About classical solutions of the path-dependent heat equation

TL;DR

This paper explores existence theorems for classical solutions to the path-dependent heat equation linked to window Brownian motion, focusing on different classes of final conditions including cylindrical and smooth functionals.

Contribution

It provides two new existence theorems for classical solutions of the path-dependent heat equation with diverse final conditions.

Findings

Established existence for cylindrical non-smooth final conditions.

Proved existence for smooth generic functionals.

Enhanced understanding of solutions related to window Brownian motion.

Abstract

This paper investigates two existence theorems for the path-dependent heat equation, which is the Kolmogorov equation related to the window Brownian motion, considered as a C([--T, 0])-valued process. We concentrate on two general existence results of its classical solutions related to different classes of final conditions: the first one is given by a cylindrical non necessarily smooth r.v., the second one is a smooth generic functional.