Linear inverse problems for Markov processes and their regularisation

TL;DR

This paper investigates inverse problems related to Markov processes, providing conditions for solutions, proposing regularisation methods, and demonstrating solutions exist under certain process modifications.

Contribution

It introduces new regularisation techniques for inverse problems involving Markov processes and characterizes conditions for their solutions, including process modifications.

Findings

Necessary and sufficient conditions for square integrable solutions.

A family of regularisation methods for inverse problems.

Existence of solutions when replacing the process with a mixture involving a jump process.

Abstract

We study the solutions of the inverse problem \[ g(z)=\int f(y) P_T(z,dy) \] for a given $g$, where $(P_t(\cdot,\cdot))_{t \geq 0}$ is the transition function of a given Markov process, $X$, and $T$ is a fixed deterministic time, which is linked to the solutions of the ill-posed Cauchy problem \[ u_t + A u=0, \qquad u(0,\cdot)=g, \] where $A$ is the generator of $X$. A necessary and sufficient condition ensuring square integrable solutions is given. Moreover, a family of regularisations for the above problems is suggested. We show in particular that these inverse problems have a solution when $X$ is replaced by $\xi X + (1-\xi)J$, where $\xi$ is a Bernoulli random variable, whose probability of success can be chosen arbitrarily close to $1$, and $J$ is a suitably constructed jump process.