RANDOM SWITCHING BETWEEN VECTOR FIELDS HAVING A COMMON ZERO

Let E be a finite set, {Fⁱ}i∈E a family of vector fields on ℝ d leaving positively invariant a compact set M and having a common zero p ∈ M. We consider a piecewise deterministic Markov process (X, I ) on M × E defined by X ˙ t = F I t ( X t ) where I is a jump process controlled by X:P(It+s = j|(Xu, Iu ) u≤t ) = aij (Xt )s+o(s) for i ≠ j on {It = i}. We show that the behaviour of (X, I) is mainly determined by the behaviour of the linearized process (Y, J) where Y ˙ t = A J t Y t , A i is the Jacobian matrix of Fⁱ at p and J is the jump process with rates (aij (p)). We introduce two quantities Λ⁻ and Λ⁺, respectively, defined as the minimal (resp., maximal) growth rate of ǁYt ǁ, where the minimum (resp., maximum) is taken over all the ergodic measures of the angular process (Θ, J) with Θ t = Y t ‖ Y t ‖ . It is shown that Λ⁺ coincides with the top Lyapunov exponent (in the sense of ergodic theory) of (Y, J) and that under general assumptions Λ⁻ = Λ⁺. We then prove that, under certain irreducibility conditions, Xt → p exponentially fast when Λ⁺ < 0 and (X, I) converges in distribution at an exponential rate toward a (unique) invariant measure supported by M \ {p} × E when Λ⁻ > 0. Some applications to certain epidemic models in a fluctuating environment are discussed and illustrate our results.