On Absolute Continuity for Stochastic Partial Differential Equations and an Averaging Principle for a Queueing Network

Abstract

The connection between elliptic stochastic diffusion processes and partial differential equations is rich and well understood. This connection is not very well understood when the stochastic differential equation takes values in an infinite dimensional space such as a function space. In this case, the diffusion is a stochastic partial differential equation (SPDE) and the notion of ellipticity is ambiguous. We establish a sufficient condition on the diffusion coefficient of a class of nonlinear SPDEs, which is analogous to the nondegeneracy condition in finite dimensions, that allows for the existence of a Markov transition density that is absolutely continuous with respect to an infinite dimensional Gaussian measure.In the second part of this work, we consider a two-station queueing network that processes K job types. The first station in this network is a polling station, and we assume that the second station is operating under any nonidling service discipline. We consider diffusion-scaled versions of many of the processes governing this system, and we show that the scaled two-dimensional total workload process converges to Brownian motion in a wedge. We also show that the scaled immediate workload process for station 2 does not converge, but admits an averaging principle.