E. B. Dynkin

A probabilistic approach to one class of nonlinear differential equations

SummaryWe establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes. As an application, we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations.

Diffusions, Superdiffusions and Partial Differential Equations

Introduction Parabolic equations and branching exit Markov systems: Linear parabolic equations and diffusions Branching exit Markov systems Superprocesses Semilinear parabolic equations and superdiffusions Elliptic equations and diffusions: Linear elliptic equations and diffusions Positive harmonic functions Moderate solutions of

Gaussian and non-Gaussian random fields associated with Markov processes

Lu=\psi(u)

Path processes and historical superprocesses

Stochastic boundary values of solutions Rough trace Fine trace Martin capacity and classes

Polynomials of the occupation field and related random fields

\mathcal{N}_1

Markov processes as a tool in field theory

and

Additive functionals of several time-reversible Markov processes

\mathcal{N}_0

Green's and Dirichlet spaces associated with fine Markov processes☆

Null sets and polar sets Survey of related results Basic facts of Markov processes and Martingales Facts on elliptic differential equations Epilogue Bibliography Subject index Notation index.

Random fields associated with multiple points of the Brownian motion

To every Markov process with a symmetric transition density, there correspond two random fields over the state space: a Gaussian field (the free field) φ and the occupation field T which describes amount of time the particle spends at each state. A relation between these two random fields is established which is useful both for the field theory and theory of Markov processes.

Branching measure-valued processes

SummaryA superprocessX over a Markov process ξ can be obtained by a passage to the limit from a branching particle system for which ξ describes the motion of individual particles.The historical process