Ergodic properties of exclusion type discrete time processes in continuum
Michael Blank (Russian Academy of Sciences, Moscow)

A new class of exclusion type processes acting in continuum with synchronous updating is introduced and studied. These Markov processes describe a countably many pure-jump stochastic or deterministic motions of particles interacting through the hard core exclusion rule. Even in the deterministic setting the behaviour of such processes is rather complicated: their topological entropy is infinite. In general without some rather restricting assumptions nothing essential can be said about the set of invariant measures of such processes (which even might be empty). Nevertheless their other important statistical feature, namely, the ergodic average of particle velocities, may be obtained and its connections to other statistical quantities, in particular to the particle density (the so-called Fundamental Diagram) may be analyzed rigorously. The main technical tool in our analysis is a (somewhat unusual) coupling construction which is applied in a nonstandard fashion: we do not prove the existence of the so-called ``successful coupling" (which even might not hold) but instead use its absence as an important ``diagnostic'' tool. Despite the fact that this approach cannot be applied to lattice systems directly, it allows to obtain (indirectly) new results for the lattice systems by embedding them to the systems in continuum. Note that a seemingly very particular class of lattice exclusion processes introduced first in 1970 by Frank Spitzer appears naturally in a very broad list of scientific fields starting from various models of traffic flows, molecular motors and protein synthesis in biology, surface growth or percolation processes in physics, and up to the analysis of Young diagrams in representation theory.

Colloquium, Department of Mathematics, Rice University
September 18, 2008, HB 227, 4-5 PM

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