Large deviations of the exit measure through a characteristic boundary for a Poisson driven SDE

Abstract

Let O the basin of attraction of the unique stable equilibrium of a dynamical system, which is the law of large numbers limit of a Poissonian SDE. We consider the law of the exit point from O of that Poissonian SDE. We adapt the approach of M. Day (1990) for the same problem for a Brownian SDE. For that purpose, we will use the Large deviation for the Poissonian SDE reflected at the boundary of O studied in our recent work Pardoux and Samegni (2018).Concatenating φ i , φ and φ j , we obtain a trajectory φ with all the required properties. Now, we define the equivalence relation "R" inŌ by zRy iff VŌ(z, y) = VŌ(y, z) = 0. 1 A pointz is called an ω−limit point of solution Y (t, z 0 ) of a dynamical system if there exists a sequence (t n ) n≥1 of time such that t n → ∞ as n → ∞, for which Y (t n , z 0 ) → ∞, n → ∞ holds. The set of all such points of Y (t, z 0 ) is called ω−limit set of Y (t, z 0 ) and denote ω(z 0 ).

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“…The proof of the next important result is a bit lengthy, and we refer to Pardoux and Samegni (2018) for it.…”