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 Levy processes, and summarizes in an epilogue the
model-theoretic genericity of hyperfinite stochastic processes
theory.
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