Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations

Abstract

In this article we prove the existence of Bernstein processes which we associate in a natural way with a class of non-autonomous linear parabolic initial-and final-boundary value problems defined in bounded convex subsets of Euclidean space of arbitrary dimension. Under certain conditions regarding their joint endpoint distributions, we also prove that such processes become reversible Markov diffusions. Furthermore we show that those diffusions satisfy two Itô equations for some suitably constructed Wiener processes, and from that analysis derive Feynman-Kac representations for the solutions to the given equations. We then illustrate some of our results by considering the heat equation with Neumann boundary conditions both in a one-dimensional bounded interval and in a twodimensional disk.

On Bernstein processes generated by hierarchies of linear parabolic systems in Rd

“…The proof of (b) is similar by using (73) in (32). It is also plain that (c) follows from (b) and thatZ τ ∈[0,T ] is non-stationary and non-Markovian for the same reasons as those given in the proof of Theorem 2 of the preceding section.…”

“…There already exists a proof of an abstract version of a related statement in [16] as well as a more analytic version of it in [32], so that we limit ourselves here to showing how the basic quantities of interest can be expressed in terms of Green's function (8):…”

“…We refer the reader for instance to [31], [32] and some of their references for an analysis of the Markovian case in various situations. Finally, in various different forms Bernstein processes have also recently appeared in applications of Optimal Transport Theory as testified for instance in [21] and in the monographs [11] and [29], and in the developments of Stochastic Geometric Mechanics as in [34].…”

“…(1) The fact that Z m τ ∈[0,T ] is both a forward and a backward Markov process is a manifestation of its reversibility in the sense of Definition 2 in [32], which is also readily apparent in (44) since the probability density of finding the process in a given region of space at a given time is expressed as the product of the forward solution (36) times the backward solution (37). As a matter of fact we can also obtain (44) either from (38) or from (41) when n = 1, and we have…”

“…Bernstein (or reciprocal) processes constitute a generalization of Markov processes and have played an increasingly important rôle in various areas of mathematics and mathematical physics over the years, particularly in view of the recent advances in the Monge-Kantorovitch formulation of Optimal Transport Theory and Stochastic Geometric Mechanics (see, e.g., [1], [6]- [9], [16], [20]- [22], [27], [32]- [34] and the many references therein for a history and other works on the subject, which trace things back to the pioneering works [5] and [28]). As such they may be intrinsically defined without any reference to partial differential equations, and may take values in any topological space countable at infinity as was shown in [16].…”