Approximation of the invariant measure with an Euler scheme for Stochastic PDE's driven by Space-Time White Noise

Charles-Edouard Bréhier (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

In this article, we consider a stochastic PDE of parabolic type, driven by a space-time white-noise, and its numerical discretization in time with a semi-implicit Euler scheme. When the nonlinearity is assumed to be bounded, then a dissipativity assumption is satisfied, which ensures that the SDPE admits a unique invariant probability measure, which is ergodic and strongly mixing - with exponential convergence to equilibrium. Considering test functions of class $\mathcal{C}^2$, bounded and with bounded derivatives, we prove that we can approximate this invariant measure using the numerical scheme, with order $1/2$ with respect to the time step.

Keywords: Stochastic Partial Differential Equations; Invariant measures and Ergodicity; Weak Approximation; Euler scheme

Domains: Numerical Analysis [math.NA]; Probability [math.PR]

Charles-Edouard Bréhier. Approximation of the invariant measure with an Euler scheme for Stochastic PDE's driven by Space-Time White Noise. Potential Analysis, 2014, 40 (1), pp.1-40. ⟨10.1007/s11118-013-9338-9⟩. ⟨hal-00669462⟩