On a new probabilistic representation for the solution of the heat equation

TL;DR

This paper introduces a novel probabilistic representation for the heat equation's solution using a new multiplication of smooth random variables based on Hida-Malliavin derivatives, offering properties contrasting with the Wick product.

Contribution

It presents a new probabilistic framework for solving the heat equation through a unique multiplication of random variables, expanding the theoretical tools available.

Findings

The new multiplication exhibits useful mathematical properties.

It contrasts with the Wick product in key aspects.

Provides a fresh perspective on probabilistic solutions to PDEs.

Abstract

We obtain a new probabilistic representation for the solution of the heat equation in terms of a product for smooth random variables which is introduced and studied in this paper. This multiplication, expressed in terms of the Hida-Malliavin derivatives of the random variables involved, exhibits many useful properties which are to some extents opposite to some peculiar features of the Wick product.