Polynomially harmonizable processes and finitely polynomially determined Lévy processes

Abstract

The sequence {Pk(t, x)} of two-variable Hermite polynomials are known to have the property that, if {Mt,t > 0} denotes the standard Brownian motion, then Pk(t,Mt) is a martingale for each k > 1. This property of standard Brownian motion vis-a-vis Hermite polynomials motivated the general notion of polynomially harmonizable processes. These are processes that admit sequences of time-space harmonic polynomials, that is, two-variable polynomials which become martingales when evaluated along the trajectory of the process. For Levy processes, this property is connected to certain properties of the associated L?vy/Kolmogorov measures. Moreover, stochastic properties of the underlying processes (like independence, stationarity of increments) turn out to be equivalent to certain algebraic/an aly tic properties of the corresponding sequence of polynomials. We first present a brief survey of these recently obtained general results and then describe necessary and sufficient conditions for certain classes of Levy processes to be uniquely determined by a finite number of time-space harmonic polynomials. AMS (1980) Subject Classification: Primary 60F05, Secondary 60J05