Approximation of the Invariant Measure with an Euler Scheme for Stochastic PDEs Driven by Space-Time White Noise

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.