Pointwise Approximation of Coupled Ornstein-Uhlenbeck Processes

We consider a stochastic evolution equation on the spatial domain D=(0,1)^d, driven by an additive nuclear or space-time white noise, so that the solution is given by an infinite-dimensional Ornstein-Uhlenbeck process. We study algorithms that approximate the mild solution of the equation, which takes values in the Hilbert space H=L_2(D), at a fixed point in time. The error of an algorithm is defined by the average distance between the solution and its approximation in H. The cost of an algorithm is defined by the total number of evaluations of one-dimensional components of the driving H-valued Wiener process W at arbitrary time nodes. We construct algorithms with an asymptotically optimal relation between error and cost. Furthermore, we determine the asymptotic behaviour of the corresponding minimal errors. We show how the minimal errors depend on the spatial dimension d, on the smoothing effect of the semigroup generated by the drift term, on the coupling of the infinite-dimensional system of scalar Ornstein-Uhlenbeck processes, which is specified by the diffusion term, and on the decay of the eigenvalues of W in case of nuclear noise. Asymptotic optimality is achieved by drift-implicit Euler-Maruyama schemes together with non-uniform time discretizations. This optimality cannot necessarily be achieved by uniform time discretizations, which are frequently analyzed in the literature. We complement our theoretical results by numerical studies