Self-stabilizing processes

Abstract

We construct “self-stabilizing” processes . These are random processes which when “localized,” that is scaled around t to a fine limit, have the distribution of an -stable process, where α is some given function on . Thus the stability index at t depends on the value of the process at t. Here, we address the case where . We first construct deterministic functions which satisfy a kind of autoregressive property involving sums over a plane point set Π. Taking Π to be a Poisson point process then defines a random pure jump process, which we show has the desired localized distributions.