Sobolev spaces of generalised smoothness and sub-Markovian semigroups
It is cornerstone in the thory of modern theory of stochastic processes that to each (regular) Dirichlet form one can associate a stochastic process. From the probabilistic point of view
there is a disadvantage to work with processes associated with
Dirichlet spaces because the process is defined only up to an exceptional set (sets of capacity zero).
We propose an L_P setting to overcome this difficulty and introduce and (investigate)
certain Bessel potential spaces associated with a sub-Markovian semigroup. We show that these spaces
are domains of definitions for pseudo-differential operators associated to a continuous negative definite function and some of them can be regarded as Triebel-Lizorkin spaces of generalized smoothness.
This allows us in particular to indicate and characterize a large class of Markov processes
starting in every point of the euclidean space.
Parts of this talk are selected from some recent joint works with N. Jacob, H.-G.Leopold
and R. Schilling.
Erich Walter Farkas (Swiss Banking Institute/University of Zurich)
May 14, 2004