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

✍ Scribed by Andreas Eberle (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 1999
• Tongue: English
• Leaves: 264
• Series: Lecture Notes in Mathematics 1718
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ 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

✦ Subjects

Probability Theory and Stochastic Processes; Partial Differential Equations; Potential Theory