RANDOM SWITCHING BETWEEN VECTOR FIELDS HAVING A COMMON ZERO

Let E be a finite set, {F-i}(i is an element of E) a family of vector fields on R-d leaving positively invariant a compact set M and having a common zero p is an element of M. We consider a piecewise deterministic Markov process (X, I) on M x E defined by (X)over dot(t)=F(I)t(X-t) where I is a jump process controlled by X: Pr(It+s = j vertical bar(X-u ,I-u)(u <= t)) = a(ij)(X-t)(s) + o(s) for i not equal j on {I-t=i}. We show that the behaviour of (X, I) is mainly determined by the behaviour of the linearized process (Y, J) where (Y)over dot(t) = A(J)t Y-t, A(i) is the Jacobian matrix of F-i at p and J is the jump process with rates (a(ij) (p)). We introduce two quantities Lambda(-) and Lambda(+), respectively, defined as the minimal (resp., maximal) growth rate of parallel to Y-t parallel to, where the minimum (resp., maximum) is taken over all the ergodic measures of the angular process (Theta, J) with Theta(t) = Y-t/parallel to Y-t parallel to. It is shown that Lambda(+) coincides with the top Lyapunov exponent (in the sense of ergodic theory) of (Y, J) and that under general assumptions Lambda(-) = Lambda(+). We then prove that, under certain irreducibility conditions, X-t -> p exponentially fast when Lambda(+) < 0 and (X, I) converges in distribution at an exponential rate toward a (unique) invariant measure supported by M \ {p} x E when Lambda(-) > 0. Some applications to certain epidemic models in a fluctuating environment are discussed and illustrate our results.