Regularity Properties for Solutions of Infinite Dimensional Kolmogorov Equations in Hilbert Spaces

Abstract

In this article we establish regularity properties for solutions of infinite dimensional Kolmogorov equations. We prove that if the nonlinear drift coefficients, the nonlinear diffusion coefficients, and the initial conditions of the considered Kolmogorov equations are n-times continuously Fréchet differentiable, then so are the generalized solutions at every positive time. In addition, a key contribution of this work is to prove suitable enhanced regularity properties for the derivatives of the generalized solutions of the Kolmogorov equations in the sense that the dominating linear operator in the drift coefficient of the Kolmogorov equation regularizes the higher order derivatives of the solutions. Such enhanced regularity properties are of major importance for establishing weak convergence rates for spatial and temporal numerical approximations of stochastic partial differential equations.

Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients

“…It is, however, not clear to us how to treat the case where F and B are globally Lipschitz continuous but with the first four derivatives growing polynomially. Furthermore, we emphasize that Theorem 1.1 solves the weak convergence problem emerged from Debussche's article (see (2.5) and Remark 2.3 in Debussche [19]) merely in the case of spatial spectral Galerkin approximations instead of other more complicated numerical approximations for the SEE (1). The method of proof of our weak convergence results, however, can be extended to a number of other kind of spatial and temporal numerical approximations for SEEs of the form (1).…”

“…To the best of our knowledge, it remained an open problem to establish essentially sharp weak convergence rates for any type of temporal, spatial, or noise numerical approximation of the SEE (1) without imposing Debussche's assumption (2). In this article we solve this problem in the case of spatial spectral Galerkin approximations for the SEE (1). This is the subject of the following theorem (Theorem 1.1), which follows immediately from Corollary 6.1 below.…”

“…As pointed out in Remark 2.3 in Debussche [19], assumption (2) is a serious restriction for SEEs of the form (1). Roughly speaking, assumption (2) imposes that the second derivative of the diffusion coefficient function vanishes and thus that the diffusion coefficient function is affine linear.…”

“…Strong convergence rates for (temporal, spatial, and noise) numerical approximations for SEEs of the form (1) are well understood. Weak convergence rates for numerical approximations of SEEs of the form (1) have been investigated for about two decades; cf., e.g., [45,24,18,20,22,25,19,32,21,35,36,33,50,34,6,48,4,5,7,49]. Except for Debussche & De Bouard [18], Debussche [19], and Andersson & Larsson [5], all of the above mentioned references assume, beside further assumptions, that the considered SEE is driven by additive noise.…”

“…The method of proof in Debussche & De Bouard [18] strongly exploits this property of the nonlinear Schrödinger equation (see [18,Section 5.2]). Therefore, the method of proof in [18] can, in general, not be used to establish weak convergence rates for the SEE (1). In Debussche's seminal article [19] (see also Andersson & Larsson [5]), essentially sharp weak convergence rates for SEEs of the form (1) are established under the hypothesis that the second derivative of the diffusion coefficient B satisfies the smoothing property that there exists a real number L ∈ [0, ∞) such that for all x, v, w ∈ H it holds that 1 B…”