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

1. Front Matter

   Pages I-VIII

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2. Introduction

   • Andreas Eberle

   Pages 1-8

3. Motivation and basic definitions: Uniqueness problems in various contexts

   • Andreas Eberle

   Pages 9-40

4. L^p uniqueness in finite dimensions

   • Andreas Eberle

   Pages 41-87

5. Markov uniqueness

   • Andreas Eberle

   Pages 89-167

6. Probabilistic aspects of L ^p and Markov uniqueness

   • Andreas Eberle

   Pages 169-184

7. First steps in infinite dimensions

   • Andreas Eberle

   Pages 185-253

8. Back Matter

   Pages 255-260

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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.

• Differential operator
• differential equation
• diffusion process
• essentially self-adjoint
• generator
• partial differential equation
• semigroup
• uniqueness
• partial differential equations

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