Second Order Partial Differential Equations in Hilbert Spaces

Abstract

This chapter is devoted to some basic results on Gaussian measures on separable Hilbert spaces, including the Cameron-Martin and Feldman-Hajek formulae. The greater part of the results are presented with complete proofs.

SLE and the free field: Partition functions and couplings

“…Absolute continuity. We begin by recalling some general results (following here [5], see also [37], I.6). Let be a Hilbert space, a trace class (symmetric, positive) covariance operator.…”

On a Class of Stochastic Semilinear PDEs

“…In the present situation (Lipschitz nonlinearities and a white noise perturbation) the result seems to be new. An extensive survey on second order partial differential operators in Hilbert spaces can be found in the monographs [1], [2], [6]. A second new result of this paper is the closability of the operator D in L 2 (H; ν) and that D(K 2 ) is included in the Sobolev space W 1,2 (H; ν).…”

“…Assume that X(t, x) is the solution of equation (6). Then it is differentiable with respect to x P-a.s., and for any h ∈ H we have…”

“…where X(t, x) is the solution of (6). For this purpose we choose a control u ∈ L 2 ([0, T ]; H) such that |y(T, x; u) − z| 2 ≤ ε/2, where y is the solution of (24).…”

“…To see this we need to observe that by (6) Dϕ n 1 ,n 2 ,n 3 ,n 4 (x) = Dϕ(x) and sup n 1 ,n 2 ,n 3 ,n 4 { ϕ n 1 , n 2 , n 3 ,n 4 b,2 + Dϕ n 1 , n 2 , n 3 ,n 4 b,2 + Lϕ n 1 , n 2 , n 3 ,n 4 b,2 } < ∞.…”