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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.

Stochastic processes --- Diffusion processes --- Differential equations, Parabolic --- Mathematical Theory --- Mathematical Statistics --- Mathematics --- Physical Sciences & Mathematics --- Differentiaalvergelijkingen [Parabolische ] --- Differential equations [Parabolic] --- Diffusieproces --- Equations differentielles paraboliques --- Processus de diffusion --- Probabilities. --- Partial differential equations. --- Group theory. --- Potential theory (Mathematics). --- Probability Theory and Stochastic Processes. --- Partial Differential Equations. --- Group Theory and Generalizations. --- Potential Theory. --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Partial differential equations --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Diffusion processes. --- Differential equations, Parabolic. --- Parabolic differential equations --- Parabolic partial differential equations --- Differential equations, Partial --- Markov processes

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This Festschrift contains six research surveys and thirty-three shorter contributions by participants of the conference ''Stochastic Partial Differential Equations and Related Fields'' hosted by the Faculty of Mathematics at Bielefeld University, October 10–14, 2016. The conference, attended by more than 140 participants, including PostDocs and PhD students, was held both to honor Michael Röckner's contributions to the field on the occasion of his 60th birthday and to bring together leading scientists and young researchers to present the current state of the art and promising future developments. Each article introduces a well-described field related to Stochastic Partial Differential Equations and Stochastic Analysis in general. In particular, the longer surveys focus on Dirichlet forms and Potential theory, the analysis of Kolmogorov operators, Fokker–Planck equations in Hilbert spaces, the theory of variational solutions to stochastic partial differential equations, singular stochastic partial differential equations and their applications in mathematical physics, as well as on the theory of regularity structures and paracontrolled distributions. The numerous research surveys make the volume especially useful for graduate students and researchers who wish to start work in the above-mentioned areas, or who want to be informed about the current state of the art.

Partial differential equations --- Differential equations --- Operational research. Game theory --- Probability theory --- Mathematics --- differentiaalvergelijkingen --- waarschijnlijkheidstheorie --- stochastische analyse --- wiskunde --- kansrekening

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