Semiflow selection and Markov selection theorems

TL;DR

This paper establishes a measurable selection theorem for semiflows in differential equations without uniqueness, extending classical Markov selection results and applying to ODEs, PDEs, and differential inclusions.

Contribution

It introduces an abstract semiflow selection theorem for non-unique solutions, inspired by Krylov's Markov selection theorem, with applications to various differential equations.

Findings

Proves a measurable semiflow selection theorem for non-unique solutions.

Connects semiflow selection to Markov selection theorems.

Provides applications to ODEs, PDEs, and differential inclusions.

Abstract

The deterministic analog of the Markov property of a time-homogeneous Markov process is the semigroup property of solutions of an autonomous differential equation. The semigroup property arises naturally when the solutions of a differential equation are unique, and leads to a semiflow. We prove an abstract result on measurable selection of a semiflow for the situations without uniqueness. We outline applications to ODEs, PDEs, differential inclusions, etc. Our proof of the semiflow selection theorem is motivated by N. V. Krylov's Markov selection theorem. To accentuate this connection, we include a new version of the Markov selection theorem related to more recent papers of Flandoli & Romito and Goldys et al.