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

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