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This monograph treats the theory of Dirichlet forms from a comprehensive point of view, using "nonstandard analysis." Thus, it is close in spirit to the discrete classical formulation of Dirichlet space theory by Beurling and Deny (1958). The discrete infinitesimal setup makes it possible to study the diffusion and the jump part using essentially the same methods. This setting has the advantage of being independent of special topological properties of the state space and in this sense is a natural one, valid for both finite- and infinite-dimensional spaces. The present monograph provides a thorough treatment of the symmetric as well as the non-symmetric case, surveys the theory of hyperfinite Lévy processes, and summarizes in an epilogue the model-theoretic genericity of hyperfinite stochastic processes theory.
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From the reviews:
"This book deals with the relationship between Dirichlet forms and stochastic processes from the perspective of nonstandard analysis. ... The book has a very extensive bibliography and gives an interesting and valuable survey of the development and the applications of the theory of Dirichlet forms. ... the book will serve as a starting point for new explorations." (Tom L. Lindström, Mathematical Reviews, Issue 2012 k)
"This book deals with the relationship between Dirichlet forms and stochastic processes from the perspective of nonstandard analysis. ... The book has a very extensive bibliography and gives an interesting and valuable survey of the development and the applications of the theory of Dirichlet forms. ... the book will serve as a starting point for new explorations." (Tom L. Lindström, Mathematical Reviews, Issue 2012 k)