Approximation of the invariant measure with an Euler scheme for Stochastic PDE's driven by Space-Time White Noise
For researchers in numerical analysis of SPDEs, this provides a rigorous convergence rate for approximating invariant measures, though the bounded nonlinearity assumption is restrictive.
The paper proves that the invariant measure of a parabolic SPDE driven by space-time white noise can be approximated by a semi-implicit Euler scheme with order 1/2 in the time step, under bounded nonlinearity and dissipativity assumptions.
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.