Discrete infinitesimal generator of the Frobenius-Perron operator semigroup associated with "outflow systems"

TL;DR

This paper introduces a discrete infinitesimal generator for the Frobenius-Perron operator semigroup in outflow systems, proving convergence of the generated semigroup to the original operator in the $L^1$ space.

Contribution

It presents a novel discrete generator for the Frobenius-Perron semigroup in systems with trajectories leaving the state space, establishing convergence results.

Findings

Discrete generator converges to the Frobenius-Perron operator

Semigroup convergence in $L^1$ space

Applicable to outflow dynamical systems

Abstract

In this technical report the $C_0$ semigroup of Frobenius-Perron operators on $L^1(X)$ is considered, where the underlying dynamical system is such that trajectories may leave the state space $X$ and terminate. We introduce a discrete infinitesimal generator and show, that the operator semigroup generated by this discrete generator converges in $L^1(X)$ pointwise to the Frobenius-Perron operator of the system.