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

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