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FIRST ORDER MODIFIED EQUATIONS AND
ERROR ANALYSIS FOR STOCHASTIC
HEAVY-BALL SCHEMES

Nassim En-Nebbazi

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Nassim En-Nebbazi. FIRST ORDER MODIFIED EQUATIONS AND ERROR ANALYSIS FOR STOCHASTIC HEAVY-BALL SCHEMES. 2025. hal-05401291

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Preprint submitted on 5 Dec 2025

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# FIRST ORDER MODIFIED EQUATIONS AND ERROR ANALYSIS FOR STOCHASTIC HEAVY-BALL SCHEMES

NASSIM EN-NEBBAZI

**ABSTRACT.** We consider a class of stochastic Heavy-Ball optimization schemes. Assuming that the objective function is strongly convex, we prove weak error estimate which is uniform in time for the error between the solution of the numerical scheme, and the solution of continuous-time modified (or high-resolution) differential equations at first order, with respect to the time-step size. At first order, the modified equation is deterministic. We go beyond existing results where the error estimates have been considered only on finite time intervals and were not uniform in time. This allows us to then provide a rigorous complexity analysis of the method in the large time and small time-step size regimes. We provide numerical experiments to illustrate the convergence result.

## 1. INTRODUCTION

To approximate the minimizers of an objective function $F: \mathbb{R}^d \to \mathbb{R}$, several numerical optimization algorithm have been introduced. One of the most famous numerical schemes is the Heavy-Ball scheme, introduced by Polyak in 1964 (see [17]), defined as follows

$$x_{n+1} = x_n - h\nabla F(x_n) + \beta(x_n - x_{n-1}),$$

where $\beta \in (0, 1)$ is momentum parameter and $h$ is a constant time-step size.

To study the asymptotic behavior of $x_N$ when $N \to \infty$, it is useful to make a connection to differential equations. Indeed, as was done by Su, Boyd and Candès, in [21], within the framework of Nesterov's accelerated gradient scheme, we can deduce a connection between the Heavy-Ball scheme and a second-order differential equation. However, to make this connection, it is important to choose the parameter $\beta = \beta_h = 1 - \alpha\sqrt{h}$ depending on $h$, as described in [22], where $\alpha > 0$ is another momentum parameter, the connection is made with the second-order differential equation

$$x''(t) + \alpha x'(t) + \nabla F(x(t)) = 0.$$

In the case where $\beta_h = \beta \in (0, 1)$ is independent of $h$, we do not obtain the equation above. Indeed, Kovachki and Stuart [14], have shown in that regime that Nesterov's and Heavy-Ball algorithms one related to the continuous-time gradient dynamics

$$x'(t) = -(1 - \beta)^{-1}\nabla F(x(t)).$$

In the sequel, we always only consider the regime $\beta = \beta_h = 1 - \alpha\sqrt{h}$ when the scheme contains a stochastic perturbation. As will be discussed below, we compare this study with that carried out in [6, 15].

In this article, we analyze the behavior of the following stochastic Heavy-Ball optimization scheme

$$\begin{cases} X_{n+1} = X_n - h\nabla F(X_n) + \beta_h(x_n - x_{n-1}) + h\sigma(t_n, X_n)\gamma_{n+1}, & \forall n \in \mathbb{N}, \\ X_0 = x_0, \end{cases}$$

where $F: \mathbb{R}^d \to \mathbb{R}$ is the objective function and $\beta_h = 1 - \alpha\sqrt{h}$ is the momentum parameter.

Connections between (discrete-time) optimization schemes and continuous-time dynamics have been studied in the literature, in deterministic and stochastic settings. Instead of considering leading

<footnote>Key words and phrases. stochastic optimization schemes; modified equations; weak and strong error analysis.</footnote>
order approximations, better understanding on the discrete-time dynamics can be obtained by the identification of more accurate continuous-time dynamics which depend on the time-step size. In the numerical analysis community for ordinary and stochastic differential equations, such more accurate approximations are referred to as modified equations, see for instance [8, 12, 15, 23]. In the optimization community, they are referred to as high-resolution equations, see for instance [7, 18, 19]. In this article, we use the terminology modified equations.

The connection between the discrete and continuous time dynamics motivates the decomposition of the analysis of the asymptotic behavior of optimization schemes into two parts: the analysis of the asymptotic behavior of the continuous-time system, and the analysis of the approximation error between the discrete and continuous time systems. On the one hand, differential and stochastic calculus techniques can be applied to show convergence of the continuous-time dynamics when time $T$ goes to infinity, by the identification of so-called Lyapunov functionals. For instance, if the objective function $F$ is strongly convex, several Lyapunov functions exist in the literature, and applying Grönwall's inequality it is straightforward to show that the solution of the continuous-time Heavy-Ball dynamics converges exponentially fast to the unique minimizer of $F$. More complex Lyapunov functions dedicated to objective functions satisfying weaker conditions have been constructed in the literature see [2, 3, 4]. On the other hand, when standard numerical analysis techniques are applied to study the approximation error between the discrete and continuous time systems, usually the error bounds with respect to the time-step size $h$ depend on the final time $T = Nh$. The dependence with respect to $T$ obtained when discrete Grönwall inequalities are employed may be exponential. Instead, it is desirable to obtain uniform in time error estimates, i.e. with upper bounds independent of the final time $T$. Combining the two types of error bounds then enables to analyze the complexity of the algorithm, i.e. to determine how to choose the parameters $h$ and $N$ of the algorithm to ensure that the error is below any arbitrary threshold.

To establish an asymptotic study of the previous scheme, we use the equivalent formulation

$$ \begin{cases} V_{n+1} = (1 - \alpha\tau)V_n - \tau\nabla F(X_n) + \tau\sigma(s_n\tau, X_n)\gamma_{n+1}, & \forall n \in \mathbb{N}, \\ X_{n+1} = X_n + \tau V_{n+1}, & \forall n \in \mathbb{N}. \end{cases} $$

where $\tau = \sqrt{h}$. We will discuss this change of time in Sections 2.2, which is already known in the deterministic case for Nesterov's algorithm, see for instance [21]. Compared with the deterministic version above, the additional term $\tau\sigma(t_n, X_n)\gamma_{n+1}$ describes random perturbations, where $(\gamma_n)_{n \ge 1}$ are independent standard $\mathbb{R}^d$-valued Gaussian random variables. We assume that the objective function $F$ is strongly convex, it thus admits a unique minimizer denoted by $x^*$. Our objective is to analyze the convergence of $X_N$ to $x^*$ when $N \to \infty$ and $\tau \to 0$. The stochastic Heavy-Ball optimization scheme above has been considered previously in the literature, see for instance [15] for the proof of weak error estimates. Moreover, Gadat, Panloup and Saadane [6] have developed convergence criteria and strong and weak error bounds for another stochastic Heavy-Ball algorithm, different from the one presented above. This algorithm follows from the deterministic continuous Heavy-Ball dynamics, developed in [10, 11]. Let us recall that when dealing with approximation of stochastic systems and associated error bounds, one may consider either strong approximation and error bounds, concerned with $E[\|X_N - x^*\|]$ or with $(E[\|X_N - x^*\|^2])^{1/2}$, or weak approximation and error bounds, concerned with $|E[\varphi(X_N)] - \varphi(x^*)|$. We refer for instance to the reference books [13, 16] on numerical methods for stochastic (partial) differential equations.

The objective of this article is to fill an important gap in the analysis of the stochastic Heavy-Ball optimization scheme above: our main contribution is to establish uniform in time weak error estimates between the solution $X_N$ of the stochastic optimization scheme and the solution at time $T = N\tau$ of associated modified equations which are either deterministic and stochastic. To establish these bounds, we impose the condition $\alpha > \frac{L_F}{\sqrt{\mu}}$ where $L_F$ and $\mu$ are defined in Assumption 2.1
in Section 2.3. For more details on this condition, we refer to the proof of Lemma 5.1 given in Appendix A. This is a major improvement compared with previous works such as [15] where the error bounds are not uniform with respect to time T and are thus not relevant for the analysis of the algorithm in the large time regime. As explained above, combined with results on the large time behavior of the continuous-time associated deterministic and stochastic dynamics, we are able to establish complexity results for the stochastic Heavy-Ball optimization scheme. We consider a deterministic modified equations, which provide first order approximation with respect to $\tau$.

For the first order approximation, we exploit the connection between the stochastic Heavy-Ball optimization scheme and the deterministic Heavy-Ball system, i.e. the ordinary differential equation

$$ \begin{cases} \frac{d}{ds} \mathcal{X}^0(s) = \mathcal{V}^0(s), & \forall s \ge 0, \\ \frac{d}{ds} \mathcal{V}^0(s) = -\alpha \mathcal{V}^0(s) - \nabla F(\mathcal{X}^0(s)), & \forall s \ge 0. \end{cases} $$

As already recalled above, assuming that $F$ is strongly convex, $\mathcal{X}^0(T)$ converges to $x^*$ exponentially fast when $T \to \infty$. Our first main result is the proof of weak error estimates

$$ |\mathbb{E}[\varphi(X_N)] - \varphi(x^*)| = O(e^{-\nu(\alpha, \mu, L_F)N\tau} + \tau), $$

see Theorem 1 for the precise statement. This follows from uniform in time weak error estimates

$$ \sup_{N \in \mathbb{N}} |\mathbb{E}[\varphi(X_N)] - \varphi(\mathcal{X}^0(N\tau))| = O(\tau). $$

Moreover, applying the weak error estimates with $\varphi = F$ and applying the following inequality

$$ \frac{\mu}{2} \|X_N - x^*\|^2 \le F(X_N) - F(x^*), $$

this follow from $\mu$-convexity condition (9) of $F$, see Assumptions 2.1 for more details. We also obtain strong error estimates

$$ (\mathbb{E}[\|X_N - x^*\|^2])^{\frac{1}{2}} = O(e^{-\nu(\alpha, \mu, L_F)N\tau} + \tau^{\frac{1}{2}}). $$

This is not a standard result in stochastic numerics, since usually weak error bounds may deduced from strong error bounds. However, this has already been done in the context of stochastic gradient descent, see [5]. To prove weak error estimates, which are uniform in time, we revisit the arguments usually employed to study approximations of invariant distributions for stochastic differential equations, in particular we need to study the regularity properties of an auxiliary mapping $u^0$ (defined by Equation (37)), which is the solution to the (Kolmogorov) partial differential equation (41) associated with the deterministic gradient system. Precisely we prove the exponential decay with respect to $s$ for the first and second order spatial partial derivatives $Du^0(s, x, v)$ and $D^2u^0(s, x, v)$, see Lemma 5.1. We also obtained a modified second-order equation in $\tau$, different for those proposed in the literature (see [15]). However, complexity analysis has shown that there is no significant gain compared to the one obtained, contrary to what is shown in the context of stochastic gradient descent [5], where the modified second-order equation provides a real gain compared to the first order. We have written a follow-up to this article [9] proposing two distinct approaches to solving this problem.

The considered stochastic Heavy-Ball optimization scheme can be interpreted as a numerical scheme applied to the stochastic differential equation

$$ \begin{cases} d\mathcal{X}^\tau(s) = \mathcal{V}^\tau(s), & \forall s \ge 0, \\ d\mathcal{V}^\tau(s) = -\alpha\mathcal{V}^\tau(s)ds - \nabla F(\mathcal{X}^\tau(s))ds + \sqrt{\tau\sigma(s\tau, \mathcal{X}^\tau(s))}dB(s), & \forall s \ge 0. \end{cases} $$
which can be interpreted as a modified equation since it depends on the time-step size $\tau$ driven by a Brownian motion. However that simple connection is not sufficient to obtain higher order approximation with respect to $\tau$.

We provide numerical experiments to illustrate the strong error estimate obtained in Theorem 1 and also to compare this study with that carried out in [5] in the context of stochastic gradient descent. We refer to Section 3 for a detailed discussion of the computational complexity of the stochastic optimisation scheme.

Let us mention several directions for future works. Study the modified equations that yield order two with respect to $\tau$; this will be developed further in [9]. It would be interesting to study the behavior of the stochastic Heavy-Ball in the cases where $\alpha \le \frac{L_F}{\sqrt{\mu}}$. Also it is interesting to study the behavior of the stochastic Heavy-Ball in the case where $\beta_h = \beta \in (0, 1)$ constant and to compare with the study of Kovachki and Stuart in [14] in the deterministic setting. The analysis may also be extended in an infinite dimensional setting.

The article is organized as follows. Section 2 is devoted to the description of the setting and the statement of the fundamental assumptions on the objective function $F$ and the diffusion coefficient $\sigma$. The main results of this article, Theorem 1, is stated in Section 3, which also contains numerical experiments and a discussion on the computational complexity. We compare this to the study of stochastic gradient descent developed in [5]. Section 4 presents some results on the stochastic Heavy-Ball optimization scheme, including uniform moment bounds given by Lemma 4.1. The proof of the first order approximation result, Theorem 1, is provided in Section 5.3. Appendix 5.4 contains the proof of technical but fundamental results, on the regularity properties of the solutions of the Kolmogorov equations associated with the first order modified equation.

## 2. SETTING

### 2.1. Notation.
We denote by $\mathbb{N}^* = \{1, \dots\}$ and $\mathbb{N} = \{0, 1, \dots\}$ the sets of positive and nonnegative integers respectively. Given $d \in \mathbb{N}^*$ be an arbitrary positive integer, the Euclidean space $\mathbb{R}^d$ is equipped with the standard inner product and norm denoted by $\langle \cdot, \cdot \rangle$ and $\| \cdot \|$ respectively. Let $\mathcal{M}_d(\mathbb{R})$ denote the space of real-valued square matrices with $d$ rows and columns. The identity matrix is denoted by $I_d$. For any matrix $M \in \mathcal{M}_d(\mathbb{R})$, $M^*$ is the transpose of $M$ and Tr($M$) is the trace of $M$. The space $\mathcal{M}_d(\mathbb{R})$ is equipped with the norm $\| \cdot \|$ defined by

$$ \|M\| = \left( \sum_{1 \le i,j \le d} |M_{i,j}|^2 \right)^{\frac{1}{2}}, \quad \forall M \in \mathcal{M}_d(\mathbb{R}). $$

If $M, N \in \mathcal{M}_d(\mathbb{R})$ are two matrices, let

$$ M : N = \sum_{1 \le i,j \le d} M_{i,j} N_{i,j} = \text{Tr}(MN^*). $$

If $\phi : \mathbb{R}^d \to \mathbb{R}$ is a real-valued function of class $C^1$, then $\nabla\phi(x) = (\partial_{x_i}\phi(x))_{1\le i\le d} \in \mathbb{R}^d$ denotes the gradient of $\phi$ at $x \in \mathbb{R}^d$. Similarly, if $\phi : \mathbb{R}^d \to \mathbb{R}$ is a function of class $C^2$, then $\nabla^2\phi(x) = (\partial_{x_i}\partial_{x_j}\phi(x))_{1\le i,j\le d} \in \mathcal{M}_d(\mathbb{R})$ denotes the Hessian matrix of $\phi$ at $x \in \mathbb{R}^d$.

