Interactions between the theory of partial differential equations of elliptic and parabolic types and the theory of stochastic processes are beneficial for both probability theory and analysis. At the beginning, mostly analytic results were used by probabilists. More recently, analysts (and physicists) took inspiration from the probabilistic approach. Of course, the development of analysis in general and of the theory of partial differential equations in particular, was motivated to a great extent by problems in physics. A difference between physics and probability is that the latter provides not only an intuition, but also rigorous mathematical tools for proving theorems. The subject of this book is connections between linear and semilinear differential equations and the corresponding Markov processes called diffusions and superdiffusions. Most of the book is devoted to a systematic presentation (in a more general setting, with simplified proofs) of the results obtained since 1988 in a series of papers of Dynkin and Dynkin and Kuznetsov. Many results obtained originally by using superdiffusions are extended in the book to more general equations by applying a combination of diffusions with purely analytic methods. Almost all chapters involve a mixture of probability and analysis. Similar to the other books by Dynkin, Markov Processes (Springer-Verlag), Controlled Markov Processes (Springer-Verlag), and An Introduction to Branching Measure-Valued Processes (American Mathematical Society), this book can become a classical account of the presented topics. Brownian And Super-brownian Motions And Differential Equations -- Exceptional Sets In Analysis And Probability -- Positive Solutions And Their Boundary Traces -- Parabolic Equations And Branching Exit Markov Systems -- Linear Parabolic Equations And Diffusions -- Fundamental Solution Of A Parabolic Equation -- Diffusions -- Poisson Operators And Parabolic Functions -- Regular Part Of The Boundary -- Green's Operators And Equation U + Lu = -[rho] -- Branching Exit Markov Systems -- Transition Operators And V-families -- From A V-family To A Bem System -- Some Properties Of Bem Systems -- Superprocesses -- Definition And The First Results -- Superprocesses As Limits Of Branching Particle Systems -- Direct Construction Of Superprocesses -- Supplement To The Definition Of A Superprocess -- Graph Of X -- Semilinear Parabolic Equations And Superdiffusions -- Connections Between Differential And Integral Equations -- Absolute Barriers -- Operators V[subscript Q] -- Boundary Value Problems -- Elliptic Equations And Diffusions -- Linear Elliptic Equations And Diffusions -- Basic Facts On Second Order Elliptic Equations -- Time Homogeneous Diffusions -- Probabilistic Solution Of Equation Lu = Au -- Positive Harmonic Functions -- Martin Boundary -- The Existence Of An Exit Point [xi Subscript [characters Not Reproducible]-] On The Martin Boundary -- H-transform -- Integral Representation Of Positive Harmonic Functions -- Extreme Elements And The Tail [sigma]-algebra -- Moderate Solutions Of Lu = [psi](u) -- From Parabolic To Elliptic Setting. E.b. Dynkin. Includes Bibliographical References (p. 225-231) And Indexes.
