On recovering parabolic diffusions from their time-averages

TL;DR

This paper investigates the recovery of parabolic diffusion processes from their time-averages without initial data, establishing well-posedness, existence, and uniqueness of solutions for the reformulated boundary value problem.

Contribution

It introduces a novel boundary value problem replacing initial conditions with time-averages and proves its well-posedness and solution regularity.

Findings

The new problem is well-posed in certain solution classes.

Existence and uniqueness of solutions are established.

Regularity results are provided for the solutions.

Abstract

The paper study a possibility to recover a parabolic diffusion from its time-average when the values at the initial time are unknown. This problem can be reformulated as a new boundary value problem where a Cauchy condition is replaced by a prescribed time-average of the solution. It is shown that this new problem is well-posed in certain classes of solutions. The paper establishes existence, uniqueness, and a regularity of the solution for this new problem and its modifications, including problems with singled out terminal values.