Given $d_1, d_2 \in \mathbb{N}^*$, if $\phi : \mathbb{R}^{d_1} \times \mathbb{R}^{d_2} \to \mathbb{R}$ is a real-valued function of class $C^1$, then $\nabla_x\phi(x, y) = (\partial_{x_i}\phi(x, y))_{1\le i\le d_1} \in \mathbb{R}^{d_1}$ and $\nabla_y\phi(x, y) = (\partial_{y_i}\phi(x, y))_{1\le i\le d_1} \in \mathbb{R}^{d_2}$ denote the gradient of $\phi$ with respect to $x \in \mathbb{R}^{d_1}$ and to $y \in \mathbb{R}^{d_2}$ respectively. Similarly, if $\phi : \mathbb{R}^{d_1} \times \mathbb{R}^{d_2} \to \mathbb{R}$ is a function of class $C^2$, then $\nabla_x^2\phi(x, y) = (\partial_{x_i}\partial_{x_j}\phi(x, y))_{1\le i,j\le d_1} \in \mathcal{M}_{d_1}(\mathbb{R})$, $\nabla_y^2\phi(x, y) = (\partial_{y_i}\partial_{y_j}\phi(x, y))_{1\le i,j\le d_2} \in \mathcal{M}_{d_2}(\mathbb{R})$ and $\nabla_{xy}^2\phi(x, y) = \nabla_{yx}^2\phi(x, y)^* = (\partial_{x_i}\partial_{y_j}\phi(x, y))_{1\le i\le d_1, 1\le j\le d_2} \in \mathcal{M}_{d_1,d_2}(\mathbb{R})$ denotes the Hessian matrix of $\phi$ at $x \in \mathbb{R}^{d_1}$, $y \in \mathbb{R}^{d_2}$ and $(x, y) \in \mathbb{R}^{d_1} \times \mathbb{R}^{d_2}$ respectively.
Given $m_1, m_2, m \in \mathbb{N}^*$, if $\phi: \mathbb{R}^{m_1} \to \mathbb{R}^{m_2}$ is a mapping of class $C^m$, then $D^m\phi(x).(k_1, \dots, k_m) \in \mathbb{R}^{m_2}$ denotes the differential of order $m$ of $\phi$ at $x \in \mathbb{R}^{m_1}$ in the directions $k_1, \dots, k_m \in \mathbb{R}^{m_1}$. In the case $m_1 = d$ and $m_2 = 1$, if $\phi: \mathbb{R}^d \to \mathbb{R}$ is of class $C^2$, with the notation introduced above one has

$$D\varphi(x).(k) = \langle \nabla\varphi(x), k \rangle, \quad D^2\varphi(x).(k_1, k_2) = \langle \nabla^2\varphi(x)k_1, k_2 \rangle, \quad \forall x \in \mathbb{R}^d \forall k, k_1, k_2 \in \mathbb{R}^d.$$

For any $m \in \mathbb{N}^*$, if $\phi: \mathbb{R}^d \to \mathbb{R}$ is of class $C^m$, then set

$$\|\varphi\|_m = \sup_{x \in \mathbb{R}^d} \sup_{k_1, \dots, k_m \in \mathbb{R}^d \setminus \{0\}} \frac{|D^m \varphi(x).(k_1, \dots, k_m)|}{\|k_1\| \dots \|k_m\|}.$$

In the sequel we consider functions $\varphi$ of class $C^2$ with bounded second order derivative. Note that its first order derivative satisfies the following upper bound: for all $x \in \mathbb{R}^d$ and $k \in \mathbb{R}^d$ one has

$$(1) \qquad |D\varphi(x).k| \le \left( \|\nabla\varphi(x^*)\| + \|\varphi\|_2 \|x - x^*\| \right) \|k\|.$$

Finally, let $(B(s))_{s \ge 0}$ be a standard $\mathbb{R}^d$-valued Wiener process defined on a probability space denoted by $(\Omega, \mathcal{F}, \mathbb{P})$, and which is adapted to a filtration $(\mathcal{F}_s)_{s \ge 0}$ which satisfies the usual conditions. If $X$ is an integrable real-valued random variable, its expectation is denoted by $\mathbb{E}[X]$. In addition, given a $\sigma$-field $\mathcal{G} \subset \mathcal{F}$, $\mathbb{E}[X|\mathcal{G}]$ denotes the conditional expectation of $X$ given $\mathcal{G}$.

In the analysis, the value of (deterministic) positive real numbers $C \in (0, \infty)$ may vary from line to line. Unless specified, the value of $C$ is uniform with respect to variables such as time $s \in \mathbb{R}^+ = [0, \infty)$, the state $x, v \in \mathbb{R}^d$ or the time-step size $h$.

## 2.2. The stochastic Heavy-Ball optimization scheme.
Let $x_0 \in \mathbb{R}^d$ be arbitrary initial values assumed to be deterministic. Considering $\mathcal{F}_0$-measurable random initial values satisfying appropriate integrability properties could be considered with straightforward minor modifications of the statements.

Let $h > 0$ denote the time-step size of the considered stochastic Heavy-Ball optimization scheme. Without loss of generality, it is assumed that $h \in (0, h_{\max})$ for some positive real number $h_{\max}$. For any nonnegative integer $n \in \mathbb{N}$, set $t_n = nh$. Let $(\gamma_n)_{n \in \mathbb{N}^*}$ be a sequence of independent standard centered $\mathbb{R}^d$-valued Gaussian random variables.

The objective of this article is to study the behavior of the stochastic Heavy-Ball optimization scheme

$$(2) \qquad \begin{cases} X_{n+1} = X_n - h\nabla F(X_n) + \beta(X_n - X_{n-1}) + h\sigma(t_n, X_n)\gamma_{n+1}, & \forall n \in \mathbb{N}, \\ X_0 = x_0, \end{cases}$$

As stated in Section 1, in order to study the asymptotic behavior of $(X_n)_{n \in \mathbb{N}}$, it is important to establish a connection to a differential equation. As mentioned in [22], to interpret this scheme an approximation of a stochastic (or deterministic) differential equation, see equations (17) and (18), it is essential to take $\beta = \beta_h = 1 - \alpha\sqrt{h}$, with $\alpha > 0$. It is assumed that $h_{\max} \le \frac{1}{\alpha^2}$, thus $\beta_h > 0$ for all $h \in (0, h_{\max})$. Moreover the mapping $F: \mathbb{R}^d \to \mathbb{R}$ is the objective function and the mapping $\sigma: [0, \infty) \times \mathbb{R}^d \to \mathcal{M}_d(\mathbb{R})$ is the diffusion coefficient. We refer to Section 2.3 for precise assumptions on $F$ en $\sigma$.

To study the scheme, we introduce the change of variables: for all $n \in \mathbb{N}$, set $V_{n+1} = \frac{X_{n+1} - X_n}{\sqrt{h}}$ with $V_0 = v_0 \in \mathbb{R}^d$. Moreover it is convenient to define an auxiliary time-step size $\tau = \sqrt{h}$. For any nonnegative integer $n \in \mathbb{N}$, set $s_n = \frac{t_n}{\tau}$. Using the notation $\sigma_\tau(s, x) = \sigma(s\tau, x)$ for all $s \in \mathbb{R}^+$ and $x \in \mathbb{R}^d$, the scheme (2) can be written as follow

$$(3) \qquad \begin{cases} X_{n+1} = X_n + \tau V_{n+1}, & \forall n \in \mathbb{N}, \\ V_{n+1} = (1 - \alpha\tau)V_n - \tau\nabla F(X_n) + \tau\sigma_\tau(s_n, X_n)\gamma_{n+1}, & \forall n \in \mathbb{N}, \\ (X_0, V_0) = (x_0, v_0). \end{cases}$$
With this formulation, we notice that the processes $(X_{n+1})$ and $(V_{n+1})_{n \in \mathbb{N}}$ depend on $(\gamma_{n+1})_{n \in \mathbb{N}}$. To simplify the proof of Theorem 1, we introduce the following change of variables.

$$ (\bar{X}_n, \bar{V}_n) = (X_n, V_{n+1}), \quad \forall n \in \mathbb{N}. $$

This transformation leads to the following formulation

$$ (4) \qquad \begin{cases} \bar{X}_{n+1} = \bar{X}_n + \tau \bar{V}_n, & \forall n \in \mathbb{N}, \\ \bar{V}_{n+1} = (1 - \alpha \tau) \bar{V}_n - \tau \nabla F(\bar{X}_{n+1}) + \tau \sigma_{\tau}(s_{n+1}, \bar{X}_{n+1}) \gamma_{n+2}, & \forall n \in \mathbb{N}, \\ (\bar{X}_0, \bar{V}_0) = (x_0, V_1) = (x_0, (1 + \alpha \tau)v_0 - \tau \nabla F(x_0) + \tau \sigma(0, x_0)\gamma_1). \end{cases} $$

This change of variables does not affect our study, because we are interested to obtain weak error estimates on $(X_n)_{n \in \mathbb{N}}$ and we have: $X_n = \bar{X}_n$ for all $n \in \mathbb{N}$. Thus the sequences of random variables $(\gamma_{n+2})_{n \in \mathbb{N}}$ and $(\gamma_{n+1})_{n \in \mathbb{N}}$ are equal in distribution, and as a result, the scheme (4) can be interpreted using the alternative formulation

$$ (5) \qquad \begin{cases} \bar{X}_{n+1} = \bar{X}_n + \tau \bar{V}_n, & \forall n \in \mathbb{N}, \\ \bar{V}_{n+1} = (1 - \alpha \tau) \bar{V}_n - \tau \nabla F(\bar{X}_{n+1}) + \sqrt{\tau \sigma_{\tau}}(s_{n+1}, \bar{X}_{n+1}) \gamma_{n+1}, & \forall n \in \mathbb{N}, \\ (\bar{X}_0, \bar{V}_0) = (x_0, V_1). \end{cases} $$

With this change of variables, we now obtain that only $(\bar{V}_{n+1})_{n \in \mathbb{N}}$ depends on $(\gamma_{n+1})_{n \in \mathbb{N}}$. Let $\Delta B_{n+1} = B(s_{n+1}) - B(s_n)$ denote the increments of the Wiener process on the interval $[s_n, s_{n+1}]$. Recall that $(\tau^{-1/2} \Delta B_{n+1})_{n \ge 0}$ are independent standard centered Gaussian $\mathbb{R}^d$-valued random variables. Thus the sequences of random variables $(\sqrt{\tau} \gamma_n)_{n \in \mathbb{N}^*}$ and $(\Delta B_n)_{n \in \mathbb{N}^*}$ are equal in distribution, and as a result, the scheme (5) can be interpreted using the alternative formulation

$$ (6) \qquad \begin{cases} \bar{X}_{n+1} = \bar{X}_n + \tau \bar{V}_n, & \forall n \in \mathbb{N}, \\ \bar{V}_{n+1} = (1 - \alpha \tau) \bar{V}_n - \tau \nabla F(\bar{X}_{n+1}) + \sqrt{\tau \sigma_{\tau}}(s_{n+1}, \bar{X}_{n+1}) \Delta B_{n+1}, & \forall n \in \mathbb{N}, \\ (\bar{X}_0, \bar{V}_0) = (x_0, V_1). \end{cases} $$

It is straightforward to check that there exists $C \in (0, \infty)$ such that

$$ (7) \qquad \mathbb{E}[\|\bar{X}_0\|^2] + \mathbb{E}[\|\bar{V}_0\|^2] \le C(1 + \|x_0\|^2 + \|v_0\|^2), $$

this will allow us to obtain error estimates in Theorem 1 that depend on $(x_0, v_0) \in \mathbb{R}^d \times \mathbb{R}^d$.

The formulation (6) suggests to interpret the scheme as a numerical method applied to a stochastic differential equation, as will be explained below.

### 2.3. Assumptions.
This section provides assumptions on the objective function $F$ and on the diffusion coefficient $\sigma$, which are necessary to justify rigorously the main properties of the stochastic Heavy-Ball optimization scheme in the large time and small time-step regime.

Let us first state the assumptions on the objective function $F$.

**Assumption 2.1 (Assumptions on the objective function $F$).** The mapping $F: \mathbb{R}^d \to \mathbb{R}$ is of class $C^3$, and it satisfies the following properties:

(i) $\nabla F: \mathbb{R}^d \to \mathbb{R}^d$ is globally Lipschitz continuous, i.e. there exists a positive real number $L_F \in (0, \infty)$ such that one has

$$ (8) \qquad \|\nabla F(x_2) - \nabla F(x_1)\| \le L_F \|x_2 - x_1\|, \quad \forall x_1, x_2 \in \mathbb{R}^d. $$

(ii) $F$ is a $\mu$-convex function, i.e. there exists a positive real number $\mu \in (0, \infty)$ such that one has

$$ (9) \qquad \langle \nabla F(x_2) - \nabla F(x_1), x_2 - x_1 \rangle \ge \mu \|x_2 - x_1\|^2, \quad \forall x_1, x_2 \in \mathbb{R}^d. $$

(iii) The third order derivative of $F$ is bounded.
**Example 2.1.** Assumption 2.1 is satisfied if F is defined as

$$F(x) = \frac{\mu}{2} \|x\|^2 + \epsilon G(x), \quad \forall x \in \mathbb{R}^d,$$

where $\mu > 0$ and the mapping $G: \mathbb{R}^d \to \mathbb{R}$ is of class $C^3$ with bounded first, second and third order derivatives, and for sufficiently small $\epsilon$.

Let us state several elementary consequences of Assumption 2.1.

When the objective function F satisfies Assumption 2.1, it is straightforward to check that $\min(F)$ exists and is attained for a unique minimizer denoted by $x^*$. Moreover the minimizer $x^*$ is characterized by the first-order optimality condition $\nabla F(x^*) = 0$.

Note that the second order derivative $\nabla^2 F$ satisfies the property

$$(10) \qquad \mu \|k\|^2 \le \langle \nabla^2 F(x) k, k \rangle \le L_F \|k\|^2, \quad \forall k \in \mathbb{R}^d \ \forall x \in \mathbb{R}^d,$$

where the upper bound follows from the Lipschitz property (8) of $\nabla F$ and the lower bound is a consequence of the $\mu$-convexity property (9). In addition, the global Lipschitz continuity condition (8) implies that $\nabla F$ has at most polynomial growth: for all $x \in \mathbb{R}^d$,

$$(11) \qquad \|\nabla F(x)\| \le L_F \|x - x^*\|.$$

Finally, note that combining the $\mu$-convexity condition (9) and the optimality condition $\nabla F(x^*) = 0$, a second-order Taylor expansion yields the following lower bound on the residual

$$(12) \qquad F(x) - F(x^*) \ge \frac{\mu}{2} \|x - x^*\|^2, \quad \forall x \in \mathbb{R}^d.$$

Similarly, one has an upper bound on the residual:

$$(13) \qquad F(x) - F(x^*) \le \frac{L_F}{2} \|x - x^*\|^2, \quad \forall x \in \mathbb{R}^d.$$

Let us now describe the assumptions on the diffusion coefficient $\sigma$.

**Assumption 2.2** (Assumptions on the diffusion coefficient $\sigma$). The mapping $\sigma: [0, \infty) \times \mathbb{R}^d \to \mathcal{M}_d(\mathbb{R})$ satisfies the following properties:

(i) The mapping $\sigma: [0, \infty) \times \mathbb{R}^d \to \mathcal{M}_d(\mathbb{R})$ is continuous.

(ii) For all $s \in [0, \infty)$, the mapping $\sigma(s, \cdot) : \mathbb{R}^d \to \mathcal{M}_d(\mathbb{R})$ is globally Lipschitz continuous and grows at most linearly : for all $s \ge 0$, there exists $\zeta(s) \in [0, \infty)$ such that one has

$$(14) \qquad \|\sigma(s, x_2) - \sigma(s, x_1)\| \le \zeta(s) \|x_2 - x_1\|, \quad \forall x_1, x_2 \in \mathbb{R}^d,$$

$$(15) \qquad \sup_{x \in \mathbb{R}^d} \frac{\|\sigma(s, x)\|}{1 + \|x - x^*\|} \le \zeta(s).$$

(iii) The mapping $\zeta: [0, \infty) \to [0, \infty)$ is assumed to be bounded. Let $\|\zeta\|_{\infty} = \sup_{s \ge 0} |\zeta(s)|$.

Finally, let us introduce the auxiliary mapping $a_\tau: [0, \infty) \times \mathbb{R}^d \to \mathcal{M}_d(\mathbb{R})$ defined by

$$(16) \qquad a_\tau(s, x) = \sigma_\tau(s, x) \sigma_\tau(s, x)^*, \quad \forall s \ge 0, \ x \in \mathbb{R}^d.$$

2.4. First order approximate ordinary differential equation. Let us first study the deterministic situation: when $\sigma = 0$, the scheme (5) can be considered as a numerical scheme of the Heavy-Ball system

$$(17) \qquad \begin{cases} \frac{d}{ds} \chi^0(s) = \mathcal{V}^0(s), & \forall s \ge 0, \\ \frac{d}{ds} \mathcal{V}^0(s) = -\alpha \mathcal{V}^0(s) - \nabla F(\chi^0(s)), & \forall s \ge 0, \\ (\chi^0(0), \mathcal{V}^0(0)) = (\bar{X}_0, \bar{V}_0). \end{cases}$$
The study of the Heavy-Ball system (17) is well known (see for instance [1, 2, 3, 4, 17, 20]). By using the $\mu$-convexity condition from Assumption 2.1 on the objective function, and due to first-order optimality condition $\nabla F(x^*) = 0$ the process $(\mathcal{X}^0(s), \mathcal{V}^0(s))_{s \ge 0}$ converge exponentially fast to $(x^*, 0)$ where $x^*$ is the unique minimizer of the objective function $F$ in the large time regime $s \to \infty$, for any initial value $\bar{X}_0, \bar{V}_0 \in L^2(\Omega, \mathbb{R}^d)$, and that the rate of convergence is given by the parameters $\mu, \alpha$ and $L_F$ (see Proposition 5.1 for details). Note that the exponential convergence obtained in Proposition 5.1 is not optimal (see for instance [17] for better rates under the $\mu$-convexity condition (9) and $L_F$-Lipschitz condition (11), and [2, 3, 4, 20] for better rates with weaker assumptions on the objective function). Nevertheless, this is not important in this context, since the value of the rate convergence does not play a significant role on the analysis of the error and of the complexity.

The equation (17) is first-order approximations, in the sense of weak convergence, of the stochastic Heavy-Ball optimization scheme (3) in the small time-step size limit $\tau \to 0$, for arbitrary fixed value of $T \in (0, \infty)$. More precisely, for any time $T \in (0, \infty)$ and any sufficiently smooth function $\varphi: \mathbb{R}^d \to \mathbb{R}$, there exists $C(T, x_0, v_0, \varphi) \in (0, \infty)$ such that for all $\tau \in (0, \tau_{\max})$ one has

$$ \sup_{n\tau \le T} |\mathbb{E}[\varphi(X_n)] - \mathbb{E}[\varphi(\mathcal{X}^0(n\tau))]| \le C(T, x_0, v_0, \varphi)\tau. $$

One of the objectives of this work is to show that under appropriate conditions the weak error estimates are uniform with respect to $T \in (0, \infty)$, which is of tremendous importance in the context of optimization where one needs to consider the large time regime $T \to \infty$. In other words, it will be proved that the weak error estimates above hold, with $\sup_{T \in (0, \infty)} C(T, x_0, v_0, \varphi) < \infty$, we refer to Theorem 1 for a more precise statement.

