Yoichi Oshima

On a construction of Markov processes associated with time dependent Dirichlet spaces

The relations between the Symmetrie regulär Dirichlet spaces and the Symmetrie Hunt processes are well known. These relations are also studied for the coersive non-symmetric Dirichlet spaces. But the time-inhomogeneous Marko v processes cannot be covered by those settings. In this work, using the results on parabolic potential theory, it is proved that the methods of the Dirichlet space theory are still effective for the construction of space-time Hunt-processes. 1991 Mathematics Subject Classfication: 60J, 31C25. §

On conservativeness and recurrence criteria for Markov processes

In the present paper, we give general criteria of conservativeness and recurrence for Markov processes associated with, not necessarily symmetric, Dirichlet spaces. The conservativeness criterion is applied to discuss a comparison theorem of conservativeness for diffusion processes. Also some sufficient conditions of conservativeness and recurrence for diffusion processes.are given.

On the Skew Product of Symmetric Diffusion Processes

Given two independent Symmetrie diffusion processes A^ and X

On a Construction of Diffusion Processes on Moving Domains

\ the skew product diffusion is the process (X^\ XA^ where At is a positive continuous additive functional of the first process. In this work, a core and an explicit expression on it of the Dirichlet form of the skew product diffusion is determined. In the proof, general theorems on the invariance of the Dirichlet form under the random time change and on the representation of martingales s stochastic integrals play basic roles. 1980 Mathematics Subject Classification (1985 Revision): 60J, 31C25. §

On the recurrence of some time inhomogeneous Markov processes

We construct time inhomogeneous diffusion processes on a space-time domain D. For each t, the t-section Dt of D is assumed to be an image of a fixed domain D0 by a homeomorphic map. After constructing a diffusion process on a fixed domain D0 by means of the general theory of time dependent Dirichlet forms, transformations by a multiplicative functional and a homeomorphic map are used to get the desired processes. In the case of reflecting difusion process on D, we also give Skorohod type representations of the process by means of the local time on the boundary of D.

Dirichlet forms and symmetric Markov processes

Abstract We introduce the notions of weak and strong sense recurrence of time inhomogeneous Markov processes and give general criteria for them. These notions of recurrence are invariant under certain time changes but the Dirichlet forms are not stable under time changes in general. We shall apply the general criterion for the strong recurrence to time inhomogeneous diffusion processes obtained by certain drift transformations of recurrent diffusion processes as well as the processes obtained by time change of Brownian motions.