Convergence rates of the splitting scheme for parabolic linear stochastic Cauchy problems

TL;DR

This paper analyzes the convergence rates of a splitting scheme applied to linear stochastic parabolic problems, establishing conditions under which the scheme converges with a specific rate in a Banach space setting.

Contribution

It provides new convergence rate results for the splitting scheme applied to stochastic Cauchy problems with analytic semigroup generators.

Findings

Convergence in Holder space with rate 1/n^ h

Conditions on parameters for convergence

Almost sure and L^p convergence results

Abstract

We study the splitting scheme associated with the linear stochastic Cauchy problem dU(t) = AU(t) dt + dW(t), where A is the generator of an analytic C_0-semigroup S={S(t)} on a Banach space E and W={W(t)} is a Brownian motion with values in a fractional domain space E_\b associated with A. We prove that if \a,\b,\g,\th \ge 0 are such that \g + \th < 1 and max[0,(\a-\b+\th)] + \g < 1/2, then the approximate solutions U_n (where n is the number of time steps) converge to the solution U in the Holder space C^\g([0,T];E_\a), both in L^p-means and almost surely, with rate 1/n^\th.