Hyperfinite Dirichlet Forms and Stochastic Processes

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 tho.

Hyperfinite Dirichlet Formsand Stochastic Processes; Preface; Contents; Chapter 1: Hyperfinite Dirichlet Forms; 1.1 Hyperfinite Quadratic Forms; 1.2 Domain of the Symmetric Part; 1.3 Resolvent of the Symmetric Part; 1.4 Weak Coercive Quadratic Forms; 1.5 Hyperfinite Dirichlet Forms; 1.6 Hyperfinite Representations; 1.7 Weak Coercive Quadratic Forms, Revisited; Chapter 2: Potential Theory of HyperfiniteDirichlet Forms; 2.1 Exceptional Sets; 2.1.1 Exceptional Sets; 2.1.2 Co-Exceptional Sets; 2.2 Excessive Functions and Equilibrium Potentials; 2.3 Capacity Theory.

2.4 Relation of Exceptionality and Capacity Theory2.5 Measures of Hyperfinite Energy Integrals; 2.6 Internal Additive Functionals and Associated Measures; 2.7 Fukushima's Decomposition Theorem; 2.7.1 Decomposition Under the Individual Probability Measures Pi; 2.7.2 Decomposition Under the Whole Measure P; 2.8 Internal Multiplicative Functionals; 2.8.1 Internal multiplicative functionals; 2.8.2 Subordinate Semigroups; 2.8.3 Subprocesses; 2.8.4 Feynman-Kac Formulae; 2.9 Alternative Expression of Hyperfinite Dirichlet Forms; 2.10 Transformations of Symmetric Dirichlet Forms.

Chapter 3: Standard Representation Theory3.1 Stand…