Duality and semi-group property for backward parabolic Ito equations

TL;DR

This paper investigates backward parabolic Ito equations, establishing existence, uniqueness, and semi-group properties, including a novel semi-group involving the diffusion term, and explores duality with forward equations.

Contribution

It introduces a new semi-group property for backward equations that incorporates the diffusion term, advancing the theoretical understanding of these stochastic PDEs.

Findings

Proved existence and uniqueness of solutions.

Established a semi-group property with anti-causality.

Analyzed duality between forward and backward equations.

Abstract

We study existence, uniqueness, semi-group property, and a priori estimates for solutions for backward parabolic Ito equations in domains with boundary. We study also duality between forward and backward equations. The semi-group for backward equations is established in the form of some anti-causality. The novelty is that the semi-group property involves the diffusion term that is a part of the solution.