Random switching between vector fields having a common zero

Let E be a finite set, {Fi}i.E a family of vector fields on .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=FIt(Xt)where I is a jump process controlled by X: ..(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=AJtYt, Ai is the Jacobian matrix of Fi at p and J is the jump process with rates (aij(p)).We introduce two quantities ..and .+, respectively, defined as the minimal (resp., maximal) growth rate of .Yt., where the minimum (resp., maximum) is taken over all the ergodic measures of the angular process (.,J)with .t=Yt.Yt..It is shown that .+ coincides with the top Lyapunov exponent (in the sense of ergodic theory) of (Y,J)and that under general assumptions ..=.+.We then prove that, under certain irreducibility conditions, Xt.p exponentially fast when .+<0 and (X,I)converges in distribution at an exponential rate toward a (unique) invariant measure supported by M.{p}.E when ..>0.Some applications to certain epidemic models in a fluctuating environment are discussed and illustrate our results.