**Remark 2.1.** Let us now take into account the stochastic perturbation. The stochastic Heavy-Ball optimization scheme (6), can be interpreted as an approximation with time-step size $\tau$ of the solution of following the stochastic differential

$$ (18) \qquad \begin{cases} d\mathcal{X}^\tau(s) = \mathcal{V}^\tau(s) \, , \quad \forall s \ge 0, \\ d\mathcal{V}^\tau(s) = -\alpha \mathcal{V}^\tau(s)ds - \nabla F(\mathcal{X}^\tau(s))ds + \sqrt{\tau}\sigma_\tau(s, \mathcal{X}^\tau(s))dB(s) \, , \quad \forall s \ge 0, \\ (\mathcal{X}^\tau(0), \mathcal{V}^\tau(0)) = (\bar{X}_0, \bar{V}_0) \, . \end{cases} $$

Note that in order to understand the behavior of the numerical scheme (3) considering the stochastic differential equation (18) instead of the Heavy-Ball system (17) is not relevant since the weak order of convergence is equal to 1 even when the noise is introduced.

### 3. MAIN RESULTS

This section is devoted to the statement of the main result of this article in Section 3.1. We then discuss the relevance of the results through numerical experiments in Section 3.2 and complexity analysis in Section 3.3. The proof of Theorem 1 is technical and requires some auxiliary results, they are postponed to Section 5.

#### 3.1. Error estimates. The main result is Theorem 1 below.

**Theorem 1.** Assume that the objective function $F$ and that the diffusion coefficient $\sigma$ satisfy the Assumption 2.1 and the Assumption 2.2 respectively. Assume also $\alpha > \frac{L_F}{\sqrt{\mu}}$.

Let us define $\nu(\alpha, \mu, L_F) \in (0, \infty)$, depending on $\alpha, \mu$ and $L_F$, as follow

$$ (19) \qquad \nu(\alpha, \mu, L_F) = \frac{\alpha \min(\mu, 1)}{2 \max(L_F + \alpha^2, \frac{3}{2})}. $$
There exist positive real numbers $\bar{\tau} \in (0, \tau_{\max})$ and $C \in (0, \infty)$ such that, for any initial values $x_0, v_0 \in \mathbb{R}^d$, for all $N \in \mathbb{N}^*$ and $\tau \in (0, \bar{\tau})$, the following error estimates are satisfied:

* weak error estimates: for any mapping $\varphi : \mathbb{R}^d \to \mathbb{R}$ of class $C^2$ with bounded second order derivative, one has

$$ (20) \qquad \left|\mathbb{E}[\varphi(X_N)] - \varphi(x^*)\right| \le C_1(\varphi, x_0, v_0)\left(e^{-\nu(\alpha, \mu, L_F)N\tau} + \tau\right), $$
where the positive real numbers $C_1(\varphi, x_0, v_0)$ is given by

$$ (21) \qquad C_1(\varphi, x_0, v_0) = C(\|\nabla\varphi(x^*)\| + \|\varphi\|_2^2)(1 + \|x_0 - x^*\|^3 + \|v_0\|^3), $$

* error estimates on the expected residual: one has

$$ (22) \qquad \mathbb{E}[F(X_N)] - F(x^*) \le C(1 + \|x_0 - x^*\|^3 + \|v_0\|^3)(e^{-2\nu(\alpha, \mu, L_F)N\tau} + \tau), $$

* strong error estimates: one has

$$ (23) \qquad \mathbb{E}[\|X_N - x^*\|] \le (\mathbb{E}[\|X_N - x^*\|^2])^{\frac{1}{2}} \le C(1 + \|x_0 - x^*\|^{\frac{3}{2}} + \|v_0\|^{\frac{3}{2}})(e^{-\nu(\alpha, \mu, L_F)N\tau} + \tau^{\frac{1}{2}}). $$

In the main result three types of errors are considered. Weak error estimates deal with $\mathbb{E}[\varphi(X_N)] - \varphi(x^*)$ for an appropriate class of functions $\varphi$, errors on the expected residual are concerned with $\mathbb{E}[F(X_N)] - F(x^*)$, and strong error estimates give upper bounds on $\mathbb{E}[\|X_N - x^*\|]$ and on $(\mathbb{E}[\|X_N - x^*\|^2])^{\frac{1}{2}}$. By the Cauchy-Schwarz inequality, one has $\mathbb{E}[\|X_N - x^*\|] \le (\mathbb{E}[\|X_N - x^*\|^2])^{\frac{1}{2}}$. The main task is to prove the weak error estimates. The error bounds on the residual will be obtained from weak error bounds when $\varphi = F$. Furthermore, the strong error estimates follow from the error bounds on the residual by a straightforward application of the inequality (12).

Error bounds are made of two contributions. The first contribution is related to the large time behavior of solution to the associated continuous-time dynamic (given by the ordinary differential equation (17)). The second contribution is related to a time-stepping error as $\tau$ goes to 0, when comparing the solutions to the associated continuous-time dynamics and the stochastic optimization scheme (3).

Note that, we assume that $\alpha > \frac{L_F}{\sqrt{\mu}}$. This assumption is necessary to obtain error estimates uniform with respect to $T \in (0, \infty)$ (see the proof of Lemma 5.1 in Appendix A)

We refer to Section 5 for a presentation of the strategy of the proof and to Section 5.4 for the details of the proofs of each of the error estimates.

The time-stepping error is of size $\tau$ in the weak error estimates (20) and the estimates on the expected residual (22), and of size $\tau^{\frac{1}{2}}$ in the strong error estimates (23).

## 3.2. Numerical experiments.
The objective of this section is investigate the behavior of the stochastic optimization scheme (3) and to verify that the terms appearing in the upper bounds from Theorem 1 are relevant. We also compare the stochastic Heavy-Ball scheme (3) to the stochastic gradient descent scheme studied in [5]. We focus only on strong error estimates (23), which may be rewritten as

$$ (24) \qquad (\mathbb{E}[\|X_N - x^*\|^2])^{\frac{1}{2}} \le C(x_0, v_0) (e^{-\nu(\alpha, \mu, L_F)T} + \sqrt{\tau}), $$

using the notation $T = N\tau$.

The numerical experiments are performed in dimension $d = 1$. In addition, the noise is additive, i.e. $\sigma$ only depends only on time $s$: one has

$$ \sigma(s, x) = \sigma(s), \quad \forall s \ge 0, x \in \mathbb{R}. $$
### 3.2.1. Quadratic objective function.
Let us assume that the objective function is given by

$$F(x) = \frac{x^2}{2},$$

in this case we have $\mu = L_F = 1$. The unique minimizer of $F$ is $x^* = 0$. In the numerical experiments, the initial value is $(x_0, v_0) = (1, 0)$ and we take $\alpha = 2$ to ensure the assumption $\alpha > \frac{L_F}{\sqrt{\mu}}$. To estimate the strong error $(\mathbb{E}[\|X_N - x^*\|^2])^{\frac{1}{2}}$, Monte Carlo averages over 100 independent realizations is computed. The time-step size $\tau$ takes values in $\{1.0 \ 10^{-2}, 5.0 \ 10^{-3}, 2.5 \ 10^{-3}, 1.25 \ 10^{-3}, 6.25 \ 10^{-4}, 3.125 \ 10^{-4}, 1.5625 \ 10^{-4}, 7.8125 \ 10^{-5}, 3.90625 \ 10^{-5}\}$.

The error is represented in logarithmic scale and a reference line with slope $1/2$ is displayed.

We consider different cases: $\sigma(s) = e^{-s}$, $\sigma(s) = \frac{1}{1+s}$ and $\sigma(s) = 1$. Figure 1 illustrates the dependence of the strong error with respect to $\tau$ for different values of the final time $T \in \{2, 4, 8, 16, 32, 64\}$. For small values of $T$ ($T = 2$ and $T = 4$), the error does not depend on $\tau$. This behavior corresponds to the observation of the error term $e^{-\nu(\alpha, \mu, L_F)T}$ in the right-hand side of (24). For large values of $T$ ($T = 32$ and $T = 64$), the error depends on $\tau$, but not in $T$, with rate $1/2$. This behavior corresponds to the observation of the error term $\sqrt{\tau}$ in the right-hand side of (24).

<figure><img src="image_3.png" /><figcaption>FIGURE 1. Illustration of strong error estimates for the quadratic objective function $F$, and with: $\sigma(s) = e^{-s}$ (top left), $\sigma(s) = \frac{1}{1+s}$ (top right) and $\sigma(s) = 1$ (bottom). The values of the final time are $T \in \{2, 4, 8, 16, 32, 64\}$.</figcaption></figure>

### 3.2.2. Comparison of Stochastic Heavy-Ball and Stochastic Gradient Descent.
The behavior of Figure 1 is very different from that established in [5] in the context of stochastic gradient descent. Figure 2 illustrates the strong errors $(\mathbb{E}[\|X_N^{\text{SHB}} - x^*\|^2])^{\frac{1}{2}}$ and $(\mathbb{E}[\|X_N^{\text{SGD}} - x^*\|^2])^{\frac{1}{2}}$ as a function of $\tau$, where $(X_N^{\text{SHB}})_{n \in \mathbb{N}}$ and $(X_N^{\text{SGD}})_{n \in \mathbb{N}}$ denote the stochastic Heavy-Ball and stochastic gradient descent schemes, respectively. In the context of stochastic gradient descent, when $\sigma(s) \to 0$, we
observe the same term $\sqrt{\tau}$ but multiplied by a function that depends only on $T$ and tends towards 0 when $T \to \infty$.

<figure><img src="image_4.png" /><figcaption>FIGURE 2. Comparison of the stochastic Heavy-Ball scheme (left) and stochastic gradient descent scheme (right) for the quadratic objective function $F$, with: $\sigma(s) = e^{-s}$ (top) and $\sigma(s) = \frac{1}{1+s}$ (bottom). The final times considered are $T \in \{2, 4, 8, 16, 32, 64\}$.</figcaption></figure>

### 3.3. Complexity analysis.
Let us perform some complexity analysis associated with the convergence result stated in Theorem 1. We only focus on the strong error estimate (23).

In order to study the computational complexity, let $\delta \in (0, 1)$ be an arbitrarily small error size, then one needs to find $\tau \in (0, \bar{\tau})$ and $N \in \mathbb{N}^*$ such that the strong error satisfies

$$ (25) \qquad (\mathbb{E}[\|X_N - x^*\|^2])^{\frac{1}{2}} \le \delta. $$

The computational cost is proportional to the number of time steps $N$. Below, the symbol $\approx$ is used to denote the size of the error, constants which are independent of the considered parameters $\delta$, $\tau$ and $N$ are omitted.

Owing to the strong error estimate (23) the upper bound (25) is satisfied if $\tau$ is chosen such that $\tau \approx \delta^2$ and if then $N$ is chosen such that $N\tau \approx \log(\delta^{-1})$. Therefore, the computational cost is of size

$$ N = N_1(\delta) \approx \delta^{-2} \log(\delta^{-1}). $$

### 4. AUXILIARY RESULTS ON THE STOCHASTIC HEAVY BALL OPTIMIZATION SCHEME

This section is devoted to studying some properties of the discrete-time stochastic processes $(X_n)_{n \in \mathbb{N}}$ and $(V_n)_{n \in \mathbb{N}}$ defined by the stochastic Heavy Ball optimization scheme (3). First, Lemma 4.1 states moment bounds, which are uniform with respect to the time-step size $\tau$, and with respect to time $n \in \mathbb{N}$. Second, for any time-step size $\tau$ an auxiliary continuous time process $(\tilde{X}^\tau(s), \tilde{V}^\tau(s))_{s \ge 0}$ is introduced and some properties are given in Lemma 4.2.
We introduce the function $\Gamma$, defined by

$$ (26) \qquad \Gamma(x, v) = F(x) - F(x^*) + \frac{\alpha^2}{4} \|x - x^*\|^2 + \frac{\alpha}{2} \langle x - x^*, v \rangle + \frac{1}{2} \|v\|^2, \quad \forall x, v \in \mathbb{R}^d. $$

The function $\Gamma$ plays a role in the proof of the bounds on the scheme in Lemma 4.1 and also on the proof of the asymptotic behavior of $(\mathcal{X}^0(s), \mathcal{V}^0(s))_{s \ge 0}$, given by (17), see Proposition 5.1 for more details.

Using the inequalities (12) and (13), by applying the Cauchy–Schwarz and Young inequalities, one shows that $\Gamma(x, v)$ can be upper and lower bounded in terms of $\|x - x^*\|^2 + \|v\|^2$:

$$ (27) \qquad \kappa_1(\mu)(\|x - x^*\|^2 + \|v\|^2) \le \Gamma(x, v) \le \kappa_2(\alpha, L_F)(\|x - x^*\|^2 + \|v\|^2), \quad \forall x, v \in \mathbb{R}^d. $$

where $\kappa_1(\mu) = \frac{1}{2}\min(\mu, \frac{1}{2})$ and $\kappa_2(\alpha, L_F) = \frac{1}{2}\max(L_F + \alpha^2, \frac{3}{2})$.

We give a Fact about the first and second derivatives of $\Gamma$ that will be useful for the following lemma and the Proposition 5.1.

**Fact.** There exists $C \in (0, \infty)$, such that for all $x, v \in \mathbb{R}^d$, $k = (k_x, k_v) \in \mathbb{R}^d \times \mathbb{R}^d$ one has

$$ (28) \qquad \begin{aligned} |D\Gamma(x, v).k| &= \left| \left\langle \nabla F(x) + \frac{\alpha^2}{2}(x - x^*) + \frac{\alpha}{2}v, k_x \right\rangle + \left\langle \frac{\alpha}{2}(x - x^*) + v, k_v \right\rangle \right| \\ &\le C(\|x - x^*\| + \|v\|)(\|k_x\| + \|k_v\|). \end{aligned} $$

$$ \text{and all } k_1 = (k_{1,x}, k_{1,v}), \quad k_2 = (k_{2,x}, k_{2,v}) \in \mathbb{R}^d \times \mathbb{R}^d \tag{29} $$

$$ \begin{aligned} |D^2\Gamma(x, v).(k_1, k_2)| &= \left| \left\langle \left(\nabla^2 F(x) + \frac{\alpha^2}{2}I_d\right)k_{1,x}, k_{2,x} \right\rangle + \left\langle k_{1,v}, k_{2,v} \right\rangle + \frac{\alpha}{2} \left( \left\langle k_{1,v}, k_{2,x} \right\rangle + \left\langle k_{1,x}, k_{2,v} \right\rangle \right) \right| \\ &\le C(\|k_1\|^2 + \|k_2\|^2). \end{aligned} $$

## 4.1. Moment bounds on the scheme.

**Lemma 4.1.** Assume that the objective function $F$ satisfies the Assumption 2.1 and that the diffusion coefficient $\sigma$ satisfies the Assumption 2.2.

There exist a sequence $(\bar{\tau}_p)_{p \in \mathbb{N}^*}$, with $\bar{\tau}_p \in (0, \tau_{\max})$ for all $p \in \mathbb{N}^*$, and a sequence $(C_p)_{p \in \mathbb{N}^*}$, such that for any initial value $x_0, v_0 \in \mathbb{R}^d$ and for any $p \in \mathbb{N}^*$ one has

$$ (30) \qquad \sup_{\tau \in (0, \bar{\tau}_p)} \sup_{n \in \mathbb{N}} \left( \mathbb{E}[\|X_n - x^*\|^{2p}] + \mathbb{E}[\|V_n\|^{2p}] \right) \le C_p (1 + \|x_0 - x^*\|^{2p} + \|v_0\|^{2p}). $$

**Remark 4.1.** It is straightforward to verify that the moment bounds (30) are satisfied by $(\bar{X}_n, \bar{V}_n)_{n \in \mathbb{N}}$, i.e.: there exist $(\bar{\tau}_p)_{p \in \mathbb{N}^*}$ and $(C_p)_{p \in \mathbb{N}^*}$, such that for any initial conditions $\bar{X}_0, \bar{V}_0 \in L^2(\Omega, \mathbb{R}^d)$ and for any $p \in \mathbb{N}^*$ one has

$$ \begin{aligned} \sup_{\tau \in (0, \bar{\tau}_p)} \sup_{n \in \mathbb{N}} \left( \mathbb{E}[\|\bar{X}_n - x^*\|^{2p}] + \mathbb{E}[\|\bar{V}_n\|^{2p}] \right) &\le C_p (1 + \|\bar{X}_0 - x^*\|^{2p} + \|\bar{V}_0\|^{2p}) \\ &\le C_p (1 + \|x_0 - x^*\|^{2p} + \|v_0\|^{2p}), \end{aligned} $$

where the last bound is given by using the inequality (7).

