Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators

Andreas Eberle (auth.)9783540666288, 3-540-66628-1

This book addresses both probabilists working on diffusion processes and analysts interested in linear parabolic partial differential equations with singular coefficients. The central question discussed is whether a given diffusion operator, i.e., a second order linear differential operator without zeroth order term, which is a priori defined on test functions over some (finite or infinite dimensional) state space only, uniquely determines a strongly continuous semigroup on a corresponding weighted Lp space. Particular emphasis is placed on phenomena causing non-uniqueness, as well as on the relation between different notions of uniqueness appearing in analytic and probabilistic contexts.

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Table of contents :
Introduction….Pages 1-8
Motivation and basic definitions: Uniqueness problems in various contexts….Pages 9-40
L p uniqueness in finite dimensions….Pages 41-87
Markov uniqueness….Pages 89-167
Probabilistic aspects of L p and Markov uniqueness….Pages 169-184
First steps in infinite dimensions….Pages 185-253