From time dependent linear Fokker-Planck equations to conservative right (strong Markov) diffusion processes
A talk in the Bielefeld Stochastic Afternoon series by
Iulian Cimpean
| Abstract: | "Given a time-dependent diffusion operator on R^d with
possibly irregular coefficients, it is a highly nontrivial task to
construct path-continuous (time-inhomogeneous) Markov solutions for the
corresponding martingale problem (m.p.). One general way to proceed is
to construct first the one-dimensional distributions of the process by
solving the corresponding Fokker-Planck equation on measures, and then
apply the superposition principle as e.g. in [1]. However, in most
general situations the Fokker-Planck equation can be uniquely solved
only in some restricted sense; this poses serious issues when one aims
to derive further key properties for the process obtained by the
superposition principle, like the strong Markov property for instance.
On the other hand, one can construct time-inhomogeneous regular (strong)
Markov processes by the theory of generalized Dirichlet forms, following
the approach from [2]. However, this second approach can be used to
construct the process only locally in space and time. The advantage in
contrast to the restricted superposition principle is that the
time-inhomogeneous m.p. can be solved locally for quasi-every space-time
starting point, and the strong Markov property is automatically
satisfied by construction; the disadvantage, besides the local nature of
the construction, is that the time-inhomogeneous m.p. can not be solved
starting from time t=0, but only from strictly positive initial times.
The aim of this talk is to discuss a result that emerges by mixing both
of the approaches described above, in such a way that all the above
issues are solved. Based on a work in progress, jointly with L. Beznea
and M. Röckner." [1] Trevisan, Dario. "Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients", Electron. J. Probab., 2016. [2] Stannat, Wilhelm. "Time-dependent diffusion operators on L^1", Journal of Evolution Equations, 2004. Within the CRC this talk is associated to the project(s): A5, B1 |