**Proof.** Let us first introduce some auxiliary notation. For any positive integer $n \in \mathbb{N}^*$, define the $\sigma$-field $\mathcal{F}_n = \sigma(\gamma_1, \dots, \gamma_n)$. Set also $\mathcal{F}_0 = \{\emptyset, \Omega\}$. Recall the notation $\mathbb{E}[\cdot|\mathcal{F}_n]$ for the conditional expectation with respect to $\mathcal{F}_n$.

For all $n \in \mathbb{N}$, using the scheme (3)

$$ \begin{aligned} \Delta X_{n+1} &= X_{n+1} - X_n = \tau V_{n+1} = \tau \left( (1 - \alpha \tau) V_n - \tau \nabla F(X_n) + \tau \sigma_{\tau}(s_n, X_n) \gamma_{n+1} \right), \\ \Delta V_{n+1} &= V_{n+1} - V_n = -\alpha \tau V_n - \tau \nabla F(X_n) + \tau \sigma_{\tau}(s_n, X_n) \gamma_{n+1}, \end{aligned} $$
Observe that the random variables $X_n, V_n$ are $\mathcal{F}_n$-measurable for all $n \in \mathbb{N}$. In addition for all $n \in \mathbb{N}$, one has

$$ (31) \qquad \begin{cases} X_{n+1} - x^* = X_n - x^* + \Delta X_{n+1}, \\ V_{n+1} = V_n + \Delta V_{n+1}. \end{cases} $$

The proof then proceeds in two steps, in order to treat the cases $p = 1$ and $p \ge 2$ separately. Step 1: the case $p = 1$.

Applying the Taylor formula on $\Gamma$, set $k_{n+1} = (\Delta X_{n+1}, \Delta V_{n+1})$ one obtains

$$ \begin{aligned} \Gamma(X_{n+1}, V_{n+1}) &= \Gamma(X_n + \Delta X_{n+1}, V_n + \Delta V_{n+1}) \\ &= \Gamma(X_n, V_n) + D\Gamma(X_n, V_n) \cdot k_{n+1} \\ &\quad + \int_0^1 (1-\theta) D^2\Gamma(X_n + \theta\Delta X_{n+1}, V_n + \theta\Delta V_{n+1}) \cdot (k_{n+1}, k_{n+1}) \, d\theta \end{aligned} $$

Using the expression (29) and the Lipschitz continuity condition (8) of $\nabla F$, one has

$$ \begin{aligned} \Gamma(X_{n+1}, V_{n+1}) &\le \Gamma(X_n, V_n) \\ &\quad + \langle \nabla F(X_n) + \frac{\alpha}{2}V_n + \frac{\alpha^2}{2}(X_n - x^*), \Delta X_{n+1} \rangle + \langle V_n + \frac{\alpha}{2}(X_n - x^*), \Delta V_{n+1} \rangle \\ &\quad + \int_0^1 (1-\theta) \langle (\nabla^2 F(X_n + \theta\Delta X_{n+1}) + \frac{\alpha^2}{2}I_d)\Delta X_{n+1}, \Delta X_{n+1} \rangle \, d\theta \\ &\quad + \int_0^1 (1-\theta) \langle \alpha\langle \Delta X_{n+1}, \Delta V_{n+1} \rangle + \|\Delta V_{n+1}\|^2 \rangle \, d\theta \\ &\le \Gamma(X_n, V_n) \\ &\quad + \langle \nabla F(X_n) + \frac{\alpha}{2}V_n + \frac{\alpha^2}{2}(X_n - x^*), \Delta X_{n+1} \rangle + \langle V_n + \frac{\alpha}{2}(X_n - x^*), \Delta V_{n+1} \rangle \\ &\quad + \frac{1}{2}((L + \alpha^2)\|\Delta X_{n+1}\|^2 + \frac{3}{2}\|\Delta V_{n+1}\|^2), \end{aligned} $$

where the last line is obtained by applying the Cauchy-Schwarz inequality. By taking the conditional expectation of the previous inequality and by using the Lipschitz condition (8) and the $\mu$-convexity (9), one has

$$ \begin{aligned} \mathbb{E}[\Gamma(X_{n+1}, V_{n+1}) | \mathcal{F}_n] &\le \Gamma(X_n, V_n) \\ &\quad + \langle \nabla F(X_n) + \frac{\alpha}{2}V_n + \frac{\alpha^2}{2}(X_n - x^*), \mathbb{E}[\Delta X_{n+1} | \mathcal{F}_n] \rangle \\ &\quad + \langle V_n + \frac{\alpha}{2}(X_n - x^*), \mathbb{E}[\Delta V_{n+1} | \mathcal{F}_n] \rangle \\ &\quad + C \mathbb{E}[\|\Delta X_{n+1}\|^2 | \mathcal{F}_n] + C \mathbb{E}[\|\Delta V_{n+1}\|^2 | \mathcal{F}_n]. \end{aligned} $$

The random variable $\gamma_{n+1}$ is independent of $\mathcal{F}_n$, thus one has

$$ \mathbb{E}[\gamma_{n+1} | \mathcal{F}_n] = \mathbb{E}[\gamma_{n+1}] = 0. $$

Considering the conditional expectation with respect to $\mathcal{F}_n$, using the definition of $\Delta X_{n+1}$ and $\Delta V_{n+1}$ and the properties of conditional expectation, since $X_n$ and $V_n$ are $\mathcal{F}_n$-measurable random variables, one has

$$ \mathbb{E}[\Delta X_{n+1} | \mathcal{F}_n] = \tau((1 - \alpha\tau)V_n - \tau\nabla F(X_n)), $$

$$ \mathbb{E}[\Delta V_{n+1} | \mathcal{F}_n] = -\tau(\alpha V_n + \nabla F(X_n)). $$
Using the Lipschitz continuity condition (8) of $\nabla F$, the definition of $\Delta X_{n+1}$ and $\Delta V_{n+1}$ and the properties of conditional expectation, there exists $C \in (0, \infty)$ such that

$$
\begin{align*}
\mathbb{E}[\|\Delta X_{n+1}\|^2 | \mathcal{F}_n] &= \tau^2 \mathbb{E}[\|V_{n+1}\|^2 | \mathcal{F}_n] \\
&= \tau^2 (\|(1 - \alpha\tau)V_n - \tau\nabla F(X_n)\|^2 + \tau^2 \|\sigma_\tau(s_n, X_n)\|^2) \\
&\le C\tau^2 \|X_n - x^*\|^2 + C\tau^2 \|V_n\|^2 + \tau^2 \|\sigma_\tau(s_n, X_n)\|^2, \\
\mathbb{E}[\|\Delta V_{n+1}\|^2 | \mathcal{F}_n] &= \|\alpha\tau V_n + \tau\nabla F(X_n)\|^2 + \tau^2 \|\sigma_\tau(s_n, X_n)\|^2 \\
&\le C\tau^2 \|X_n - x^*\|^2 + C\tau^2 \|V_n\|^2 + \tau^2 \|\sigma_\tau(s_n, X_n)\|^2.
\end{align*}
$$

Combining the results above, there exists $C \in (0, \infty)$ such that for all $n \in \mathbb{N}$ one obtains

$$
\begin{align*}
\mathbb{E}[\Gamma(X_{n+1}, V_{n+1}) | \mathcal{F}_n] &\le \Gamma(X_n, V_n) \\
&\quad + \tau \langle \nabla F(X_n) + \frac{\alpha}{2}V_n + \frac{\alpha^2}{2}(X_n - x^*), (1 - \alpha\tau)V_n - \tau\nabla F(X_n) \rangle \\
&\quad + \langle V_n + \frac{\alpha}{2}(X_n - x^*), -\alpha\tau V_n - \tau\nabla F(X_n) \rangle \\
&\quad + C\mathbb{E}[\|\Delta X_{n+1}\|^2 | \mathcal{F}_n] + C\mathbb{E}[\|\Delta V_{n+1}\|^2 | \mathcal{F}_n] \\
&\le \Gamma(X_n, V_n) - \left(\alpha\tau - \frac{\alpha}{2}\tau(1 - \alpha\tau)\right) \|V_n\|^2 \\
&\quad - \left(\frac{\alpha\tau}{2} + \frac{\alpha^2\tau^2}{2}\right) \langle \nabla F(X_n), X_n - x^* \rangle - \tau^2 \|\nabla F(X_n)\|^2 \\
&\quad - \frac{\alpha^3}{2}\tau^2 \langle X_n - x^*, V_n \rangle - \frac{3\alpha}{2}\tau^2 \langle \nabla F(X_n), V_n \rangle \\
&\quad + C\tau^2 \|V_n\|^2 + C\tau^2 \|X_n - x^*\|^2 + C\tau^2 \|\sigma_\tau(s_n, X_n)\|^2,
\end{align*}
$$

using the $\mu$-convexity (9) of the objective function $F$, one has for all $n \in \mathbb{N}$

$$
\begin{align*}
\mathbb{E}[\Gamma(X_{n+1}, V_{n+1}) | \mathcal{F}_n] &\le \Gamma(X_n, V_n) - \left(\alpha\tau - \frac{\alpha}{2}\tau(1 - \alpha\tau)\right) \|V_n\|^2 \\
&\quad - \left(\frac{\alpha\mu}{2}\tau + \frac{\alpha^2\mu}{2}\tau^2 + \mu^2\tau^2\right) \|X_n - x^*\|^2 \\
&\quad - \frac{\alpha^3}{2}\tau^2 \langle X_n - x^*, V_n \rangle - \frac{3\alpha}{2}\tau^2 \langle \nabla F(X_n), V_n \rangle \\
&\quad + C\tau^2 \|V_n\|^2 + C\tau^2 \|X_n - x^*\|^2 + C\tau^2 \|\sigma_\tau(s_n, X_n)\|^2.
\end{align*}
$$

By applying successively the Cauchy–Schwarz and Young inequalities and using the Lipschitz condition (8) and the linear growth condition (15) of $\sigma$, one obtains

$$
\mathbb{E}[\Gamma(X_{n+1}, V_{n+1}) | \mathcal{F}_n] \le \Gamma(X_n, V_n) - \left(\frac{\alpha}{2}\tau - C\tau^2\right) \|V_n\|^2 - \left(\frac{\alpha\mu}{2}\tau - C\tau^2\right) \|X_n - x^*\|^2 + C\tau^2
$$

Due to the second upper bound in (27), one has

$$
\mathbb{E}[\Gamma(X_{n+1}, V_{n+1}) | \mathcal{F}_n] \le \rho_1(\tau) \Gamma(X_n, V_n) + C\tau^2,
$$

where for all $\tau \in (0, \tau_{\max})$

$$
\rho_1(\tau) = 1 - \frac{\alpha\tau}{2\kappa_2(\alpha, L_F)} \min(\mu, 1) + C\tau^2.
$$

Applying the tower property of conditional expectation, one has

$$
\mathbb{E}[\Gamma(X_{n+1}, V_{n+1})] \le \rho_1(\tau) \mathbb{E}[\Gamma(X_n, V_n)] + C\tau^2.
$$
By definition of $\rho_1$, there exists $\bar{\tau}_1 \in (0, \tau_{\max})$ such that $\rho_1(\tau) \in (0, 1)$ for all $\tau \in (0, \bar{\tau}_1)$. Applying the discrete Grönwall inequality, one obtains for all $n \ge 0$ and for all $\tau \in (0, \bar{\tau}_1)$

$$ \mathbb{E}[\Gamma(X_n, V_n)] \le \rho_1(\tau)^n \Gamma(X_0, V_0) + \frac{C\tau^2}{1 - \rho_1(\tau)} = \rho_1(\tau)^n \Gamma(X_0, V_0) + \frac{C\tau}{\frac{\alpha \min(\mu, 1)}{2\kappa_2(\alpha, L_F)} + C\tau}. $$

Finally by applying the inequalities (27), we obtain

$$ \sup_{\tau \in (0, \bar{\tau}_1)} \sup_{n \in \mathbb{N}} (\mathbb{E}[\|X_n - x^*\|^2] + \mathbb{E}[\|V_n\|^2]) \le C(1 + \|x_0 - x^*\|^2 + \|v_0\|^2). $$

Step 2: the case $p \ge 2$.

Applying the Taylor formula and the chain rule on the function $\Gamma^p$, set $k_{n+1} = (\Delta X_{n+1}, \Delta V_{n+1})$ one obtains

$$ \begin{aligned} \Gamma^p(X_{n+1}, V_{n+1}) &= \Gamma^p(X_n, V_n) + D(\Gamma^p)(X_n, V_n) \cdot k_{n+1} \\ &\quad + \int_0^1 (1-\theta) D^2(\Gamma^p)(X_n + \theta \Delta X_{n+1}, V_n + \theta \Delta V_{n+1}) \cdot (k_{n+1}, k_{n+1}) \, d\theta, \\ &= \Gamma^p(X_n, V_n) + p \Gamma^{p-1}(X_n, V_n) D \Gamma(X_n, V_n) \cdot k_{n+1} \\ &\quad + \int_0^1 (1-\theta) p \Gamma^{p-1}(X_n + \theta \Delta X_{n+1}, V_n + \theta \Delta V_{n+1}) D^2 \Gamma(X_n + \theta \Delta X_{n+1}, V_n + \theta \Delta V_{n+1}) \cdot (k_{n+1}, k_{n+1}) \, d\theta \\ &\quad + \int_0^1 (1-\theta) p (p-1) \Gamma^{p-2}(X_n + \theta \Delta X_{n+1}, V_n + \theta \Delta V_{n+1}) (D \Gamma(X_n + \theta \Delta X_{n+1}, V_n + \theta \Delta V_{n+1}) \cdot k_{n+1})^2 \, d\theta. \end{aligned} $$

Using the inequalities (28) and (29), there exists $C_p \in (0, \infty)$ such that

$$ \begin{aligned} \Gamma^p(X_{n+1}, V_{n+1}) &\le \Gamma^p(X_n, V_n) + p \Gamma^{p-1}(X_n, V_n) D \Gamma(X_n, V_n) \cdot (\Delta X_{n+1}, \Delta V_{n+1}) \\ &\quad + C_p \int_0^1 (1-\theta) \Gamma^{p-1}(X_n + \theta \Delta X_{n+1}, V_n + \theta \Delta V_{n+1}) (\|\Delta X_{n+1}\|^2 + \|\Delta V_{n+1}\|^2) \, d\theta \\ &\quad + C_p \int_0^1 (1-\theta) \Gamma^{p-2}(X_n + \theta \Delta X_{n+1}, V_n + \theta \Delta V_{n+1}) (\|X_n - x^*\|^2 + \|V_n\|^2) (\|\Delta X_{n+1}\|^2 + \|\Delta V_{n+1}\|^2) \, d\theta. \end{aligned} $$

Applying the right-hand inequality of (27) and the convexity inequality $(a+b)^q \le 2^{q-1}(a^q + b^q)$ infers the upper bound

$$ \begin{aligned} \Gamma^p(X_{n+1}, V_{n+1}) &\le \Gamma^p(X_n, V_n) + p \Gamma^{p-1}(X_n, V_n) D \Gamma(X_n, V_n) \cdot (\Delta X_{n+1}, \Delta V_{n+1}) \\ &\quad + C_p \int_0^1 (\|X_n + \theta \Delta X_{n+1}\|^2 + \|V_n + \theta \Delta V_{n+1}\|^2)^{(p-1)} (\|\Delta X_{n+1}\|^2 + \|\Delta V_{n+1}\|^2) \, d\theta \\ &\quad + C_p \int_0^1 (\|X_n + \theta \Delta X_{n+1}\|^2 + \|V_n + \theta \Delta V_{n+1}\|^2)^{(p-2)} (\|\Delta X_{n+1}\|^2 + \|\Delta V_{n+1}\|^2) (\|X_n - x^*\|^2 + \|V_n\|^2) \, d\theta \\ &\le \Gamma^p(X_n, V_n) + p \Gamma^{p-1}(X_n, V_n) D \Gamma(X_n, V_n) \cdot (\Delta X_{n+1}, \Delta V_{n+1}) \\ &\quad + C_p \int_0^1 (\|X_n - x^*\|^{2(p-1)} + \|V_n\|^{2(p-1)}) (\|\Delta X_{n+1}\|^2 + \|\Delta V_{n+1}\|^2) \, d\theta \\ &\quad + C_p \int_0^1 (\|\Delta X_{n+1}\|^{2(p-1)} + \|\Delta V_{n+1}\|^{2(p-1)}) (\|\Delta X_{n+1}\|^2 + \|\Delta V_{n+1}\|^2) \, d\theta. \end{aligned} $$

Considering the conditional expectation with respect to $\mathcal{F}_n$, by using the previous computation, one obtains

$$ \begin{aligned} \mathbb{E}[\Gamma^p(X_{n+1}, V_{n+1}) | \mathcal{F}_n] \\ \le \Gamma^p(X_n, V_n) + p \Gamma^{p-1}(X_n, V_n) D \Gamma(X_n, V_n) \cdot \mathbb{E}[\Delta X_{n+1} | \mathcal{F}_n] + C_p \tau^2 (\|X_n - x^*\|^{2p} + \|V_n\|^{2p}) + C_p \tau^2 \end{aligned} $$
$$ \leq \Gamma^p(X_n, V_n) - p\Gamma^{p-1}(X_n, V_n)\left(\left(\frac{\alpha\mu}{2}\tau - C_p\tau^2\right)\|X_n - x^*\|^2 + \left(\frac{\alpha}{2}\tau - C_p\tau^2\right)\|V_n\|^2\right) + C_p\tau^2. $$

