Mehler formula and capacities for infinite dimensional Ornstein-Uhlenbeck processes with general linear drift
This article essentially decomposes in the following three main results, that are refinements of previous results by Sugita, Feyel and La Pradelle, Shigekawa, Röckner. Firstly consider a locally convex vector space \(E\) endowed with a centered Gaussian measure and its Mehler semigroup \(p_t\); let \(C_{r,p}\) be the \((r,p)\)-capacity associated with the Gamma transform of \(p_t\); it has been known, for \(E\) Banach, that these \(C_{r,p}\) are tight on compact sets. Here it is shown that they are tight on metrizable compact sets. Secondly consider a positive selfadjoint operator \(A\) on a separable Hilbert space \(H\). A bigger separable Banach space \(E\) carrying a centered Gaussian probability \(\gamma\) (with given covariance commuting with \(A)\) is constructed, with the property that \(e^{- tA}\) extends to a strongly continuous contraction semigroup on \(E\). Then the generalized Mehler semigroup \(p^A_t\) with kernel \((e^{- tA} x + \sqrt {1 - e^{- 2tA}}y)\) and measure \(\gamma\) makes sense on \(E\). Now the third main result is that the capacity \(C^A_{1,2}\) associated (like above) with \(p^A_t\) is tight iff \(p^A_t\) is the semigroup of a diffusion \(X_t\) on \(E\) (that weakly solves \(dX_t = dW_t + A(X_t) dt\), \(W_t\) being Brownian on \(E\) with the covariance of \(\gamma)\).
scientific article; zbMATH DE number 880171