Stochastic processes with reflections

Stochastic billiards in unbounded domains

Generally speaking, billiards models are dynamical systems describing the motion of a particle in a region with reflection rules at the boundary. In the early 1900s, Knudsen undertook a series of experiments studying the flow of rarefied gases through tubes. If the mean free path length of the gas is much bigger than the diameter of the tube, then collisions between gas particles are much rarer than collisions of particles with the tube boundary, and the bulk behaviour is described via single-particle dynamics. This Knudsen regime of ideal gas dynamics leads to the study of billiards processes, in which a particle moves with constant velocity until it hits the boundary.

Deterministic reflection leads to classical billiards models. The presence of microscopic irregularities in the domain boundary motivates considering random reflections and hence stochastic billiards: what appears to be a single reflection at the boundary is comprised of a rapid sequence of reflections whose cumulative effect is essentially random.

In unbounded domains, stochastic billiards may be recurrent or transient, depending on whether the process returns infinitely often to a bounded region. It is of interest to classifiy recurrence according to the shape of the domain and the reflection law at the boundary. For a natural class of generalized parabolic domains, we have investigated this problem in the case of independent, identically distributed reflections, and also in the more general case of Markovian reflections, when the distribution of the outgoing angle at a collision is determined by the incoming angle and a Markov kernel.

               

Reflecting random walks and diffusions

If the ideal gas described above has a higher density of particles, then intermolecular collisions can no longer be neglected. If the density is high enough, then the ensemble action of collisions with other particles will mean that a single particle is now observed to undertake Brownian motion (or some other diffusion) in the interior of the domain. Reflecting diffusions are also motivated from models of queueing and communications systems, from mathematical finance, and other applications.

       

Conservative interacting particle systems with mutual reflections

Consider a system of $N+1$ particles on the real line (or the one-dimensional integer lattice). Suppose that each particle would, in free space, perform an independent random walk (or diffusion), but interacts with other particles via some form of mutual reflection, or exclusion, which enforces that the ordering of the particles is maintained. In discrete space, the exclusion process is one example of dynamics of this type; in continuous space, models of this type include diffusions with rank-based interactions and the Atlas model of financial mathematics and its relatives. Considering the vector in the $N$-dimensional orthant describing the spacings between the particles, this class of models can be translated into reflecting random walks or diffusions in the orthant.

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