Due to the second upper bound in (27), one has

$$ \mathbb{E}[\Gamma^p(X_{n+1}, V_{n+1}) | \mathcal{F}_n] \leq \rho_p(\tau) \Gamma_1^p(X_n, V_n) + C_p \tau^2, $$

where for all $\tau \in (0, \tau_{\max})$

$$ \rho_p(\tau) = 1 - \frac{\alpha \tau p}{2\kappa_2(\alpha, L_F)} \min(\mu, 1) + C_p \tau^2. $$

Applying the tower property of conditional expectation, one has

$$ \mathbb{E}[\Gamma^p(X_{n+1}, V_{n+1})] \leq \rho_p(\tau) \mathbb{E}[\Gamma^p(X_n, V_n)] + C_p \tau^2. $$

By definition of $\rho_p$, there exists $\bar{\tau}_p \in (0, \tau_{\max})$ such that $\rho_p(\tau) \in (0, 1)$ for all $\tau \in (0, \bar{\tau}_p)$. Applying the discrete Grönwall inequality, one obtains for all $n \geq 0$ and for all $\tau \in (0, \bar{\tau}_p)$

$$ \mathbb{E}[\Gamma^p(X_n, V_n)] \leq \rho_p(\tau)^n \Gamma^p(X_0, V_0) + \frac{C_p \tau^2}{1 - \rho_p(\tau)} = \rho_p(\tau)^n \Gamma(X_0, V_0) + \frac{C_p \tau}{\frac{\alpha p \min(\mu, 1)}{2\kappa_2(\alpha, L_F)} + C_p \tau}. $$

Finally by applying the inequalities (27) and the convexity inequality $(a+b)^q \leq 2^{q-1}(a^q + b^q)$, we obtain

$$ \sup_{\tau \in (0, \bar{\tau}_p)} \sup_{n \in \mathbb{N}} \left( \mathbb{E}[\|X_n - x^*\|^{2p}] + \mathbb{E}[\|V_n\|^{2p}] \right) \leq C_p (1 + \|x_0 - x^*\|^{2p} + \|v_0\|^{2p}). $$

□

4.2. **Auxiliary processes.** Given any time-step size $\tau \in (0, \tau_{\max})$, let us introduce auxiliary stochastic processes $(\tilde{X}^\tau(s), \tilde{V}^\tau(s))_{s \geq 0}$ defined as follows: for any integer $n \geq 0$, and for all $s \in [s_n, s_{n+1}]$ one has

$$ (32) \qquad \begin{cases} \tilde{X}^\tau(s) = \bar{X}_n + (s - s_n)\bar{V}_n \\ \tilde{V}^\tau(s) = \bar{V}_n - \alpha(s - s_n)\bar{V}_n - (s - s_n)\nabla F(\bar{X}_{n+1}) + \sqrt{\tau}\sigma_\tau(s_{n+1}, \bar{X}_{n+1})(B(s) - B(s_n)). \end{cases} $$

Note that $(\tilde{X}^\tau(s_n), \tilde{V}^\tau(s_n)) = (\bar{X}_n, \bar{V}_n)$ for all $n \in \mathbb{N}$, i.e. the auxiliary processes interpolates the stochastic Heavy Ball optimization scheme (6) at the grid times $s_n = n\tau$. Note the interest of introducing the change of variables $(\bar{X}_n, \bar{V}_n) = (X_n, V_{n+1})$ for all $n \in \mathbb{N}$, because in this case we obtain that the quadratic variation of $(\tilde{X}^\tau(s), \tilde{V}^\tau(s))_{s \geq 0}$ is zero, which would not be the case if we had not considered the change of variable, and this simplifies the proof of Proposition 5.2. In addition, the stochastic processes $(\tilde{X}^\tau(s), \tilde{V}^\tau(s))_{s \geq 0}$ have continuous trajectories. Finally, for any integer $n \in \mathbb{N}$, on the interval $[s_n, s_{n+1}]$ the processes $s \mapsto (\tilde{X}^\tau(s), \tilde{V}^\tau(s))$ are the solution to the stochastic differential equations

$$ (33) \qquad \begin{cases} d\tilde{X}^\tau(s) = \bar{V}_n \, ds, & s \in [s_n, s_{n+1}], \\ d\tilde{V}^\tau(s) = -(\alpha\bar{V}_n + \nabla F(\bar{X}_{n+1})) \, ds + \sqrt{\tau}\sigma_\tau(s_{n+1}, \bar{X}_{n+1}) \, dB(s), & s \in [s_n, s_{n+1}], \\ (\tilde{X}^\tau(s_n), \tilde{V}^\tau(s_n)) = (\bar{X}_n, \bar{V}_n). \end{cases} $$

The auxiliary processes $(\tilde{X}^\tau(s), \tilde{V}^\tau(s))_{s \geq 0}$ play a technical role in the proofs of the weak error estimates. The following properties, which follow from Lemma 4.1, are employed in the error analysis.
**Lemma 4.2.** Assume that the objective F satisfies the basic properties of Assumption 2.1 and that the diffusion coefficient σ satisfies Assumption 2.2. For all $p \in \mathbb{N}^*$, there exists $\bar{\tau}_p \in (0, \tau_{\max})$ and $C_p \in (0, \infty)$ such that one has the moment bounds

$$ (34) \quad \sup_{\tau \in (0, \bar{\tau}_p)} \sup_{s \ge 0} \mathbb{E}[\|\tilde{X}^\tau(s) - x^*\|^{2p}] + \mathbb{E}[\|\tilde{V}^\tau(s)\|^{2p}] \le C_p(1 + \|x_0 - x^*\|^{2p} + \|v_0\|^{2p}), $$

and such that one has the increment bounds: for all $\tau \in (0, \bar{\tau}_p)$,

$$ (35) \quad \sup_{n \in \mathbb{N}} \sup_{s \in [s_n, s_{n+1}]} \mathbb{E}[\|\tilde{X}^\tau(s) - \bar{X}_n\|^{2p}] + \mathbb{E}[\|\tilde{V}^\tau(s) - \bar{V}_n\|^{2p}] \le C_p(1 + \|x_0 - x^*\|^{2p} + \|v_0\|^{2p})\tau^{2p}. $$

*Proof of Lemma 4.2.* Let $p \in \mathbb{N}^*$ and let $\bar{\tau}_p$ be given by Lemma 4.1.

*Proof of the moments estimates (34).* For all $\tau \in (0, \bar{\tau}_p)$, for any integer $n \ge 0$, owing to the definition (32) of the interpolating process $(\tilde{X}^\tau(s), \tilde{V}^\tau(s))_{s \ge 0}$, one has for all $s \in [s_n, s_{n+1}]$

$$ \begin{aligned} & (\mathbb{E}[\|\tilde{X}^\tau(s) - x^*\|^{2p}])^{\frac{1}{2p}} + (\mathbb{E}[\|\tilde{V}^\tau(s)\|^{2p}])^{\frac{1}{2p}} \\ & \le (\mathbb{E}[\|\bar{X}_n - x^*\|^{2p}])^{\frac{1}{2p}} + (s - s_n)(\mathbb{E}[\|\bar{V}_n\|^{2p}])^{\frac{1}{2p}} \\ & \quad + (\mathbb{E}[\|\tilde{V}^\tau(s)\|^{2p}])^{\frac{1}{2p}} \\ & \le (\mathbb{E}[\|\bar{X}_n - x^*\|^{2p}])^{\frac{1}{2p}} + C_p(\mathbb{E}[\|\bar{V}_n\|^{2p}])^{\frac{1}{2p}} \\ & \quad + (s - s_n)(\mathbb{E}[\|\nabla F(\bar{X}_{n+1})\|^{2p}])^{\frac{1}{2p}} \\ & \quad + \sqrt{\tau}(\mathbb{E}[\|\sigma_\tau(s_{n+1}, \bar{X}_{n+1})(B(s) - B(s_n))\|^{2p}])^{\frac{1}{2p}} \end{aligned} $$

Then by the linear growth conditions (11) and (15),

$$ \begin{aligned} & (\mathbb{E}[\|\tilde{X}^\tau(s) - x^*\|^{2p}])^{\frac{1}{2p}} + (\mathbb{E}[\|\tilde{V}^\tau(s)\|^{2p}])^{\frac{1}{2p}} \\ & \le (\mathbb{E}[\|\bar{X}_n - x^*\|^{2p}])^{\frac{1}{2p}} + C_p (\mathbb{E}[\|\bar{V}_n\|^{2p}])^{\frac{1}{2p}} \\ & \quad + L_F \tau (\mathbb{E}[\|\bar{X}_{n+1} - x^*\|^{2p}])^{\frac{1}{2p}} \\ & \quad + C_p \tau \|\zeta\|_{\infty} (1 + (\mathbb{E}[\|\bar{X}_{n+1} - x^*\|^{2p}])^{\frac{1}{2p}}), \end{aligned} $$

using the property

$$ (\mathbb{E}[\|B(s) - B(s_n)\|^{2p}])^{\frac{1}{2p}} = C_p(s - s_n)^{\frac{1}{2}} \le C_p \tau^{\frac{1}{2}}. $$

Applying moment bounds on $(\bar{X}_n, \bar{V}_n)_{n \in \mathbb{N}}$ given in Remark 4.1 together with the convexity inequality $(a+b)^q \le 2^{q-1}(a^q + b^q)$ then imply that for all $\tau \in (0, \bar{\tau}_p)$ one has

$$ \begin{aligned} \sup_{s \ge 0} (\mathbb{E}[\|\tilde{X}^\tau(s) - x^*\|^{2p}])^{\frac{1}{2p}} + (\mathbb{E}[\|\tilde{V}^\tau(s)\|^{2p}])^{\frac{1}{2p}} &= \sup_{n \ge 0} \sup_{s \in [s_n, s_{n+1}]} (\mathbb{E}[\|\tilde{X}^\tau(s) - x^*\|^{2p}])^{\frac{1}{2p}} + (\mathbb{E}[\|\tilde{V}^\tau(s)\|^{2p}])^{\frac{1}{2p}} \\ &\le C_p \left(1 + \sup_{n \ge 0} (\mathbb{E}[\|\bar{X}_n - x^*\|^{2p}])^{\frac{1}{2p}} + (\mathbb{E}[\|\bar{V}_n\|^{2p}])^{\frac{1}{2p}}\right) \\ &\le C_p \left(1 + \|x_0 - x^*\| + \|v_0\|\right). \end{aligned} $$

This concludes the proof of the moment bounds (34).

*Proof of the increments bounds (35).* One proceeds by using similar arguments. Let $\tau \in (0, \bar{\tau}_p)$ and $n \ge 0$. Then for all $s \in [s_n, s_{n+1}]$ owing to (32) and (35) one has

$$ \begin{aligned} & (\mathbb{E}[\|\tilde{X}^\tau(s) - \bar{X}_n\|^{2p}])^{\frac{1}{2p}} + (\mathbb{E}[\|\tilde{V}^\tau(s) - \bar{V}_n\|^{2p}])^{\frac{1}{2p}} \\ & \le (s - s_n)(\mathbb{E}[\|\bar{V}_n\|^{2p}])^{\frac{1}{2p}} + \alpha(s - s_n)(\mathbb{E}[\|\bar{V}_n\|^{2p}])^{\frac{1}{2p}} \\ & \quad + (s - s_n)(\mathbb{E}[\|\nabla F(\bar{X}_{n+1})\|^{2p}])^{\frac{1}{2p}} \\ & \quad + \sqrt{\tau}(\mathbb{E}[\|\sigma_\tau(s_{n+1}, \bar{X}_{n+1})(B(s) - B(s_n))\|^{2p}])^{\frac{1}{2p}} \\ & \le (1 + \alpha)\tau(\mathbb{E}[\|\bar{V}_n\|^{2p}])^{\frac{1}{2p}} + L_F \tau(\mathbb{E}[\|\bar{X}_{n+1} - x^*\|^{2p}])^{\frac{1}{2p}} \end{aligned} $$
$$
\begin{align*}
&+ C_p \tau \|\zeta\|_{\infty} \left( 1 + \left( \mathbb{E}[\|\bar{X}_{n+1} - x^*\|^{2p}] \right)^{\frac{1}{2p}} \right) \\
&\le C_p \tau \left( 1 + \sup_{n \ge 0} \left( \mathbb{E}[\|\bar{X}_{n+1} - x^*\|^{2p}] \right)^{\frac{1}{2p}} + \left( \mathbb{E}[\|\bar{V}_n\|^{2p}] \right)^{\frac{1}{2p}} \right) \\
&\le C_p \tau \left( 1 + \|x_0 - x^*\| + \|v_0\| \right).
\end{align*}
$$

□

5. PROOF OF THEOREM 1

Let us explain the strategy of the proof. Let $\varphi : \mathbb{R}^d \to \mathbb{R}$ be a mapping of class $C^2$. The error is decomposed as

$$
(36) \quad \mathbb{E}[\varphi(X_N)] - \varphi(x^*) = \mathbb{E}[\varphi(\mathcal{X}^0(N\tau))] - \varphi(x^*) + \mathbb{E}[\varphi(X_N)] - \mathbb{E}[\varphi(\mathcal{X}^0(N\tau))],
$$

where $(\mathcal{X}^0(s), \mathcal{V}^0(s))_{s \ge 0}$ is the solution of Heavy Ball system (17) with random initial value $(\bar{X}_0, \bar{V}_0)$.
On the one hand, the treatment of the first error term $\mathbb{E}[\varphi(\mathcal{X}^0(N\tau))] - \varphi(x^*)$ follows from an upper
bound on $\mathbb{E}[\|\mathcal{X}^0(s)-x^*\|]$ which is stated in Proposition 5.1 below. On the other hand, the treatment
of the second error term $\mathbb{E}[\varphi(X_N)] - \mathbb{E}[\varphi(\mathcal{X}^0(N\tau))]$ requires more effort. For this part of the error,
we prove upper bounds depending on $\tau$ which are uniform with respect to $N$. Introduce the auxiliary
function $u^0: [0, \infty) \times \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$ given by

$$
(37) \qquad u^0(s, x, v) = \varphi(\mathcal{X}_{x,v}^0(s)),
$$

where $(\mathcal{X}_{x,v}^0(s), \mathcal{V}_{x,v}^0(s))_{s \ge 0}$ denotes the solution of (17) with arbitrary deterministic initial value
$(\mathcal{X}_{x,v}^0(0), \mathcal{V}_{x,v}^0(0)) = (x,v) \in \mathbb{R}^d \times \mathbb{R}^d$. As a result, the second error term is written as, we recall
$s_N = N\tau$ and for all $n \in \mathbb{N}$, we have $(\bar{X}_n, \bar{V}_n) = (X_n, V_{n+1})$

$$
\begin{align}
(38) \qquad \mathbb{E}[\varphi(X_N)] - \mathbb{E}[\varphi(\mathcal{X}^0(N\tau))] &= \mathbb{E}[\varphi(\bar{X}_N)] - \mathbb{E}[\varphi(\mathcal{X}^0(N\tau))] \nonumber \\
&= \mathbb{E}[u^0(0, \bar{X}_N, \bar{V}_N)] - \mathbb{E}[u^0(s_N, \bar{X}_0, \bar{V}_0)] \tag*{=}
\end{align}
$$

Recalling that $(\bar{X}_n, \bar{V}_n) = (\tilde{X}^\tau(s_n), \tilde{V}^\tau(s_n))$ for all $n \in \{0, \dots, N\}$, where the auxiliary continuous time processes $(\tilde{X}^\tau(s), \tilde{V}^\tau(s))_{s \ge 0}$ are defined by (32). A standard telescoping sum argument gives us the following decomposition

$$
(39) \quad \mathbb{E}[u^0(0, \bar{X}_N, \bar{V}_N)] - \mathbb{E}[u^0(s_N, \bar{X}_0, \bar{V}_0)] = \sum_{n=0}^{N-1} \epsilon_n,
$$

where for all $n \in \{0, \dots, N-1\}$ the error term $\epsilon_n$ is defined by

$$
(40) \quad \epsilon_n = \mathbb{E}[u^0(s_N - s_{n+1}, \tilde{X}^\tau(s_{n+1}), \tilde{V}^\tau(s_{n+1}))] - \mathbb{E}[u^0(s_N - s_n, \tilde{X}^\tau(s_n), \tilde{V}^\tau(s_n))].
$$

Then one proceeds by applying the Itô formula, using (33). Error bounds for $\epsilon_n$ are obtained in the
proof of Proposition 5.2. To analyze the error terms, it is important to observe that the function
$u^0$ is solution to the linear partial differential equation

$$
(41) \quad \begin{cases} \partial_s u^0(s, x, v) = \langle \nabla_x u^0(s, x, v), v \rangle - \langle \nabla_v u^0(s, x, v), \alpha v + \nabla F(x) \rangle, & \forall s \ge 0, \forall x, v \in \mathbb{R}^d, \\ u^0(0, x, v) = \varphi(x), & \forall x, v \in \mathbb{R}^d. \end{cases}
$$

The remainder of this section is organized as follows. The large time behavior of the solution
$\mathcal{X}^0(T) = \mathcal{X}_{\bar{X}_0, \bar{V}_0}^0(T)$ when $T \to \infty$ is then studied in Section 5.1, this is the key ingredient to treat
the first error term appearing in the right-hand side of (36). Regularity properties of the function $u^0$
are stated in Section 5.2, with proofs postponed to Appendix A. The application of those results to
obtain upper bounds on the second error term appearing on the right-hand side of (36), following
the strategy outlined above, is then presented in Section 5.3. Finally, Section 5.4 combines the upper bounds on the error terms and provides the proof of Theorem 1.

