Approximation of the invariant distribution for a class of ergodic SPDEs using an explicit tamed exponential Euler scheme

Charles-Edouard Bréhier (1, 2)

1 PSPM - Probabilités, statistique, physique mathématique
2 ICJ - Institut Camille Jordan

We consider the long-time behavior of an explicit tamed exponential Euler scheme applied to a class of parabolic semilinear stochastic partial differential equations driven by additive noise, under a one-sided Lipschitz continuity condition. The setting encompasses nonlinearities with polynomial growth. First, we prove that moment bounds for the numerical scheme hold, with at most polynomial dependence with respect to the time horizon. Second, we apply this result to obtain error estimates, in the weak sense, in terms of the time-step size and of the time horizon, to quantify the error to approximate averages with respect to the invariant distribution of the continuous-time process. We justify the efficiency of using the explicit tamed exponential Euler scheme to approximate the invariant distribution, since the computational cost does not suffer from the at most polynomial growth of the moment bounds. To the best of our knowledge, this is the first result in the literature concerning the approximation of the invariant distribution for SPDEs with non-globally Lipschitz coefficients using an explicit tamed scheme.

Mots clés: Stochastic partial differential equations; exponential integrators; tamed scheme; invariant distribution; Stochastic partial differential equations

Domaines: Mathématiques [math]

Charles-Edouard Bréhier. Approximation of the invariant distribution for a class of ergodic SPDEs using an explicit tamed exponential Euler scheme. ESAIM: Mathematical Modelling and Numerical Analysis, 2022, 56 (1), pp.151-175. ⟨10.1051/m2an/2021089⟩. ⟨hal-03560973⟩