\sigma$-finite measure $\varphi$ on $\overline{\mathcal{D}}$ such that for any $B \in \mathcal{B}(\mathcal{D}_1)$,

$$ \varphi(B) > 0 \implies \mathbb{E}_z \left[ \int_0^{+\infty} \mathbf{1}_{Z_t \in B} dt \right] > 0 \quad \forall z \in \overline{\mathcal{D}}. $$

Assume that there exists $z \in \mathcal{D}$ and $B \in \mathcal{B}(\mathcal{D}_1)$ such that $\mathbb{E}_z \left[ \int_0^{+\infty} \mathbf{1}_{Z_t \in B} dt \right] = 0$. If $B$ has a non empty interior, then $\exists A \subset B$, $A$ a non empty open subset of $\mathcal{D}$. Define $P_t^D(z, \cdot) := P(Z_t^z \in \cdot, 0 \le t < \tau_1^z(Z))$ with $\tau_1^z(Z)$ the first positive time the process reaches the boundary. Let $\tilde{Z}$ denote the absorbed process as defined in [11] (see also [10]). Note that for any $t \ge 0$, $Z_t^z \mathbf{1}_{t < \tau_1^z(Z)}$ has the same probability distribution as $\tilde{Z}_t^z \mathbf{1}_{t < \tau_1^z(\tilde{Z})}$. We thus know from [10, Lemma 6.6] and the definition of (C5) in [10, p. 3052] that there exists $t_0 > 0$ such that $\forall t \ge t_0$, $P_t^D(z, A) > 0$ so that:

$$ \mathbb{E}_z \left[ \int_0^{+\infty} \mathbf{1}_{Z_t \in B} dt \right] = \int_0^{+\infty} P_t(z, B) dt \\ \ge \int_0^{+\infty} P_t(z, A) dt \ge \int_0^{+\infty} P_t^D(z, A) dt \ge \int_{t_0}^{+\infty} P_t^D(z, A) dt > 0. $$

So we have a contradiction. Thus $\mathbb{E}_z \left[ \int_0^{+\infty} \mathbf{1}_{Z_t \in B} dt \right] = 0$ implies that the interior of $B$ is empty thus $\text{Leb}(B \cap \mathcal{D}_1) = 0$, with $\text{Leb}(\cdot)$ Lebesgue measure on $\mathbb{R}^d \times \mathbb{R}^d$.

Note that the proof that there exists $t_0 > 0$ such that $\forall t \ge t_0$, $P_t^D(z, A) > 0$ provided in [10, Proof of Lemma 6.6] can easily be extended to the case $z \in \Gamma_-$. Such a case was not considered in the original statement as the authors were interested by the absorbed diffusion process.

It concludes the proof of $\varphi$-irreducibility. Lebesgue measure restricted to $\mathcal{D}_1$ is then the maximal irreducibility measure $\varphi_0$ in the sense of [9, p. 1674]. In particular it implies that the invariant probability measure, if it exists, and $\text{Leb}(\cdot \cap \mathcal{D}_1)$ are mutually absolutely continuous.

Then to prove aperiodicity of the process $(Z_t^z, t \ge 0)$ we have to prove that for some small set $C \in \mathcal{B}^+(\mathcal{D}_1) := \{A \in \mathcal{B}(\mathcal{D}_1) \text{ with } \psi(A) > 0\}$, there exists $T$ such that $P_t(z, C) > 0$ for all $t \ge T$ and all $z \in C$ (see [9, p. 1675]). Note that $(Z_t^z, t \ge 0)$ is a non-explosive right process. It is clearly a right process as it has càdlàg trajectories. Moreover it is non explosive. Indeed, for $z \in \mathcal{D}_1$, let $\tau_i^z(Z), i \ge 1$ the sequence of successive positive times at which the process $Z$ reaches the boundary. Between two consecutive such times, $\tau_i^z(Z) \le t < \tau_{i+1}^z(Z)$, the reflected process behaves as the diffusion process on
$\mathbb{R}^{2d}$ starting from the point $Z_{\tau_i(Z)}^z$ at the boundary and we know from [17] that this last process is non explosive. Then all compact sets are petite for some skeleton chain (see [13, Section 5]). Thus choosing any compact $C$ in $\mathcal{D}$ with non empty interior, we can prove aperiodicity as from [10, Lemma 6.6], we have the existence of $t_0 > 0$ such that $P_t(z, C) > 0$ for all $t \ge t_0$ and all $z \in C$. $\square$

