# Error estimates of the backward Euler-Maruyama method for multi-valued stochastic differential equations

## TL;DR

This paper establishes error estimates for the backward Euler-Maruyama method applied to multi-valued stochastic differential equations, including those with non-smooth convex potentials, demonstrating convergence of order at least 1/4.

## Contribution

It provides the first rigorous error analysis for the backward Euler-Maruyama method in the context of multi-valued SDEs with non-smooth potentials, extending existing techniques.

## Key findings

- Method is well-defined and converges with order at least 1/4
- Applicable to stochastic gradient flows with discontinuous gradients
- Verified on overdamped Langevin and stochastic p-Laplace equations

## Abstract

In this paper, we derive error estimates of the backward Euler-Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable, but assumed to be convex. We show that the backward Euler-Maruyama method is well-defined and convergent of order at least $1/4$ with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in [Nochetto, Savar\'e, and Verdi, Comm.\ Pure Appl.\ Math., 2000]. We verify that our setting applies to an overdamped Langevin equation with a discontinuous gradient and to a spatially semi-discrete approximation of the stochastic $p$-Laplace equation.