Zeit/Raum: Freitag, 19.01.2019, 14:15-15:45, Raum A 3.339
Gast: Prof. Dr. Anja Sturm, Universität Göttingen
Vortragstitel: On pathwise dualities for Markov processes and applications to interacting particle systems
Abstract: Duality between two Markov processes is a very useful tool for studying the distribution (at a fixed time) of one process via the distribution of another process. Pathwise duality refers to a much stronger connection of the two processes in the sense that it gives a coupling on the same probability space, and thus allows for almost sure statements over time intervals (with the dual process running in reversed time).
There is little general theory on determining (pathwise) dual processes to Markov processes with respect to appropriate duality functions. In this talk, we present a systematic treatment of monotonicity-based pathwise dualities for Markov processes taking values in (finite) partially ordered sets. We show that every such Markov process whose generator can be represented via monotone maps (also referred to as realizable monotonicity) has a pathwise dual process.
We also show that the dual simplifies greatly when the state space is a lattice and all monotone maps satisfy an additivity condition. On distributive lattices this leads to a percolation representation of the processes and their dual. This setting includes the case of additive interacting particle systems and the dual may correspond to describing genealogies of the particles.
Our unified treatment thus includes many well known dualities including also Gray's duality for attractive spin-systems and Siegmund's duality for processes with a totally ordered state spaces as well as new possibilities.
This talk is based on joint work with Jan Swart (UTIA Prague).