## 3.2 Construction of a Lyapunov function under fluctuation-dissipation relation for $d=1$

In this section and in the following, we work in the fluctuation-dissipation framework. We consider the dynamics described in (4) with $\beta = 2$ and specular reflection. However, all our results extend straightforwardly to any $\beta > 0$. To prove exponential ergodicity in this framework, we have to exhibit a Lyapunov function for the reflected diffusion.

Let $G \in \mathcal{C}^1([-1, 1])$ with $G(1) = G(-1) = 0$, we define the Liapunov function as

$$ \Psi(x, y) = e^{F(x, y) - \inf_{\mathcal{D}} F(x, y)} \ge 1, $$

where

$$ F(x, y) = a\left(\frac{1}{2}y^2 + V(x)\right) + bG(x)y, \quad a, b > 0. $$

Condition $G(1) = G(-1) = 0$ implies that for $x = \pm 1$, $y \mapsto F(x, y)$ thus $y \mapsto \Psi(x, y)$ is an even function so that $F$ and $\Psi$ satisfy the boundary condition. In the following, we choose $G$ such that $\|G\|_{\infty} \le 1$.

Straightforward computations lead to:

$$ -\mathcal{L}F = a\left[\frac{1}{2}\Sigma^2(x) - \Sigma^2(x)y^2\right] + b\left[y^2\partial_x G(x) - (\Sigma^2(x)y + \partial_x V(x))G(x)\right]. $$

thus

$$ \frac{\mathcal{L}\Psi}{\Psi} = \mathcal{L}(F) - \frac{1}{2}\sigma^2(x)(a(y+bG(x))^2). \qquad (12) $$

Now, let $K = [-1, 1] \times [-L, L]$ be a compact subset of $\mathcal{D}$, with $L > 0$ to be chosen in the following. We want to prove that there exists $L > 0$ such that, for some positive constants $\varepsilon, C$, the following inequality holds:

$$ \frac{\mathcal{L}\Psi}{\Psi} \ge \varepsilon \mathbf{1}_{K^c} - C \mathbf{1}_K. \qquad (13) $$

Let us fix some notation:

$$ \Sigma_{\infty} := \|\sigma\|_{\infty}, \quad M_0 := \|\partial_x G\|_{\infty}, \quad M := \|(\partial_x V)G\|_{\infty}, \quad \Sigma_0 := \inf_{x \in [-1, 1]} |\sigma(x)|. $$

Then

$$ \begin{aligned} \frac{\mathcal{L}\Psi}{\Psi} &\ge -\Sigma_{\infty}^2(a^2y^2 + b^2) + (a\Sigma_0^2 - bM_0)y^2 - b|y|\Sigma_{\infty}^2 - \frac{a}{2}\Sigma_{\infty}^2 - bM \\ &= y^2(-a^2\Sigma_{\infty}^2 + a\Sigma_0^2 - bM_0) - b\Sigma_{\infty}^2|y| - (\Sigma_{\infty}^2b^2 + \frac{a}{2}\Sigma_{\infty}^2 + bM) \\ &=: cy^2 - b\Sigma_{\infty}^2|y| - d. \end{aligned} \qquad (14) $$

We now have to choose $a, b > 0$ such that $c > 0$. To achieve this, we first choose $0 < a < \Sigma_0^2/\Sigma_{\infty}^2$ so that $-a^2\Sigma_{\infty}^2 + a\Sigma_0^2 > 0$. Then we choose $0 < b < \frac{a(\Sigma_0^2 - a\Sigma_{\infty}^2)}{M_0}$ such that $c > 0$. Now, to choose $L$ defining $K$, we have to study the table of variations of polynomials $cy^2 - b\Sigma_{\infty}^2y - d$ and to conclude by using the
symmetry of $y \mapsto cy^2 - b\Sigma_{\infty}^2|y| - d$. Indeed, choosing $L > L_0$ with $L_0$ the positive root of polynomials $cy^2 - b\Sigma_{\infty}^2y - d$, that is $L_0 = \frac{b\Sigma_{\infty}^2 + \sqrt{b^2\Sigma_{\infty}^4 + 4cd}}{2c}$, we get for $(x, y) \in K^c$:

$$ \frac{\mathcal{L}\Psi}{\Psi} \geq cy^2 - b\Sigma_{\infty}^2|y| - d \geq cL^2 - b\Sigma_{\infty}^2L - d =: \varepsilon > 0. \qquad (15) $$

