Non-local logistic equations from the probability viewpoint

TL;DR

This paper explores solutions to non-local logistic equations using probabilistic methods, providing new representations especially for the fractional case, which has been challenging to solve explicitly on the entire real line.

Contribution

It introduces a probabilistic representation for non-local logistic equations, including the fractional case, and extends solutions to the entire real line.

Findings

Probabilistic representation for non-local logistic equations

Explicit solutions on compact sets using Euler's numbers

Extension of solutions to the whole real line

Abstract

We investigate the solution to the logistic equation involving non-local operators in time. In the linear case such operators lead to the well-known theory of time changes. We provide the probabilistic representation for the non-linear logistic equation with non-local operators in time. The so-called fractional logistic equation has been investigated by many researchers, the problem to find the explicit representation of the solution on the whole real line is still open. In our recent work the solution on compact sets has been written in terms of Euler's numbers.