Uniqueness problems of diffusion operators on Euclidean space and on abstract Wiener space

Abstract

The central question discussed in this thesis is whether a given diffusion operators,<br/> i.e., a second order linear elliptic differential operator without zeroth order term,<br/> which is a priori only defined on test functions over some (finite or infinite dimensional<br/> ) state space, uniquely determines a strongly continuous semigroup on a corresponding<br/> weighted L^p space.<br/> On the first part of the thesis, we are mainly focus on equivalence of different definitions<br/> of capacities, and removability of singularities. More precisely, let L be either<br/> a fractional powers of Laplacian of order less than one whose domain is smooth compactly<br/> supported functions on R^d ∖ Σ of a given compact set Σ ⊂ R^d of zero Lebesgue<br/> measure or integral powers of Ornstein-Uhlenbeck operator defined on suitable algebras<br/> of functions vanishing in a neighborhood of a given closed set Σ of zero Gaussian<br/> measure in abstract Wiener space. Depending on the size of Σ, the operator under<br/> consideration, may or may not be L^p unique. We give descriptions for the critical<br/> size of Σ in terms of capacities and Hausdorff measures. In addition, we collect some known results for certain multi-parameter stochastic processes.<br/> On the second part of this thesis, we are mainly focus on Neumann problems<br/> on L^p(U, µ), where U ⊂ R^d is an open set. More precisely, let L be a nonsymmetric<br/> operator of type Lu = ∑ aij∂i∂ju+∑ bi∂iu, whose domain is C^2_0,Neu(U). We give some<br/> results about Markov uniqueness, L^p-uniqueness, relation of L^1-uniqueness and conservativeness,<br/> uniqueness of invariant measures, elliptic regularity, etc under certain<br/> assumption on µ and on the coefficients of L.