Pathwise Hölder convergence of the implicit-linear Euler scheme for semi-linear SPDEs with multiplicative noise

Abstract

In this article we prove pathwise Hölder convergence with optimal rates of the implicit Euler scheme for the abstract stochastic Cauchy problemHere A is the generator of an analytic C 0 -semigroup on a umd Banach space X, W H is a cylindrical Brownian motion in a Hilbert space H, and the functions F : [0, T ] × X → X θ F and G : [0, T ] × X → L (H, X θ G ) satisfy appropriate (local) Lipschitz conditions. The results are applied to a class of second order parabolic SPDEs driven by multiplicative space-time white noise. γ ([0,T ];X) 1 p ≤ Cn −δ (1 + x 0 L p (Ω;Xη ) ).

Pathwise space approximations of semi-linear parabolic SPDEs with multiplicative noise

“…Recall that in order to obtain a fully discretized scheme from the results presented here and maintain pathwise convergence, one would have to combine our results with pathwise convergence results for a time discretization scheme. We mentioned already that such results may be found in [5] or [23]. We refer to [17] for a recent overview of such results.…”

“…The pathwise convergence results of Theorems 1.2 and 1.3 remain valid if F and G are merely locally Lipschitz and satisfy linear growth conditions. The argument by which this is demonstrated is entirely analogous to the argument presented in [5], and we provide it here only for the reader's convenience.…”

“…Note that by the equivalence of strong and mild solutions in the finite-dimensional case the process U (n) satisfying (8) is precisely the solution to (5) in Theorem 2.4 if we take H 0 = H n , P 0 = P n and A 0 = A n .…”

“…By combining these results with e.g. the time discretization results in [5] or [23] one can obtain pathwise convergence of a fully discretized scheme 1 .…”

“…This is demonstrated in Section 5, where we extend the Theorems 1.2 and 1.3 to the case that F and G are locally Lipschitz and x 0 ∈ L 0 (Ω, F 0 , H A η ). This advantage of pathwise convergence has already been investigated in [15] for equations with additive noise, and applied again in [5] for time discretizations of equations with multiplicative noise. The approach taken in [5] and [15] translates directly to the results obtained in this article.…”