On non autonomous Ornstein-Uhlenbeck evolution operators in infinite dimension

Abstract

The present thesis concerns non autonomous evolution equations driven by time depending Ornstein-Uhlenbeck operators in an infinite dimensional separable Hilbert space. They arise as Kolmogorov equations of linear stochastic evolution PDEs with Gaussian noise, rewritten as stochastic evolution equation in suitable Hilbert spaces. In the autonomous case, stochastic PDEs and their Kolmogorov equations are widely studied; one of the most fruitful approaches is the Da Prato-Zabczyk one, that exploits semigroup theory. In the non autonomous case, semigroups of operators are replaced by evolution operators whose theory is not complete and satisfying as in the autonomous case. In fact, the study of non autonomous Kolmogorov equations in infinite dimensions is just at the beginning. In the first chapter of this thesis we recall some preliminaries about functional analysis, infinite dimensional analysis and about abstract non autonomous parabolic problems. In chapter two we consider backward linear non autonomous Kolmogorov equations and their mild solutions, expressed through the Ornstein-Uhlenbeck evolution operator Ps,t. Moreover we study smoothing properties of Ps,t along suitable directions and in chapter three we use these results to prove maximal Hölder regularity for the mild solutions. In chapter four we study the behavior of Ps,t on a linear space of cylindrical and smooth functions that are essential for further results. In chapter five we extend Ps,t to a continuous and bounded operator between Lp spaces with respect to suitable probability measures. To be more precise, under natural assumptions we show the existence of an evolution system of Gaussian measures, namely a family tγ u of Gaussian r rPR measures such that for every s ď t and for every continuous and bounded f : X ÞÑ R we have żż Ps,tf dγs “ f dγt. XX In this case, Ps,t can be extended to a linear bounded operator from LppX,γtq to LppX,γsq, with s ď t. We prove a family of Logarithmic-Sobolev inequalities, and we use them to prove a hypercontractivity result for the evolution operator. In chapter six we prove a representation formula for Ps,t as a second quantization operator and we use it to prove further properties of Ps,t .