Transport Equation on Semidiscrete Domains and Poisson-Bernoulli Processes

TL;DR

This paper studies a scalar transport equation on mixed domains, showing solutions can represent stochastic counting processes and distributions, with implications for understanding convergence and applications in probability theory.

Contribution

It introduces a unified analysis of transport equations on semidiscrete domains, linking solutions to stochastic processes and exploring their probabilistic properties.

Findings

Solutions can conserve sign and integrals over time and space.

Solutions correspond to counting stochastic processes and distributions.

The framework suggests new ways to modify and analyze stochastic processes.

Abstract

In this paper we consider a scalar transport equation with constant coefficients on domains with discrete space and continuous, discrete or general time. We show that on all these underlying domains solutions of the transport equation can conserve sign and integrals both in time and space. Detailed analysis reveals that, under some initial conditions, the solutions correspond to counting stochastic processes and related probability distributions. Consequently, the transport equation could generate various modifications of these processes and distributions and provide some insights into corresponding convergence questions. Possible applications are suggested and discussed.