Infinite-dimensional stochastic differential equations obtained by subordination and related Dirichlet forms

Abstract

New results related to the decomposition theorem of additive functionals associated to quasi-regular Dirichlet forms are presented. A characterization of subordinate processes associated to quasi-regular symmetric Dirichlet forms in terms of the unique solutions of the corresponding martingale problems is obtained. The subordinate of (generalized) Ornstein–Uhlenbeck processes are exhibited explicitly in terms of generators, Dirichlet forms, and unique pathwise solutions of stochastic differential equations (SDEs). In the case where the state space is infinite dimensional as, e.g. in Euclidean quantum field theory, the construction provides a characterization of the processes in terms of projections on the topological dual space, and corresponding finite-dimensional SDEs.