Markov Selection and $W$-strong Feller for 3D Stochastic Primitive Equations

TL;DR

This paper investigates the analytical properties of weak solutions to 3D stochastic primitive equations, establishing the Markov property and regularity of the transition semigroup under additive noise, with implications for solution dependence on initial conditions.

Contribution

It demonstrates the existence of Markov solutions satisfying the Markov property via an abstract selection principle and extends regularity results to arbitrary times.

Findings

Existence of a family of Markov solutions for the stochastic primitive equations.

Extension of transition semigroup regularity from small to arbitrary times.

Continuous dependence of solutions on initial conditions under additive noise.

Abstract

This paper studies some analytical properties of weak solutions of 3D stochastic primitive equations with periodic boundary conditions. The martingale problem associated to this model is shown to have a family of solutions satisfying the Markov property, which is achieved by means of an abstract selection principle. The Markov property is crucial to extend the regularity of the transition semigroup from small times to arbitrary times. Thus, under a regular additive noise, every Markov solution is shown to have a property of continuous dependence on initial conditions, which follows from employing the weak-strong uniqueness principle and the Bismut-Elworthy-Li formula.