Stochastic derivative and heat type PDEs

TL;DR

This paper explores the relationship between multitime Brownian sheets and heat-type PDEs, revealing volumetric solution properties and extending integrability theory to multitime stochastic systems.

Contribution

It introduces new results on volumetric solutions, stochastic derivatives, and multitime processes, extending the theory of heat PDEs and stochastic differential systems.

Findings

Solutions of forward/backward heat PDEs are volumetric.

Forward mean value of Brownian process solves heat PDE.

Backward mean value of Brownian process solves heat PDE.

Abstract

In this paper we address again the problem of the connection between multitime Brownian sheet and heat type PDEs. The main results include: the volumetric character of the solutions of the forward (backward) diffusion-like PDEs; the forward mean value of a Brownian process as the solution of the forward heat PDE; the backward mean value of a Brownian process as the solution of the backward heat PDE; the multitime stochastic processes with volumetric dependence; the stochastic partial derivative of a stochastic process with respect to a multitime Wiener process; Hermite polynomials stochastic processes; union of Tzitzeica hypersurfaces (constant level sets of multitime stochastic processes with volumetric dependence). The original results permit to extend the complete integrability theory to multitime stochastic differential systems, using path independent curvilinear integrals and volumetric dependence.