Infinitesimal generators for a class of polynomial processes

Abstract

Abstract. We study the infinitesimal generators of evolutions of linear mappings on the space of polynomials, which correspond to a special class of Markov processes with polynomial regressions called quadratic harnesses. We relate the infinitesimal generator to the unique solution of a certain commutation equation, and we use the commutation equation to find an explicit formula for the infinitesimal generator of free quadratic harnesses. This is an expanded (arxiv) version of the paper.

Asymmetric Simple Exclusion Process with Open Boundaries and Quadratic Harnesses

“…The infinitesimal generator (4.8) for the Markov process in this case can be made more explicit by specializing [14]; to get the bi-Poisson process with q = 0, we take τ = σ = 0, and use (4.3) for the other two parameters. To avoid atoms, we restrict the range of parameters α, β.…”

“…Taking q = 0 in (4.10), we see that the expression for τ k N − τ k+1 N involves the product of two integrals with respect to P 0,1 (θ, dy) and π 1 (dx). The first step is to read out from [14,Remark 4.2] that for q = 0 we have P 0,1 (θ, dy) = (1 + θx)π 1 (dy).…”