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

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Abstract

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.

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  1. ENS Cachan Bretagne - IRMAR, Université Rennes 1, Avenue Robert Schuman, 35170, Bruz, France

    Charles-Edouard Bréhier

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  1. Charles-Edouard Bréhier
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Bréhier, CE. Approximation of the Invariant Measure with an Euler Scheme for Stochastic PDEs Driven by Space-Time White Noise. Potential Anal 40, 1–40 (2014). https://doi.org/10.1007/s11118-013-9338-9

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  • DOI: https://doi.org/10.1007/s11118-013-9338-9

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