Also, for $(x, y) \in K$:

$$ \frac{\mathcal{L}\Psi}{\Psi} \geq \min_{y \in [-L, L]} (cy^2 - b\Sigma_{\infty}^2|y| - d) =: -C \quad \text{with } C > 0. \qquad (16) $$

From (15) and (16) we get (13).

**Existence of a function G** Let $0 < \kappa < 1$. We are searching for a symmetric function $G$ on $[-1, 1]$, with $G(x) = 1$ on $[0, 1-\kappa)$, $G(x) = \alpha x^3 + \beta x^2 + \gamma x + \delta$ on $(1-\kappa, 1]$ and constraint $G(1) = 0$. For $\bar{G}$ to be continuously differentiable we add the constraint $G'(1-\kappa) = 0$ and $G'(1) = 0$. All the above conditions are satisfied by choosing

$$ \alpha = \frac{2}{9}(-2u^3 - 4u^2 - 3u + 5) \frac{1}{(u+1)^2}, \quad \beta = \frac{1}{3(u+1)}(2u^3 + 4u^2 + 3u - 5), $$

$$ \gamma = \frac{-2u}{3(u+1)^2}(2u^3 + 4u^2 + 3u - 5), \quad \delta = \frac{3u-1}{9(u+1)^2}(2u^3 + 4u^2 + 3u - 5), $$

with $u = 1 - \kappa$.

### 3.3 Extension to any dimension $d \geq 1$

We assume that $\mathcal{O}$ is a non-empty bounded open set of $\mathbb{R}^d$ with $\mathcal{C}^2$ boundary, $\partial\mathcal{O}$. We still assume that the fluctuation-dissipation condition holds true. For any $x$ in $\bar{\mathcal{O}}$ we define $\Pi(x)$ the projection of $x$ on $\partial\mathcal{O}$. We also define $G$, continuously differentiable on $\bar{\mathcal{O}}$, as $G(x) := (x - \Pi(x))/2 \sup_{x \in \bar{\mathcal{O}}} \|x\|$, with $\| \cdot \|$ the euclidean norm on $\mathbb{R}^d$. Then, for any $x \in \partial\mathcal{O}$, $G(x) = 0$, and $\|G(x)\| \leq 1$ for any $x \in \bar{\mathcal{O}}$. Following the path of Section 3.2, we define the Liapunov function as

$$ \Psi(x, y) = e^{F(x, y) - \inf_{\bar{\mathcal{O}}} F(x, y)} \geq 1, $$

where

$$ F(x, y) = a \left( \frac{1}{2} \|y\|^2 + V(x) \right) + b <G(x), y>, \quad a, b > 0. $$

By construction of $G$ we get that for any $(x, y) \in \Gamma_+$, $F(x, y) = F(x, y - (y \cdot n(x)) \cdot n(x))$, so that $\Psi$ satisfies boundary conditions. Also, we can prove that

$$ \begin{aligned} \frac{\mathcal{L}\Psi}{\Psi} &= \mathcal{L}(F) - \frac{1}{2} \|\Sigma(x) \nabla_y F\|^2 \\ &= \mathcal{L}(F) - \frac{1}{2} \|\Sigma(x) (ay + bG(x))\|^2. \end{aligned} $$