## 5.1. Large time behavior of the solution of (17).

**Proposition 5.1.** Assume the Assumption 2.1 be satisfied.

Let $\nu(\alpha, \mu, L_F)$ be defined by (19), and let

$$C(\alpha, \mu, L_F) = \frac{\kappa_2(\alpha, L_F)}{\kappa_1(\mu)},$$

where $\kappa_1$ and $\kappa_2$ are defined as follow

$$\kappa_1(\mu) = \frac{1}{2}\min(\mu, \frac{1}{2}), \quad \kappa_2(\alpha, L_F) = \frac{1}{2}\max(L_F + \alpha^2, \frac{3}{2}).$$

For any initial value $(\bar{X}_0, \bar{V}_0) \in L^2(\Omega, \mathbb{R}^d) \times L^2(\Omega, \mathbb{R}^d)$ and any $T \in (0, \infty)$, one has

$$(42) \quad \mathbb{E}[\|\mathcal{X}^0(T) - x^*\|^2] + \mathbb{E}[\|\mathcal{V}^0(T)\|^2] \le C(\alpha, \mu, L_F)e^{-2\nu(\alpha, \mu, L_F)T} (\mathbb{E}[\|\bar{X}_0 - x^*\|^2] + \mathbb{E}[\|\bar{V}_0\|^2]),$$

where we recall that $(\mathcal{X}^0(s), \mathcal{V}^0(s))_{s \ge 0}$ denote the solution of Heavy Ball system (17), with initial value $(\mathcal{X}^0(0), \mathcal{V}^0(0)) = (\bar{X}_0, \bar{V}_0) \in L^2(\Omega, \mathbb{R}^d) \times L^2(\Omega, \mathbb{R}^d)$

**Remark 5.1.** Note that for the previous proposition, we did not assume $\alpha > \frac{L_F}{\sqrt{\mu}}$ and that this assumption is only useful for the following lemma.

*Proof.* The proof is based on the function $\Gamma$ introduced in (26). Let $\bar{X}_0, \bar{V}_0 \in L^2(\Omega, \mathbb{R}^d)$. We begin by proving that the function $\Gamma$ is non-increasing along the trajectories $(\mathcal{X}^0(s), \mathcal{V}^0(s))$ of the Heavy Ball equation (17) with initial condition $(\mathcal{X}^0(0), \mathcal{V}^0(0)) = (\bar{X}_0, \bar{V}_0)$. Then, applying the Grönwall's lemma then yields an exponential decay estimate, which, combined with the inequalities in (27), completes the proof.

By applying chain rule to the function $s \mapsto \Gamma(\mathcal{X}^0(s), \mathcal{V}^0(s))$, one has

$$\begin{align*}
\frac{d}{ds}\mathbb{E}[\Gamma(\mathcal{X}^0(s), \mathcal{V}^0(s))] &= \mathbb{E}[\langle\nabla_x\Gamma(\mathcal{X}^0(s), \mathcal{V}^0(s)), \frac{d}{ds}\mathcal{X}^0(s)\rangle] + \mathbb{E}[\langle\nabla_v\Gamma(\mathcal{X}^0(s), \mathcal{V}^0(s)), \frac{d}{ds}\mathcal{V}^0(s)\rangle] \\
&= \mathbb{E}[\langle\nabla F(\mathcal{X}^0(s)) + \frac{\alpha^2}{2}(\mathcal{X}^0(s) - x^*) + \frac{\alpha}{2}\mathcal{V}^0(s), \mathcal{V}^0(s)\rangle] \\
&\quad + \mathbb{E}[\langle\frac{\alpha}{2}(\mathcal{X}^0(s) - x^*) + \mathcal{V}^0(s), -\alpha\mathcal{V}^0(s) - \nabla F(\mathcal{X}^0(s))\rangle] \\
&= -\frac{\alpha}{2}\mathbb{E}[\langle\nabla F(\mathcal{X}^0(s)), \mathcal{X}^0(s) - x^*\rangle] - \frac{\alpha}{2}\mathbb{E}[\|\mathcal{V}^0(s)\|^2].
\end{align*}$$

Applying the $\mu$-convexity condition (10) on the objective function $F$ from Assumption 2.1, one obtains

$$\frac{d}{ds}\mathbb{E}[\Gamma(\mathcal{X}^0(s), \mathcal{V}^0(s))] \le -\frac{\alpha\mu}{2}\mathbb{E}[\|\mathcal{X}^0(s) - x^*\|^2] - \frac{\alpha}{2}\mathbb{E}[\|\mathcal{V}^0(s)\|^2].$$

Then, using the right-hand side inequality (27), one has

$$\frac{d}{ds}\mathbb{E}[\Gamma(\mathcal{X}^0(s), \mathcal{V}^0(s))] \le -2\nu(\alpha, \mu, L_F)\mathbb{E}[\Gamma(\mathcal{X}^0(s), \mathcal{V}^0(s))],$$

where $\nu(\alpha, \mu, L_F)$ is given by (19). Let $T \in (0, \infty)$, applying the Grönwall inequality, one has

$$\mathbb{E}[\Gamma(\mathcal{X}^0(T), \mathcal{V}^0(T))] \le e^{-2\nu(\alpha, \mu, L_F)T} \mathbb{E}[\Gamma(\bar{X}_0, \bar{V}_0)].$$

Then, using the inequalities (27), we obtain

$$\mathbb{E}[\|\mathcal{X}^0(T) - x^*\|^2] + \mathbb{E}[\|\mathcal{V}^0(T)\|^2] \le \frac{\max(L_F + \alpha^2, \frac{3}{2})}{\min(\mu, \frac{1}{2})} e^{-2\nu(\alpha, \mu, L_F)T} (\mathbb{E}[\|\bar{X}_0 - x^*\|^2] + \mathbb{E}[\|\bar{V}_0\|^2]).$$
This concludes the proof of Proposition 5.1.

□

As we specified earlier, this bound is not optimal; it is possible to do better, see for instance [17, 2, 3, 4, 20].

**5.2. Regularity properties of the auxiliary function $u^0$.** In this section, we provide a result to obtain uniform time bounds for (36). Recall that the auxiliary function $u^0$ is defined by (37).

**Lemma 5.1.** Assume that the objective function $F$ satisfies the Assumptions 2.1. Assume also $\alpha > \frac{L_F}{\sqrt{\mu}}$.

Let us define $\gamma$ as follow

$$ \gamma = \gamma(\alpha, \mu, L_F) = \frac{\alpha^2 \mu - L_F^2}{2 \max(\alpha^2, \frac{3}{2})} \min\left(\frac{1}{2\alpha\mu}, \frac{\alpha\mu}{L_F^2 + \alpha^2\mu}\right). $$

Let $\varphi : \mathbb{R}^d \to \mathbb{R}$ be of class $C^2$ with bounded second order derivative. For all $s \ge 0$ the mapping $u^0(s, \cdot, \cdot)$ defined by (37) is of class $C^2$. Moreover, there exists a positive real number $C^0 \in (0, \infty)$ such that the first order derivative of $u^0$ satisfies for all $s \ge 0$

$$ (43) \quad \sup_{x,v \in \mathbb{R}^d} \sup_{k \in \mathbb{R}^{2d} \setminus \{0\}} \frac{|Du^0(s, x, v).k|}{(1 + \|x - x^*\| + \|v\|)\|k\|} \le C^0 (\|\nabla \varphi(x^*)\| + \|\varphi\|_2) e^{-\gamma s}, $$

and such that the second order derivative satisfies for all $s \ge 0$

$$ (44) \quad \sup_{x,v \in \mathbb{R}^d} \sup_{k_1, k_2 \in \mathbb{R}^{2d} \setminus \{0\}} \frac{|D^2 u^0(s, x, v) (k_1, k_2)|}{(1 + \|x - x^*\| + \|v\|) \|k_1\| \|k_2\|} \le C^0 (\|\nabla \varphi(x^*)\| + \|\varphi\|_2) e^{-\gamma s}. $$

**Remark 5.2.** The assumption $\alpha > \frac{L_F}{\sqrt{\mu}}$ ensures that $\gamma > 0$, and this assumption is essential only for this Lemma, i.e., to obtain uniform time bounds for (36).

The proof of Lemma 5.1 is given in Appendix A.

**5.3. Proof of weak error estimates.** We now focus on the main task of the proof of Theorem 1, which consists in proving error estimates for $\mathbb{E}[\varphi(X_N)] - \mathbb{E}[\varphi(X^0(N\tau))]$, uniformly with respect to $N$, with order of convergence 1 with respect to $\tau$.

**Proposition 5.2.** Assume that the objective function $F$ satisfies the Assumption 2.1, that the diffusion coefficient $\sigma$ satisfies the Assumption 2.2. Assume that $\alpha > \frac{L_F}{\sqrt{\mu}}$.

There exist positive real numbers $\bar{\tau} \in (0, \tau_{\max})$ and $C \in (0, \infty)$ such that for any $C^2$ mapping $\varphi : \mathbb{R}^d \to \mathbb{R}$ with bounded second order derivative, for any initial value $x_0, v_0 \in \mathbb{R}^d$, and for any $\tau \in (0, \bar{\tau})$, one has

$$ (45) \quad \sup_{N \in \mathbb{N}^*} \left| \mathbb{E}[\varphi(X_N)] - \mathbb{E}[\varphi(X^0(N\tau))] \right| \le C_1(\varphi, x_0, v_0)\tau, $$

where the positive real number $C_1(\varphi, x_0, v_0)$ is given by (21).

In order to prove Proposition 5.2, we follow a standard approach employed to prove weak error estimates for numerical methods applied to stochastic differential equations, sketched at the beginning of Section 5 above. Having exponential decrease with respect to $s$ in the bounds (43) and (44) of the first and second derivatives with respect to $(x, v)$ of the solution of the partial differential equation (37) is fundamental in order to obtain a right-hand side in (45) which is independent of $N$. The value of $\bar{\tau}$ is imposed to apply some moment bounds on the solution of the stochastic optimization Heavy Ball scheme $(X_n, V_n)_{n \ge 0}$ given by Lemma 4.1 and on the processes $(\tilde{X}^\tau(s), \tilde{V}^\tau(s))_{s \ge 0}$ given by Lemma 4.2.
To prove Proposition 4.2, we adopt the same approach as the one used in [5], relying on the exponential decay properties established in inequalities (43) and (44) for the first and second derivatives with respect to $(x, v)$ of the solution to the partial differential equation (41).

**Proof of Proposition 5.2.** Let the function $\varphi : \mathbb{R}^d \to \mathbb{R}$ and the initial value $x_0, v_0 \in \mathbb{R}^d$ be given. Combining the expressions (38) and (39) for the weak error $\mathbb{E}[\varphi(X_N)] - \mathbb{E}[\varphi(X^0(N\tau))]$, it is sufficient to prove the following upper bound: for all $N \in \mathbb{N}^*$ and $\tau \in (0, \bar{\tau})$, for all $n \in \{0, \dots, N-1\}$, one has

$$ (46) \qquad |\epsilon_n| \le C_1(\varphi, x_0, v_0)\tau \int_{s_n}^{s_{n+1}} e^{-\gamma(s_N-s)} ds, $$

with $C_1(\varphi, x_0, v_0)$ defined by (21) (up to a modification of the multiplicative constant $C \in (0, \infty)$)

Let us prove the claim (46) above. For all $N \in \mathbb{N}^*$ and $n \in \{0, \dots, N-1\}$, recall that $s \in [s_n, s_{n+1}] \mapsto (\tilde{X}^\tau(s), \tilde{V}^\tau(s))$ is solution to the stochastic differential system (33). Therefore, applying Itô's formula on the interval $[s_n, s_{n+1}]$ yields the following decomposition of the error term $\epsilon_n$: one has

$$ \begin{aligned} \epsilon_n = & \int_{s_n}^{s_{n+1}} \mathbb{E}[-\partial_s u^0(s_N - s, \tilde{X}^\tau(s), \tilde{V}^\tau(s))] ds \\ & + \int_{s_n}^{s_{n+1}} \mathbb{E}[\langle \nabla_x u^0(s_N - s, \tilde{X}^\tau(s), \tilde{V}^\tau(s)), \bar{V}_n \rangle] ds \\ & - \int_{s_n}^{s_{n+1}} \mathbb{E}[\langle \nabla_v u^0(s_N - s, \tilde{X}^\tau(s), \tilde{V}^\tau(s)), \alpha \bar{V}_n + \nabla F(\bar{X}_{n+1}) \rangle] ds \\ & + \frac{\tau}{2} \int_{s_n}^{s_{n+1}} \mathbb{E}[\nabla_v^2 u^0(s_N - s, \tilde{X}^\tau(s), \tilde{V}^\tau(s)) : a_\tau(s_{n+1}, \tilde{X}^\tau(s_{n+1}))] ds \end{aligned} $$

Recall that the function $u^0$ is the solution to the partial differential equation (41). This allows us to decompose $\epsilon_n$ as

$$ \epsilon_n = \epsilon_{n,1} + \epsilon_{n,2} + \epsilon_{n,3} + \epsilon_{n,4}, $$

where for all $n \in \{0, \dots, N-1\}$ one defines

$$ \begin{aligned} \epsilon_{n,1} &= \int_{s_n}^{s_{n+1}} \mathbb{E}[\langle \nabla_x u^0(s_N - s, \tilde{X}^\tau(s), \tilde{V}^\tau(s)), \bar{V}_n - \tilde{V}^\tau(s) \rangle] ds \\ \epsilon_{n,2} &= \alpha \int_{s_n}^{s_{n+1}} \mathbb{E}[\langle \nabla_v u^0(s_N - s, \tilde{X}^\tau(s), \tilde{V}^\tau(s)), \tilde{V}^\tau(s) - \bar{V}_n \rangle] ds \\ \epsilon_{n,3} &= \int_{s_n}^{s_{n+1}} \mathbb{E}[\langle \nabla_v u^0(s_N - s, \tilde{X}^\tau(s), \tilde{V}^\tau(s)), \nabla F(\tilde{X}^\tau(s)) - \nabla F(\bar{X}_{n+1}) \rangle] ds \\ \epsilon_{n,4} &= \frac{\tau}{2} \int_{s_n}^{s_{n+1}} \mathbb{E}[\nabla_v^2 u^0(s_N - s, \tilde{X}^\tau(s), \tilde{V}^\tau(s)) : a_\tau(s_{n+1}, \tilde{X}^\tau(s_{n+1}))] ds \end{aligned} $$

The claim (46) follows from the bounds for $\epsilon_{n,1}$, $\epsilon_{n,2}$, $\epsilon_{n,3}$ and $\epsilon_{n,4}$ which are obtained below.

**Upper bounds for $\epsilon_{n,1}$.** Applying the inequality (43) from Lemma 5.1 on the first order derivative of $u(s_N - s, \cdot, \cdot)$, one has

$$ |\epsilon_{n,1}| \le C(\|\nabla\varphi(x^*)\| + \|\varphi\|_2) \int_{s_n}^{s_{n+1}} e^{-\gamma(s_N-s)} \mathbb{E}[(1 + \|\tilde{X}^\tau(s) - x^*\| + \|\tilde{V}^\tau(s)\|)\|\tilde{V}^\tau(s) - \tilde{V}^\tau(s_n)\|] ds. $$

Assume that $\bar{\tau} \le \bar{\tau}_1$ where $\bar{\tau}_1 \in (0, \tau_{\max})$ is given by Lemma 4.2, then applying the Cauchy-Schwarz inequality, using the moment bounds (34) and the increment bounds (35) from Lemma 4.2 for the
auxiliary processes $(\tilde{X}^{\tau}(s), \tilde{V}^{\tau}(s))_{s \ge 0}$, one obtains the upper bounds

$$|\epsilon_{n,1}| \le C(\|\nabla\varphi(x^*)\| + \|\varphi\|_2)\tau(1 + \|x_0 - x^*\|^2 + \|v_0\|^2) \int_{s_n}^{s_{n+1}} e^{-\gamma(s_N-s)} ds \\ \le C_1(\varphi, x_0, v_0)\tau \int_{s_n}^{s_{n+1}} e^{-\gamma(s_N-s)} ds,$$

for all $N \in \mathbb{N}^*$, all $n \in \{0, \dots, N-1\}$ and all $\tau \in (0, \bar{\tau})$.

