Convergence rates of the splitting scheme for parabolic linear stochastic Cauchy problems
Provides theoretical convergence guarantees for numerical solutions of parabolic stochastic PDEs, relevant to researchers in stochastic analysis and numerical methods.
The paper proves convergence rates for a splitting scheme applied to linear stochastic Cauchy problems with an analytic semigroup generator, showing convergence in Hölder spaces with rate 1/n^þ under specific parameter conditions.
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,þ\ge 0 are such that \g + þ< 1 and max[0,(\a-\b+þ)] + \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^þ.