Thus

$$ \frac{\mathcal{L}\Psi}{\Psi} \geq -a^2 \|\Sigma(x)y\|^2 - b^2 \|\Sigma(x)G(x)\|^2 - a \left( \frac{1}{2} d \Sigma^2(x) - \langle \Sigma^2(x)y, y \rangle \right) \\ - b \left( \sum_{i=1}^{d} \sum_{j=1}^{d} \partial_{x_j} G_i(x) y_i y_j - \langle y, \Sigma^2(x) G(x) \rangle - \langle \nabla V(x), G(x) \rangle \right). $$
We recall that from Assumption (AΣ) there exist Σ₀, Σ∞ > 0 such that for any x in O, Σ₀ ≤ Σ(x) ≤ Σ∞. We know, as O is bounded and V, G are continuously differentiable, that there exist M > 0 such that for any x ∈ O, <∇V(x), G(x)> ≥ −M and we define M₀ as M₀ := ||A(x)||∞, with A(x) := (∂xⱼGᵢ(x))ᵢⱼ. We then get, similarly to what happens for d = 1,

$$
\begin{align*}
\frac{\mathcal{L}\Psi}{\Psi} &\ge \|y\|^2(-a^2\Sigma_{\infty}^2 + a\Sigma_0^2 - bM_0) - b\Sigma_{\infty}^2\|y\| - (\Sigma_{\infty}^2 b^2 + \frac{a}{2}\Sigma_{\infty}^2 + bM) \\
&=: c\|y\|^2 - b\Sigma_{\infty}^2\|y\| - d.
\end{align*}
$$

Then, arguing as in dimension $d = 1$, we define $K := \bar{\Omega} \times \{y, \|y\| \le L\}$ with $L > L_0 = \frac{b\Sigma_{\infty}^2 + \sqrt{b^2\Sigma_{\infty}^4 + 4cd}}{2c}$, we get Lyapunov inequality (13).

3.4 Exponential convergence to equilibrium

In this section, we state the exponential convergence to equilibrium.

**Theorem 2.** Let $\mathcal{O}$ be a non-empty bounded domain in $\mathbb{R}^d$ such that $\partial\mathcal{O}$ is $C^2$. Let us consider the Langevin dynamics described by (3) with specular reflection at the boundary. We assume that Hypotheses (Av1), (Ac1) and (AΣ) are satisfied and that the reflected-dissipation relation holds true, so that the dynamics may be re-written as (4).

• Let $P_t$ be the associated semi-group. Then there exist a unique invariant probability measure $\pi$ on $D_1 = D \cup \Gamma_+$, constants $D < \infty$, $\rho < 1$ such that for all $t \ge 0$:

$$
\|P_t(z, \cdot) - \pi(\cdot)\|_{\Psi} \le \Psi(z) D \rho^t
$$

with, for any signed measure $\mu$, $\|\mu\|_{\Psi} := \sup_{|g| \le \Psi} |\int \mu(dz)g(z)|$.

• Moreover, $\pi$ admits a density with respect to Lebesgue probability measure on $D_1$ given by:

$$
f_{\beta}(x, y) = \frac{1}{K} \exp(-\beta H(x, y))
$$

with

$$
H(x, y) := \frac{1}{2} \sum_{i=1}^{d} y_i^2 + V(x)
$$

and K the normalizing constant.

**Proof.** In Sections 3.2 and 3.3 we have proven the process $(Z_t, t \ge 0)$ is aperiodic, $\phi_0$-irreducible and as we have exhibited a Lyapunov function, then from [9, Theorem 5.2] it is $\Psi$-uniformly ergodic, that is there exist a unique invariant probability measure $\pi$ on $D_1$, constants $D < \infty$, $\rho < 1$ such that for all $t \ge 0$:

$$
\|P_t(z, \cdot) - \pi(\cdot)\|_{\Psi} \le \Psi(z) D \rho^t
$$

with, for any signed measure $\mu$, $\|\mu\|_{\Psi} := \sup_{|g| \le \Psi} |\int \mu(dz)g(z)|$. Moreover, we know from Section 3.1 that $\pi$ and $\text{Leb}(D_1 \cap \cdot)$ are mutually absolutely continuous. $\square$

**Remark 1.** Note that convergence to equilibrium stated in [15] is different in nature as it is written in terms of a weighted $\mathbb{L}^2$ norm.

**Remark 2.** The proof of the first item in Theorem 2 above is based on the existence of a Lyapunov function compatible with boundary conditions. Therefore to extend this result to more general frameworks, that is frameworks where the fluctuation-dissipation relation does not hold true and with different boundary conditions, the main challenge is to exhibit a Lyapunov function compatible with boundary conditions.
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