Upper bounds for $\epsilon_{n,2}$. The treatment of this term exactly the same approach as for the term $\epsilon_{n,1}$. Using the inequality (43) from Lemma 5.1 on the first order derivative of $u(s_N - s, \cdot, \cdot)$, one has

$$|\epsilon_{n,2}| \le C(\|\nabla\varphi(x^*)\| + \|\varphi\|_2) \int_{s_n}^{s_{n+1}} e^{-\gamma(s_N-s)} \mathbb{E}[(1 + \|\tilde{X}^{\tau}(s) - x^*\| + \|\tilde{V}^{\tau}(s)\|)\|\tilde{V}^{\tau}(s) - \tilde{V}^{\tau}(s_n)\|] ds.$$

Applying the Cauchy-Schwarz inequality and using the moment bounds (34) together with the increment bounds (35) from Lemma 4.2 for the auxiliary processes $(\tilde{X}^{\tau}(s), \tilde{V}^{\tau}(s))_{s \ge 0}$, one obtains the upper bounds

$$|\epsilon_{n,2}| \le C(\|\nabla\varphi(x^*)\| + \|\varphi\|_2)\tau(1 + \|x_0 - x^*\|^2 + \|v_0\|^2) \int_{s_n}^{s_{n+1}} e^{-\gamma(s_N-s)} ds \\ \le C_1(\varphi, x_0, v_0)\tau \int_{s_n}^{s_{n+1}} e^{-\gamma(s_N-s)} ds,$$

for all $N \in \mathbb{N}^*$, all $n \in \{0, \dots, N-1\}$ and all $\tau \in (0, \bar{\tau})$.

Upper bounds for $\epsilon_{n,3}$.

Recall that $\nabla F: \mathbb{R}^d \to \mathbb{R}^d$ is a Lipschitz continuous mapping (see Equation (8) from Assumption 2.1). Therefore, using the inequality (43) from Lemma 5.1 on the first order derivative of $u(s_N - s, \cdot, \cdot)$, one has

$$|\epsilon_{n,3}| \le C(\|\nabla\varphi(x^*)\| + \|\varphi\|_2) \int_{s_n}^{s_{n+1}} e^{-\gamma(s_N-s)} \mathbb{E}[(1 + \|\tilde{X}^{\tau}(s) - x^*\| + \|\tilde{V}^{\tau}(s)\|)\|\tilde{X}^{\tau}(s) - \tilde{X}^{\tau}(s_{n+1})\|] ds.$$

Applying the Cauchy-Schwarz inequality, using the moment bounds (34) and the increment bounds (35) from Lemma 4.2 for the auxiliary process $(\tilde{X}^{\tau}(s), \tilde{V}^{\tau}(s))_{s \ge 0}$, one obtains the upper bounds

$$|\epsilon_{n,3}| \le C(\|\nabla\varphi(x^*)\| + \|\varphi\|_2)\tau(1 + \|x_0 - x^*\|^2 + \|v_0\|^2) \int_{s_n}^{s_{n+1}} e^{-\gamma(s_N-s)} ds \\ \le C_1(\varphi, x_0, v_0)\tau \int_{s_n}^{s_{n+1}} e^{-\gamma(s_N-s)} ds,$$

for all $N \in \mathbb{N}^*$, all $n \in \{0, \dots, N-1\}$ and all $\tau \in (0, \bar{\tau})$.

Upper bounds for $\epsilon_{n,4}$.

Recall that the mapping $(s, x) \mapsto \sigma(s, x)$ has at most linear growth with respect to $x \in \mathbb{R}^d$, uniformly with respect to $s \ge 0$ (see Equation (15) from Assumption 2.2). Recall also that $a$ is defined by (16). Therefore, using the inequality (44) from Lemma 5.1 on the second order derivative of $u(s_N - s, \cdot, \cdot)$, one has

$$|\epsilon_{n,4}| \le C(\|\nabla\varphi(x^*)\| + \|\varphi\|_2)\tau\zeta(s_{n+1}\tau)^2 \int_{s_n}^{s_{n+1}} e^{-\gamma(s_N-s)} \mathbb{E}[(1 + \|\tilde{X}^{\tau}(s) - x^*\| + \|\tilde{V}^{\tau}(s)\|)^3] ds.$$

Recall that $\zeta(s_{n+1}\tau) \le \|\zeta\|_{\infty}$ for all $n \in \mathbb{N}$. Assume that $\bar{\tau} \le \bar{\tau}_2$ where $\bar{\tau}_2 \in (0, \tau_{\max})$ is given by Lemma 4.2. Applying the moment bounds (34) from Lemma 4.2, one obtains the upper bounds

$$|\epsilon_{n,4}| \le C(\|\nabla\varphi(x^*)\| + \|\varphi\|_2)\tau(1 + \|x_0 - x^*\|^3 + \|v_0\|^3) \int_{s_n}^{s_{n+1}} e^{-\gamma(s_N-s)} ds$$
$$ \leq C_1(\varphi, x_0, v_0)\tau \int_{s_n}^{s_{n+1}} e^{-\gamma(s_N-s)} ds, $$

for all $N \in \mathbb{N}^*$, all $n \in \{0, \dots, N-1\}$ and all $\tau \in (0, \bar{\tau})$.

*Conclusion.* Gathering the upper bounds on $|\epsilon_{n,1}|, |\epsilon_{n,2}|, |\epsilon_{n,3}|$ and $|\epsilon_{n,4}|$ obtained above then yields the claim (46), under the condition $\bar{\tau} \le \bar{\tau}_2$.

The proof of the weak error estimate (45) is then straightforward: for all $N \in \mathbb{N}^*$ one has

$$ \begin{aligned} |\mathbb{E}[\varphi(X_N)] - \mathbb{E}[\varphi(\mathcal{X}^0(N\tau))]| &\le \sum_{n=0}^{N-1} |\epsilon_n| \\ &\le C_1(\varphi, x_0, v_0)\tau \sum_{n=0}^{N-1} \int_{s_n}^{s_{n+1}} e^{-\gamma(s_N-s)} ds \\ &\le C_1(\varphi, x_0, v_0)\tau \int_0^{s_N} e^{-\gamma(s_N-s)} ds \\ &\le \frac{C_1(\varphi, x_0, v_0)\tau}{\gamma}. \end{aligned} $$

This concludes the proof of Proposition 5.2.

<figure><img src="image_5.png" /></figure>

### 5.4. Proof of Theorem 1.
We are now in a position to provide the proof of Theorem 1.

*Proof of Theorem 1.* First, let us establish the weak error estimates (20). Recall that owing to (36) the error term appearing in the left-hand side of (20) can be decomposed as

$$ \mathbb{E}[\varphi(X_N)] - \varphi(x^*) = \mathbb{E}[\varphi(\mathcal{X}^0(N\tau))] - \varphi(x^*) + \mathbb{E}[\varphi(X_N)] - \mathbb{E}[\varphi(\mathcal{X}^0(N\tau))]. $$

On the one hand, using the inequalities (1), (7) and applying the upper bound (42) from Proposition 5.1, one has

$$ \begin{aligned} |\mathbb{E}[\varphi(\mathcal{X}^0(N\tau))] - \varphi(x^*)| &\le C \mathbb{E}[(\|\nabla\varphi(x^*)\| + \|\varphi\|_2\|\mathcal{X}^0(N\tau) - x^*\|)\|\mathcal{X}^0(N\tau) - x^*\|] \\ &\le \mathbb{E}[C_1(\varphi, \bar{X}_0, \bar{V}_0)]e^{-\nu(\alpha, \mu, L_F)N\tau} \\ &\le C_1(\varphi, x_0, v_0)e^{-\nu(\alpha, \mu, L_F)N\tau}. \end{aligned} $$

On the other hand, applying the uniform in time weak error estimates (45) from Proposition 5.2, one has for all $N \in \mathbb{N}$ and all $\tau \in (0, \bar{\tau})$

$$ |\mathbb{E}[\varphi(X_N)] - \mathbb{E}[\varphi(\mathcal{X}^0(N\tau))]| \le C_1(\varphi, x_0, v_0)\tau, $$

where $C_1(\varphi, x_0, v_0)$ is given by (21). Gathering the two upper bounds yields the weak error estimates (20).

Second, let us establish the error estimates on the expected residual (22). Using the decomposition (36) with $\varphi = F$, one has

$$ \mathbb{E}[F(X_N)] - F(x^*) = \mathbb{E}[F(\mathcal{X}^0(N\tau))] - F(x^*) + \mathbb{E}[F(X_N)] - \mathbb{E}[F(\mathcal{X}^0(N\tau))]. $$

On the one hand, using the inequality (13) and applying the upper bound (42), one has

$$ \mathbb{E}[F(\mathcal{X}^0(N\tau))] - F(x^*) \le C \mathbb{E}[\|\mathcal{X}^0(N\tau) - x^*\|^2] \le C e^{-2\nu(\alpha, \mu, L_F)N\tau} (\|x_0 - x^*\|^2 + \|v_0\|^2) $$

On the other hand, applying the uniform in time weak error estimates (45) from Proposition 5.2 with $\varphi = F$, one has for all $N \in \mathbb{N}$ and all $\tau \in (0, \bar{\tau})$

$$ |\mathbb{E}[F(X_N)] - \mathbb{E}[F(\mathcal{X}^0(N\tau))]| \le C(1 + \|x_0 - x^*\|^3 + \|v_0\|^3)\tau. $$

Gathering the two upper bounds yields the error estimates on the expected residual (22).
Finally, let us establish the strong error estimates (23). Owing to the inequality (12), one obtains

$$ \mathbb{E}[\|X_N - x^*\|^2] \le \frac{2}{\mu}(\mathbb{E}[F(X_N)] - F(x^*)) $$

and it then suffices to apply the error estimates on the expected residual (22) to get (23).

The proof of Theorem 1 is thus completed.

<figure><img src="image_6.png" /></figure>

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APPENDIX A. PROOF OF LEMMA 5.1

The objective of this section is to prove the estimates (43) and (44) on the first and second order derivatives of the mapping $u^0(s, \cdot, \cdot)$ defined by (37). Let us start with some preliminary results.

Repeating the proof of the inequality (42) with an arbitrary initial value $(x, v) \in \mathbb{R}^d \times \mathbb{R}^d$, it is straightforward to obtain the following bound on the solution $(\mathcal{X}_{x,v}^0(s), \mathcal{V}_{x,v}^0(s))_{s \ge 0}$ of the ordinary differential equation (17) with arbitrary initial values $(\mathcal{X}_{x,v}^0(0), \mathcal{V}_{x,v}^0(0)) = (x, v) \in \mathbb{R}^d \times \mathbb{R}^d$: one has

$$ (47) \qquad \sup_{s \ge 0} \sup_{x,v \in \mathbb{R}^d} \frac{\|\mathcal{X}_{x,v}^0(s)\| + \|\mathcal{V}_{x,v}^0(s)\|}{1 + \|x - x^*\| + \|v\|} < \infty. $$

In this section, we let $\varphi : \mathbb{R}^d \to \mathbb{R}$ be a mapping of class $\mathcal{C}^2$ with a bounded second order derivative. In general its first order derivative is not bounded, however it has at most linear growth, see (1).

To simplify the notation in the proof below, let $b(x) = -\nabla F(x)$ for all $x \in \mathbb{R}^d$. As a result one has $Db(x).k = -\nabla^2 F(x)k$ for all $x \in \mathbb{R}^d$ and $k \in \mathbb{R}^d$. Applying the equivalent formulation (10) of the $\mu$-convexity condition (9) from Assumption 2.1, then one has the inequality

$$ (48) \qquad \langle Db(x).k, k \rangle \le -\mu \|k\|^2, \quad \forall x \in \mathbb{R}^d, k \in \mathbb{R}^d. $$

In this section, we consider the function $\tilde{\Gamma}$, defined as follows

$$ \tilde{\Gamma}(x, v) = \frac{1}{2} \|v\|^2 + \frac{\alpha}{2} \langle x, v \rangle + \frac{\alpha^2}{4} \|x\|^2, \quad \forall x, v \in \mathbb{R}^d. $$

By applying the same methods used for the function $\Gamma$ to obtain the bounds (27), we obtain the following inequalities

$$ (49) \qquad \tilde{\kappa}_1(\alpha)(\|x\|^2 + \|v\|^2) \le \tilde{\Gamma}(x, v) \le \tilde{\kappa}_2(\alpha)(\|x\|^2 + \|v\|^2), \quad \forall x, v \in \mathbb{R}^d, $$

with $\tilde{\kappa}_1(\alpha) = \frac{1}{2} \min(\frac{\alpha^2}{3}, \frac{1}{2})$ and $\tilde{\kappa}_2(\alpha) = \frac{1}{2} \max(\alpha^2, \frac{3}{2})$. The proof of Lemma 5.1 proceeds by proving the inequalities (43) and (44).

*Proof of the inequality* (43). Owing to the definition (37) of the mapping $u^0$, for all $s \ge 0$, $(x, v) \in \mathbb{R}^d \times \mathbb{R}^d$ and $k = (k^x, k^v) \in \mathbb{R}^d \times \mathbb{R}^d$, one has

$$ Du^0(s, x, v).k = D\varphi(\mathcal{X}_{x,v}^0(t)).\eta_{k^x, k^v}^{0,x}(s), $$

where $s \ge 0 \mapsto (\eta_{k^x, k^v}^{0,x}(s), \eta_{k^x, k^v}^{0,v}(s)) \in \mathbb{R}^d \times \mathbb{R}^d$ is the solution of the linear differential equation

$$ \left\{ \begin{array}{ll} \displaystyle \frac{d}{ds}\eta_{k^x, k^v}^{0,x}(s) = \eta_{k^x, k^v}^{0,v}(s), & \forall s \ge 0, \\ \\ \displaystyle \frac{d}{ds}\eta_{k^x, k^v}^{0,v}(s) = -\alpha\eta_{k^x, k^v}^{0,v}(s) + Db(\mathcal{X}_{x,v}^0(s)).\eta_{k^x, k^v}^{0,x}(s), & \forall s \ge 0, \\ \\ \displaystyle (\eta_{k^x, k^v}^{0,x}(0), \eta_{k^x, k^v}^{0,v}(0)) = (k^x, k^v) = k. \end{array} \right. $$

Owing to the inequality (1)

$$ (50) \qquad |Du^0(s, x, v).k| \le \left( \|\nabla\varphi(x^*)\| + \|\varphi\|_2 \|\mathcal{X}_{x,v}^0(s) - x^*\| \right) \|\eta_{k^x, k^v}^{0,x}(s)\|, \quad \forall s \ge 0. $$

One thus needs to prove an upper bound for $\|\eta_{k^x, k^v}^{0,x}(s)\|$. Employing the inequality (48) and the chain rule on the function $s \mapsto \tilde{\Gamma}(\mathcal{X}_{x,v}^0(s), \mathcal{V}_{x,v}^0(s))$ where $(\mathcal{X}_{x,v}^0(s), \mathcal{V}_{x,v}^0(s))$ are solution of ordinary differential system (17) for initial condition $(\mathcal{X}_{x,v}^0(0), \mathcal{V}_{x,v}^0(0)) = (x, v) \in \mathbb{R}^d \times \mathbb{R}^d$ yields, for all $s \ge 0$,

$$ \frac{d}{ds} \tilde{\Gamma}(\eta_{k^x, k^v}^{0,x}(s), \eta_{k^x, k^v}^{0,v}(s)) = \langle \nabla_x \tilde{\Gamma}(\eta_{k^x, k^v}^{0,x}(s), \eta_{k^x, k^v}^{0,v}(s)), \frac{d}{ds} \eta_{k^x, k^v}^{0,x}(s) \rangle + \langle \nabla_v \tilde{\Gamma}(\eta_{k^x, k^v}^{0,x}(s), \eta_{k^x, k^v}^{0,v}(s)), \frac{d}{ds} \eta_{k^x, k^v}^{0,x}(s) \rangle $$
$$
\begin{align*}
&= \left\langle \frac{\alpha}{2} \eta_{k^x, k^v}^{0,v}(s) + \frac{\alpha^2}{2} \eta_{k^x, k^v}^{0,x}(s), \eta_{k^x, k^v}^{0,v}(s) \right\rangle \\
&\quad + \left\langle \eta_{k^x, k^v}^{0,v}(s) + \frac{\alpha}{2} \eta_{k^x, k^v}^{0,x}(s), -\alpha \eta_{k^x, k^v}^{0,x}(s) + Db(\mathcal{X}_{x,v}^0(s)), \eta_{k^x, k^v}^{0,x}(s) \right\rangle \\
&= -\frac{\alpha}{2} \left\langle Db(\mathcal{X}_{x,v}^0(s)), \eta_{k^x, k^v}^{0,x}(s), \eta_{k^x, k^v}^{0,x}(s) \right\rangle - \frac{\alpha}{2} \|\eta_{k^x, k^v}^{0,v}(s)\|^2 \\
&\quad - \left\langle Db(\mathcal{X}_{x,v}^0(s)), \eta_{k^x, k^v}^{0,x}(s), \eta_{k^x, k^v}^{0,v}(s) \right\rangle.
\end{align*}
$$

By applying the $\mu$-convexity condition (48) and Cauchy-Schwarz inequality we obtain for all $s \ge 0$

$$
\frac{d}{ds} \tilde{\Gamma}(\eta_{k^x, k^v}^{0,x}(s), \eta_{k^x, k^v}^{0,v}(s)) \le -\frac{\alpha \mu}{2} \|\eta_{k^x, k^v}^{0,x}(s)\|^2 - \frac{\alpha}{2} \|\eta_{k^x, k^v}^{0,v}(s)\|^2 + L \|\eta_{k^x, k^v}^{0,x}(s)\| \|\eta_{k^x, k^v}^{0,v}(s)\|.
$$

Using Young inequality, we obtain for all $s \ge 0$ and $\epsilon > 0$

$$
\frac{d}{ds} \tilde{\Gamma}(\eta_{k^x, k^v}^{0,x}(s), \eta_{k^x, k^v}^{0,v}(s)) \le \frac{L_F^2 - \alpha \mu \epsilon}{2\epsilon} \|\eta_{k^x, k^v}^{0,x}(s)\|^2 + \frac{\epsilon - \alpha}{2} \|\eta_{k^x, k^v}^{0,v}(s)\|^2.
$$

Here, we want to apply Gröwall's lemma to obtain inequality (43). To do so, we will use inequali-
ties (49). However, before that, it is necessary to impose $L_F^2 - \alpha\mu\epsilon < 0$ and $\epsilon - \alpha < 0$ in order to
obtain exponentially decreasing bounds, this is equivalent to choose $\epsilon \in (\frac{L_F^2}{\alpha\mu}, \alpha)$. We specify that
this interval is not empty because we assumed $\alpha > \frac{L_F}{\sqrt{\mu}}$. Taking $\epsilon = \frac{\alpha}{2} + \frac{L_F^2}{2\alpha\mu}$

$$
\frac{d}{ds} \tilde{\Gamma}(\eta_{k^x, k^v}^{0,x}(s), \eta_{k^x, k^v}^{0,v}(s)) \le -\frac{\alpha \mu}{2(\alpha^2 \mu + L_F^2)} (\alpha^2 \mu - L_F^2) \|\eta_{k^x, k^v}^{0,x}(s)\|^2 - \frac{1}{4} (\alpha - \frac{L_F^2}{\alpha \mu}) \|\eta_{k^x, k^v}^{0,v}(s)\|^2, \quad \forall s \ge 0.
$$

We introduce the parameter $\gamma$, defined as follows

$$
(51) \qquad \gamma(\alpha, \mu, L_F) = \frac{\alpha^2 \mu - L_F^2}{4 \tilde{\kappa}_2(\alpha)} \min \left( \frac{1}{2\alpha\mu}, \frac{\alpha\mu}{L_F^2 + \alpha^2\mu} \right).
$$

where $\tilde{\kappa}_2(\alpha)$ is given by the right-hand side inequality (49).

