Markov processes on function spaces

Question

Is there any reference on Continuous time Markov process whose state space is infinite dimensional function spaces, such as the space of continuous functions $C(R^d)$?

It seems Dirichlet Form is a more natural way to do the construction, because the generator involves some kind of functional derivative whose domain is delicate. Thanks!

Answer 1

A fundamental theorem of Fukushima says that a sufficient condition for a Dirichlet form to be associated to a Markov process is that it be regular: in particular, this condition requires the state space to be locally compact, and doesn't cover infinite dimensional state spaces. However, Albeverio, Ma and Röckner have given a necessary and sufficient condition which they call quasi-regular, and it allows for state spaces which are not locally compact.

See:

MR1200639 Albeverio, S., Ma, Z. M., Röckner, M. Quasi-regular Dirichlet forms and Markov processes. J. Funct. Anal. 111 (1993), no. 1, 118–154.

MR1214375 Ma, Zhi Ming, Röckner, Michael. Introduction to the theory of (nonsymmetric) Dirichlet forms. Universitext. Springer-Verlag, Berlin, 1992. vi+209 pp. ISBN: 3-540-55848-9 60J40