Geometrical structures induced by continuous strong markov processes associated with generalized second order elliptic differential operators

Abstract

It is shown that, under mild conditions, a continuous strong Markov process X on a d-dimensional domain can be associated with a generalized second order elliptic differential operator Ψ of the form up to a transformation of the state space. HereB* is the adjoint of a (possibly degenerate) second order elliptic differential operator Bk M is a locally finite measure on and D Ψ denotes the domain of definition of Ψ. Based on this representation, an intrinsic space-time geometrical structure is introduced in such a manner that the process X can be regarded as a Brownian motion with respect to the geometrical structure