Smooth invariant densities for random switching on the torus

Abstract

We consider a random dynamical system obtained by switching between the flows generated by two smooth vector fields on the 2d-torus, with the random switchings happening according to a Poisson process. Assuming that the driving vector fields are transversal to each other at all points of the torus and that each of them allows for a smooth invariant density and no periodic orbits, we prove that the switched system also has a smooth invariant density, for every switching rate. Our approach is based on an integration by parts formula inspired by techniques from Malliavin calculus.

Singularities of invariant densities for random switching between two linear ODEs in 2D

“…Then, there exist two distinct vectors (s 1 , t 1 ), (s 2 , t 2 ) ∈ R 0 (x) such that Ψ (s1,t1) 0 (x) = Ψ (s2,t2) 0 (x) =: y. This implies that the u 0 trajectory through x and the u 1 trajectory through y intersect in two distinct points z (1) := Ψ t1 0 (x) and z (2) := Ψ t2 0 (x) that both lie in Γ r . For any two points x, y ∈ Γ • , the u 0 trajectory through x and the u 1 trajectory through y intersect in at most two distinct points, so z (1) and z (2) are the only points of intersection.…”

“…This implies that the u 0 trajectory through x and the u 1 trajectory through y intersect in two distinct points z (1) := Ψ t1 0 (x) and z (2) := Ψ t2 0 (x) that both lie in Γ r . For any two points x, y ∈ Γ • , the u 0 trajectory through x and the u 1 trajectory through y intersect in at most two distinct points, so z (1) and z (2) are the only points of intersection. Since z (1) , z (2) ∈ Γ r , Lemma 2 implies that det U (z (i) ) > 0 for i ∈ {0, 1}.…”

“…For any two points x, y ∈ Γ • , the u 0 trajectory through x and the u 1 trajectory through y intersect in at most two distinct points, so z (1) and z (2) are the only points of intersection. Since z (1) , z (2) ∈ Γ r , Lemma 2 implies that det U (z (i) ) > 0 for i ∈ {0, 1}. As a result, one trajectory crosses the other in the same direction at both points of intersection, which is impossible.…”

“…We started an exploration of higher dimensions in [2], where we considered a class of switching systems on the two-dimensional torus that is devoid of these obstacles (the contraction is subexponential and all points are elliptic). For this class, we showed that the invariant densities belong to C ∞ and that there are no singularities.…”

Global threshold dynamics of a stochastic epidemic model incorporating media coverage

“…The other goal is to further study the invariant density of process I(t). Many longterm asymptotic properties of dynamical systems or random dynamical systems can be described in terms of invariant measure [27] and the density function with respect to Lebesgue measure of the marginals of an invariant measure that can be called an invariant density [28]. If invariant density is L 1 on a set Ω, it satisfies the Fokker-Planck equations (FPE) in the interior of Ω [29].…”