On some Gaussian Bernstein processes in and the periodic Ornstein–Uhlenbeck process

Pierre-A Vuillermot,Jean-C Zambrini

doi:10.1080/07362994.2016.1156547

Abstract

ABSTRACTIn this article, we prove new results regarding the existence of Bernstein processes associated with the Cauchy problem of certain forward–backward systems of decoupled linear deterministic parabolic equations defined in Euclidean space of arbitrary dimension , whose initial and final conditions are positive measures. We concentrate primarily on the case where the elliptic part of the parabolic operator is related to the Hamiltonian of an isotropic system of quantum harmonic oscillators. In this situation there are many Gaussian processes of interest whose existence follows from our analysis, including N-dimensional stationary and non-stationary Ornstein–Uhlenbeck processes, as well as Bernstein bridges which may be interpreted as Markovian loops in a particular case. We also introduce a new class of stationary non-Markovian processes, which we eventually relate to the N-dimensional periodic Ornstein–Uhlenbeck process, which is generated by a one-parameter family of non-Markovian probability measures. In this case, our construction requires the consideration of an infinite hierarchy of pairs of forward–backward heat equations associated with the pure point spectrum of the elliptic part, rather than just one pair in the Markovian case. We finally stress the potential relevance of these new processes to statistical mechanics, the random evolution of loops, and general pattern theory.

Highlights

Introduction and outlineLet us consider the two adjoint parabolic Cauchy problems1 @tu(x; t) = 2 xu(x; t)V (x)u(x; t); (x; t) 2 RN (0; T ] ; u(x; 0) = '0(x); x 2 RN (1) and1 @tv(x; t) = 2 xv(x; t)V (x)v(x; t); (x; t) 2 RN [0; T ) ; v(x;T ) = T (x); x 2 RN ; (2)where T 2 (0; +1) is arbitrary
Below we show that under very general conditions on V; '0 and T, we can associate with (1)(2) a class of the so-called Bernstein or reciprocal processes, denoted by Z 2[0;T ]. These are processes that constitute a generalization of Markov processes which have played an increasingly important rôle in various areas of mathematics and mathematical physics over the years
We introduce a new class of Bernstein processes which we can eventually relate to the so-called periodic Ornstein-Uhlenbeck process, and which is generated by a one-parameter family of non-Markovian probability measures

Summary

Introduction and outline

In these equations, x denotes Laplace’s operator with respect to the spatial variable, V is a real-valued function while '0 and T are positive measures on RN. We show there that the construction of the ’s given by (5) requires the consideration of a hierarchy of in...nitely many pairs of problems of the form (3)-(4) associated with the whole pure point spectrum of the elliptic operator on the right-hand side, rather than just one pair in the Markovian case. In an appendix we prove an important series expansion for the Green function associated with(3)-(4)

A class of Bernstein processes in RN

Two Ornstein-Uhlenbeck processes and a Bernstein bridge