By using the right-hand side inequality of (49), we have for all $s \ge 0$

$$
\frac{d}{ds} \tilde{\Gamma}(\eta_{k^x, k^v}^{0,x}(s), \eta_{k^x, k^v}^{0,v}(s)) \le -2\gamma(\alpha, \mu, L_F) \tilde{\Gamma}(\eta_{k^x, k^v}^{0,x}(s), \eta_{k^x, k^v}^{0,v}(s)),
$$

and applying the Grönwall lemma, one then obtains the upper bound

$$
\tilde{\Gamma}(\eta_{k^x, k^v}^{0,x}(s), \eta_{k^x, k^v}^{0,v}(s)) \le e^{-2\gamma(\alpha, \mu, L_F)s} \tilde{\Gamma}(k^x, k^v), \quad \forall s \ge 0.
$$

Finally, using the inequalities (49), there exists $C \in (0, \infty)$ such that we have the upper bound

$$
(52) \qquad \| \eta_{k^x, k^v}^{0,x}(s) \| \le C e^{-\gamma(\alpha, \mu, L_F)s} \|k\|, \quad \forall s \ge 0.
$$

Combining the inequalities (47) and (52) with the upper bound (50) for $|Du^0(s, x, y).k|$ above, there
exists a positive real number $C \in (0, \infty)$ such that for all $x, v \in \mathbb{R}^d, k = (k^x, k^v) \in \mathbb{R}^d \times \mathbb{R}^d$ and all
$s \ge 0$ one has

$$
|Du^0(s, x, v).k| \le C (\|\nabla\varphi(x^*)\| + \|\varphi\|_2) (1 + \|x - x^*\| + \|v\|) e^{-\gamma(\alpha, \mu, L_F)s} \|k\|.
$$

The proof of the inequality (43) is thus completed.

<figure><img src="image_7.png" /></figure>

Proof of the inequality (44). Owing to the definition (37) of the mapping $u^0$, for all $s \ge 0$, $x, v \in \mathbb{R}^d$
and $k_1 = (k_1^x, k_1^v)$, $k_2 = (k_2^x, k_2^v) \in \mathbb{R}^d \times \mathbb{R}^d$, one has

$$
D^2 u^0(s, x, v).(k_1, k_2) = D\varphi(\mathcal{X}_{x,v}^0(s)).\zeta_{(k_1^x, k_1^v),(k_2^x, k_2^v)}^{0,x}(s) + D^2\varphi(\mathcal{X}_{x,v}^0(s)).(\eta_{k_1^x, k_1^v}^{0,x}(s), \eta_{k_2^x, k_2^v}^{0,x}(s)),
$$
where $s \ge 0 \mapsto (\zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s), \zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s)) \in \mathbb{R}^d \times \mathbb{R}^d$ is the solution of the linear differential equation

$$
(53) \quad \left\{
\begin{array}{ll}
\displaystyle \frac{d}{ds} \zeta_{(k_1^x, k_1^v)}^{0,x} (s) = \zeta_{(k_1^x, k_1^v)}^{0,v} (s), & \forall s \ge 0 \\[2ex]
\displaystyle \frac{d}{ds} \zeta_{(k_1^x, k_1^v)}^{0,v} (s) = -\alpha \zeta_{(k_1^x, k_1^v)}^{0,v} (s) + Db(\mathcal{X}_{x,v}^0(s)). \zeta_{(k_1^x, k_1^v)}^{0,x} (s) \\
\phantom{\displaystyle \frac{d}{ds} \zeta_{(k_1^x, k_1^v)}^{0,v} (s) = -\alpha \zeta_{(k_1^x, k_1^v)}^{0,v} (s) + Db(\mathcal{X}_{x,v}^0(s))} + D^2b(\mathcal{X}_{x,v}^0(s)). (\eta_{k_1^x, k_1^v}^{0,x}(s), \eta_{k_2^x, k_2^v}^{0,x}(s)), & \forall s \ge 0, \\[2ex]
(\zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (0), \zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (0) = 0.
\end{array}
\right.
$$

Since $\varphi$ is of class $\mathcal{C}^2$ with bounded second order derivative, using the inequality (1), one obtains
(54)

$$
|D^2 u^0(s, x, v) (k_1, k_2)| \le \left( \|\nabla \varphi(x^*)\| + \|\varphi\|_2 \|\mathcal{X}_{x,v}^0(s) - x^*\| \right) \|\zeta_{(k_1^x, k_1^v)}^{0,x} - \zeta_{(k_2^x, k_2^v)}^{0,x}\|(s) + \|\varphi\|_2 \|\eta_{k_1^x, k_1^v}^{0,x}(s) - \eta_{k_2^x, k_2^v}^{0,x}(s)\|.
$$

Upper bounds for $\|\mathcal{X}_{x,v}^0(s) - x^*\|$ and $\|\eta_{k_j^x, k_j^v}^{0,x}(s)\|$ ($j=1,2$) are given by (47) and (52) respectively.

It thus remains to prove an upper bound for $\|\zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s)\|$. For all $s \ge 0$ by the chain rule one
has

$$
\begin{align*}
\frac{d}{ds} \tilde{\Gamma}(\zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s), \zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s))
&= \langle \nabla_x \tilde{\Gamma}(\zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s), \zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s) \rangle, \frac{d}{ds} \zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s) \rangle \\
&\quad + \langle \nabla_y \zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s), \zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s) \rangle, \frac{d}{ds} \zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s) \rangle \\
&= \langle \frac{\alpha}{2} \zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s) + \frac{\alpha^2}{2} \zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s), \zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s) \rangle \\
&\quad + \langle \zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s) + \frac{\alpha}{2} \zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s), -\alpha \zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s) + Db(\mathcal{X}_{x,v}^0(s)). \zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s) \rangle \\
&\quad + \langle \zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s) + \frac{\alpha}{2} \zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s), D^2b(\mathcal{X}_{x,v}^0(s)). (\eta_{k_1^x, k_1^v}^{0,x}(s), \eta_{k_2^x, k_2^v}^{0,x}(s)) \rangle \\
&= -\frac{\alpha}{2} \langle Db(\mathcal{X}_{x,v}^0(s)). \zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s), \zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s) \rangle - \frac{\alpha}{2} \|\zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s)\|^2 \\
&\quad - \langle Db(\mathcal{X}_{x,v}^0(s)). \zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s), \zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s) \rangle \\
&\quad - \langle D^2b(\mathcal{X}_{x,v}^0(s)). (\eta_{k_1^x, k_1^v}^{0,x}(s), \eta_{k_2^x, k_2^v}^{0,x}(s)), \zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s) \rangle \\
&\quad - \langle D^2b(\mathcal{X}_{x,v}^0(s)). (\eta_{k_1^x, k_1^v}^{0,x}(s), \eta_{k_2^x, k_2^v}^{0,x}(s)), \zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s) \rangle.
\end{align*}
$$

Recall that the second order derivative of $b$ (the third order derivative of $F$ is bounded, owing to Assumption 2.1). By applying the $\mu$-convexity condition (48) and Cauchy-Schwarz inequality, there exists $C \in (0, \infty)$ such that for all $s \ge 0$ one has

$$
\begin{align*}
\frac{d}{ds} \tilde{\Gamma}(\zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s), \zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s)) \le & -\frac{\alpha \mu}{2} \|\zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s)\|^2 - \frac{\alpha}{2} \|\zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s)\|^2 \\
& + L_F \|\zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s)\| \|\zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s)\| \\
& + C \|\eta_{k_1^x, k_1^v}^{0,x}(s)\| \|\eta_{k_2^x, k_2^v}^{0,x}(s)\| \|\zeta_{(k_1^x, k_1^v)}^{0,x}, (k_2^x, k_2^v)) (s)\| \\
& + C \|\eta_{k_1^x, k_1^v}^{0,v}(s)\| \|\eta_{k_2^x, k_2^v}^{0,x}(s)\| \|\zeta_{(k_1^x, k_1^v)}^{0,v}, (k_2^x, k_2^v)) (s)\|.
\end{align*}
$$
Using Young inequality, we obtain for all $s \ge 0$

$$
\begin{aligned}
& \frac{d}{ds} \tilde{\Gamma}(\zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,x}(s), \zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,v}(s)) \\
& \le \frac{L_F^2 - \alpha \mu \epsilon}{2\epsilon} \|\zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,x}(s)\|^2 + \frac{\epsilon - \alpha}{2} \|\zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,v}(s)\|^2 \\
& \quad + C \|\eta_{k_1^x, k_1^v}^{0,x}(s)\| \|\eta_{k_2^x, k_2^v}^{0,x}(s)\| \|\zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,x}(s)\| + C \|\eta_{k_1^x, k_1^v}^{0,v}(s)\| \|\eta_{k_2^x, k_2^v}^{0,x}(s)\| \|\zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,v}(s)\|
\end{aligned}
$$

Repeating the previous argument, we require $L_F^2 - \alpha\mu\epsilon < 0$ and $\epsilon - \alpha < 0$, which is equivalent to
have $\epsilon \in (\frac{L_F^2}{2\alpha\mu}, \alpha)$. By choosing $\epsilon = \frac{\alpha}{2} + \frac{L_F^2}{2\alpha\mu}$, we obtain for all $s \ge 0$

$$
\begin{aligned}
& (55) \quad \frac{d}{ds} \tilde{\Gamma}(\zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,x}(s), \zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,v}(s)) \\
& \le -2\gamma(\alpha, \mu, L_F) \tilde{\Gamma}(\zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,x}(s), \zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,v}(s)) \\
& \quad + C \|\eta_{k_1^x, k_1^v}^{0,x}(s)\| \|\eta_{k_2^x, k_2^v}^{0,x}(s)\| \|\zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,x}(s)\| + C \|\eta_{k_1^x, k_1^v}^{0,v}(s)\| \|\eta_{k_2^x, k_2^v}^{0,x}(s)\| \|\zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,v}(s)\|.
\end{aligned}
$$

By applying Young inequality, one obtains for all $s \ge 0$

$$
\begin{aligned}
& \frac{d}{ds} \tilde{\Gamma}(\zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,x}(s), \zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,v}(s)) \\
& \le -\gamma(\alpha, \mu, L_F) \tilde{\Gamma}(\zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,x}(s), \zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,v}(s)) \\
& \quad + C \|\eta_{k_1^x, k_1^v}^{0,v}(s)\|^2 \|\eta_{k_2^x, k_2^v}^{0,x}(s)\|^2
\end{aligned}
$$

As a result, applying the Grönwall inequality and using the inequality (52), for all $s \ge 0$ one obtains

$$
\tilde{\Gamma}(\zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,x}(s), \zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,v}(s)) \le C \int_0^s e^{-\gamma(\alpha, \mu, L_F)(s-r)} e^{-4\gamma(\alpha, \mu, L_F)r} dr \|k_1\|^2 \|k_2\|^2.
$$

Here we can compute the integral and obtain an exponentially decreasing bound, but with a worse
exponent than that of inequality (52). However, we can do better; indeed by using the left–hand
side inequality (49) which gives the following inequalities: for all $s \ge 0$

$$
\begin{aligned}
\| \zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,x}(s) \|^2 + \| \zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,v}(s) \|^2 &\le C \int_0^s e^{-\gamma(\alpha, \mu, L_F)(s-r)} e^{-4\gamma(\alpha, \mu, L_F)r} dr \|k_1\|^2 \|k_2\|^2 \\
&\le C e^{-\gamma(\alpha, \mu, L_F)s} \|k_1\|^2 \|k_2\|^2,
\end{aligned}
$$

this implies for all $s \ge 0$

$$
\| \zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,x}(s) \| + \| \zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,v}(s) \| \le C e^{-\gamma(\alpha, \mu, L_F)s} \|k_1\| \|k_2\|.
$$

Using this inequality and the inequality (52) in (55), one then obtains another upper bound: for all $s \ge 0$ one has

$$
\begin{aligned}
& \frac{d}{ds} \tilde{\Gamma}(\zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,x}(s), \zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,v}(s)) \\
& \le -2\gamma(\alpha, \mu, L_F) \tilde{\Gamma}(\zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,x}(s), \zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,v}(s)) \\
& \quad + C e^{-(2\gamma(\alpha, \mu, L_F) + \frac{\gamma(\alpha, \mu, L_F)}{2})t} \|k_1\|^2 \|k_2\|^2.
\end{aligned}
$$

Applying again the Grönwall inequality, for all $t \ge 0$ one obtains

$$
\begin{aligned}
\tilde{\Gamma}(\zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,x}(s), \zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,v}(s)) &\le C \int_0^s e^{-2\gamma(\alpha, \mu, L_F)(s-r)} e^{-2\gamma(\alpha, \mu, L_F) + \frac{\gamma(\alpha, \mu, L_F)}{2}} r dr \|k_1\|^2 \|k_2\|^2 \\
&\le C e^{-2\gamma(\alpha, \mu, L_F)s} \int_0^s e^{-\gamma(\alpha, \mu, L_F)r} dr \|k_1\|^2 \|k_2\|^2 \\
&\le C e^{-2\gamma(\alpha, \mu, L_F)s} \|k_1\|^2 \|k_2\|^2
\end{aligned}
$$

Using the left–hand side inequality (49), one obtains the following inequality: for all $s \ge 0$

$$
(56) \qquad \| \zeta_{(k_1^x, k_1^v), (k_2^x, k_2^v)}^{0,x}(s) \| \le C e^{-\gamma(\alpha, \mu, L_F)s} \|k_1\| \|k_2\|.
$$
Combining the inequalities (47), (52) and (56) with the upper bound (54) for $|D^2 u^0(t, x, v).(k_1, k_2)|$ above, as a result there exists a positive real number $C \in (0, \infty)$ such that for all $x, v \in \mathbb{R}^d$, $k_1, k_2 \in \mathbb{R}^{2d}$ one has

$$|D^2 u^0(s, x, v).(k_1, k_2)| \le C (\|\nabla \varphi(x^*)\| + \|\varphi\|_2) (1 + \|x - x^*\| + \|v\|) e^{-\gamma(\alpha, \mu, L_F)s} \|k_1\| \|k_2\|.$$

The proof of the inequality (44) is thus completed.

□

## DECLARATIONS

**Funding.** N. En-Nebbazi was funded by a PhD grant of the Région Nouvelle Aquitaine and of the Université de Pau et des Pays de l'Adour. C-E. Béhier, M. Dambrine and N. En-Nebbazi have been funded by the ANR grant ANR-24-CE40-3786 Schémas d'optimisation stochastique - dynamiques inertielles et en dimension infinie – SOS2ID.

UNIVERSITE DE PAU ET DES PAYS DE L'ADOUR, E2S UPPA, CNRS, LMAP, PAU, FRANCE

Email address: nenebbazi@univ-pau.fr