Convergence Rates of the Splitting Scheme for Parabolic Linear Stochastic Cauchy Problems

We study the splitting scheme associated with the linear stochastic Cauchy problem $dU(t)=AU(t)\,dt+dW(t)$, t∈[0,T], $U(0)=x$. Here A is the generator of an analytic C₀-semigroup S=(t)\ₜ₀ on a Banach space E and W=(t)\ₜ₀ is a Brownian motion with values in a fractional domain space Eβ associated with A. We prove that if α,β,γ,θ0 are such that γ+θ<1 and (α-β+θ)⁺+γ<1/2, then the approximate solutions U⁽ⁿ⁾ converge to the solution U in the space Cγ([0,T];Eα), both in Lᵖ-means and almost surely, with rate 1/nθ.