Large deviations results for the exit problem with characteristic boundary

Random perturbations of a periodically driven nonlinear oscillator: escape from a resonance zone

“…We restrict ourselves to the case that h 0 lies between H| sk and H| sd . Theorem 3.1 shows that, for κ = 1 the distribution of H ε converges to that of H 0 given by the stochastic differential equation (SDE) (29), and for κ > 1 the distribution converges to that of the deterministic equation d dt H 0 t = B(H 0 t ). For small ε the distribution of H 0 can be used to approximate the dynamics of H ε .…”

Stochastic Dynamic Bifurcations and Excitability

“…We did not discuss the distribution of time intervals between spikes, for the reason that not much is known about it. Unlike the case of exit from a potential well, we have to deal here with the problem of noise-induced escape through a characteristic boundary (Day, 1990a;Day, 1992), which does not necessarily follow an exponential law. If for instance the boundary is a periodic orbit, cycling occurs and the exit location depends logarithmically on the noise intensity (Day, 1990b;Day, 1994;Day, 1996;Berglund and Gentz, 2004).…”

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

“…In other words, the aim is to study the probability that a trajectory of Z N exits O in the neighborhood of a point y ∈ ∂O, P(|Z N,z (τ N O ) − y| < δ) for large N. Here τ N,z O denotes the first time of exit of the process Z N,z (t) from O. We adopt the approach of Day [2]. First we define a reflected Poissonian SDE for which the large deviation principle is satisfied, with the same rate function as the original one defined by (1).…”

“…First we define a reflected Poissonian SDE for which the large deviation principle is satisfied, with the same rate function as the original one defined by (1). We then follow the arguments of Day [2] to obtain our results.…”

“…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.…”