Orthogonal Intertwiners for Infinite Particle Systems On The Continuum:
Interacting particle systems are studied using powerful tools, including
duality. Recently, dualities have been explored for inclusion processes,
exclusion processes, and independent random walkers on discrete sets
using univariate orthogonal polynomials. This talk generalizes these
dualities to intertwiners for particle systems on more general spaces,
including the continuum. Instead of univariate orthogonal polynomials,
the talk dives into the theory of infinite-dimensional polynomials
related to chaos decompositions and multiple stochastic integrals. The
new framework is applied to consistent particle systems containing a
finite or infinite number of particles, including sticky and correlated
Brownian motions.
Spectral gap of the symmetric inclusion process:
In this talk, we consider the symmetric inclusion process on a general finite graph. Our main result establishes universal upper and lower bounds for the spectral gap of this interacting particle system in terms of the spectral gap of the random walk on the same graph. In the regime in which the gamma-like reversible measures of the particle system are log-concave, our bounds match, yielding a version for the symmetric inclusion process of the celebrated Aldous' spectral gap conjecture --- originally formulated for the interchange process and proved by Caputo, Liggett and Richthammer (JAMS 2010). Finally, by means of duality techniques, we draw analogous conclusions for an interacting diffusion-like unbounded conservative spin system known as Brownian energy process, which may be interpreted as a spatial version of the Wright-Fisher diffusion with mutation. Based on a joint work with Seonwoo Kim (SNU, South Korea).