Abstract

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.

Highlights

  • In the last 25 years, the analysis of numerical methods for stochastic partial differential equations (SPDEs) has been a very active research field

  • Pionnering works have focused on the so-called strong convergence of numerical schemes for equations with Lipschitz continuous nonlinearities, and in the last decade many results concerning convergence of schemes for equations with non-globally Lipschitz continuous nonlinearities, and weak convergence, have been obtained

  • We study the question of approximating the invariant distribution μ‹ using a numerical scheme

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Summary

Introduction

In the last 25 years, the analysis of numerical methods for stochastic partial differential equations (SPDEs) has been a very active research field. Since the convergence to equilibrium is exponentially fast (with respect to T ) in the models considered here, the analysis of the computational cost (see Cor. 4.3) reveals that there is no loss in the efficiency, when compared with a situation where uniform moment bounds and error estimates would hold (for instance, if the nonlinearity is globally Lipschitz as in [3], or if an implicit integrator is employed as in [12]).

Setting
Linear operator
Nonlinearity
Wiener process and the stochastic convolution
Preliminary results
Tamed exponential Euler scheme and main results
M q when
Some useful regularity results
Full Text
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