Brownian processes in infinite dimension
Consider a zero-mean Gaussian measure \(\xi\) on some locally convex space \(E\), and let \(H\) be the Cameron-Martin space of \((E,\xi)\). The authors construct first a Brownian motion \(W\) with values in \(E\) and covariance \(\xi\), and then the stochastic integral \(\int^ t_ 0 \langle \Phi_ s, dW\rangle\) for a generic predictable process \(\Phi\) with values in \(H\) and such that \(\int^ t_ 0 \mathbb{E}(| \Phi_ s |^ 2)ds < \infty\). They prove that every centered square integrable variable which is measurable with respect to \(W\), can be expressed as a stochastic integral, and also obtain a version of Clark's formula. The main purpose of the paper is to solve a stochastic Cauchy problem \(dX_ t = \sqrt {2a} dW_ t - AX_ t dt\), where \(A\) is the infinitesimal generator of an analytic contracting semigroup \((C_ t)\) of operators of \(H\), and \(a\) a closed Hermitian positive operator such that \(a\) and \(A\) commute. Under some technical conditions, the solution of the preceding SDE is \(X_ t (x) = C_ t x + \sqrt{2a} \int^ t_ 0 C_{t-s}dW_ s\).
scientific article; zbMATH DE number 739491