Cauchy Noise and Affiliated Stochastic Processes

TL;DR

This paper develops a detailed framework for constructing conditional and perturbed Markov processes driven by Cauchy noise, extending the Schrödinger interpolation problem to jump processes and their approximations.

Contribution

It introduces a novel approach to modeling stochastic processes with Cauchy noise, expanding the Schrödinger interpolation framework beyond Gaussian assumptions.

Findings

Extended Schrödinger interpolation to jump processes

Constructed probabilistic solutions for Cauchy-driven processes

Provided step process approximants for these processes

Abstract

By departing from the previous attempt (Phys. Rev. {\bf E 51}, 4114, (1995)) we give a detailed construction of conditional and perturbed Markov processes, under the assumption that the Cauchy law of probability replaces the Gaussian law (appropriate for the Wiener process) as the model of primordial noise. All considered processes are regarded as probabilistic solutions of the so-called Schr\"{o}dinger interpolation problem, whose validity is thus extended to the jump-type processes and their step process approximants.