Dynamical system models with delayed dynamics and small noise arise in a variety of applications in science and engineering. In many applications, stable equilibrium or periodic behavior is critical to a well functioning system. Sufficient conditions for the stability of equilibrium points or periodic orbits of certain deterministic dynamical systems with delayed dynamics are known and it is of interest to understand the sample path behavior of such systems under the addition of small noise. We consider a small noise stochastic delay differential equation (SDDE) with coefficients that depend on the history of the process over a finite delay interval. We obtain asymptotic estimates, as the noise vanishes, on the time it takes a solution of the stochastic equation to exit a bounded domain that is attracted to a stable equilibrium point or periodic orbit of the corresponding deterministic equation. To obtain these asymptotics, we prove a sample path large deviation principle (LDP) for the SDDE that is uniform over initial conditions in bounded sets. The proof of the uniform sample path LDP uses a variational representation for exponential functionals of strong solutions of the SDDE. We anticipate that the overall approach may be useful in proving uniform sample path LDPs for a broad class of infinite-dimensional small noise stochastic equations.
The study of exit time asymptotics for finite-dimensional SDEs is a classical subject in the theory of sample path large deviations, beginning with the work of Freidlin and Wentzell @cite_40 @cite_35 , which culminated in the books @cite_10 @cite_7 . There have been numerous other works related to exit time asymptotics for SDEs, including @cite_11 @cite_3 @cite_8 @cite_46 @cite_9 @cite_53 @cite_39 @cite_45 . In [Chapter 12] daPrato1992 , da Prato and Zabczyk detail a general approach for estimating exit time asymptotics for a class of small noise SPDEs with additive noise. As mentioned above, in @cite_34 @cite_21 @cite_20 the authors obtain exit time asymptotics for a variety of SPDEs with multiplicative noise and in @cite_12 the authors develop a general approach for proving a uniform LDP over bounded sets for a broad class of SPDEs with multiplicative noise and compact semigroups.
@cite_35 @cite_7 @cite_8 @cite_9 @cite_53 @cite_21 @cite_3 @cite_39 @cite_40 @cite_45 @cite_46 @cite_34 @cite_10 @cite_12 @cite_20 @cite_11
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1.Random Perturbations.- 2.Small Random Perturbations on a Finite Time Interval.- 3.Action Functional.- 4.Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point.- 5.Perturbations Leading to Markov Processes.- 6.Markov Perturbations on Large Time Intervals.- 7.The Averaging Principle. Fluctuations in Dynamical Systems with Averaging.- 8.Random Perturbations of Hamiltonian Systems.- 9. The Multidimensional Case.- 10.Stability Under Random Perturbations.- 11.Sharpenings and Generalizations.- References.- Index. Systems with wide bandwidth noise inputs are a common occurrence in stochastic control and communication theory and elsewhere, e.g., tracking or synchronization systems such as phase locked loops (PLL). One is often interested in calculating such quantities as the probability of escape from a desired “error” set, in some time interval, or the mean time for such escape. Diffusion approximations (the system obtained in the limit bandwidth as the @math ) are often used for this since they are easier to analyze. When the noise effects in the physical system are small, one is tempted to do an asymptotic analysis (noise intensity @math ) on the diffusion approximation, and use this for the desired estimates on the original system. Such a procedure does not work in general: the double limit bandwidth @math , intensity @math is not always justified. Under quite broad conditions on the noise processes, it is justified for the systems studied here. We study a particular form of the PLL owing to... We consider diffusion random perturbations of a dynamical systemSt in a domainG⊂Rm which, in particular, may be invariant under the action ofSt. Continuing the study of [K1-K4] we find the asymptotic behavior of the principal eigenvalue of the corresponding generator when the diffusion term tends to zero. In this paper we study the effect on a dynamical system of small random perturbations of the type of white noise: where is the -dimensional Wiener process and as . We are mainly concerned with the effect of these perturbations on long time-intervals that increase with the decreasing . We discuss two problems: the first is the behaviour of the invariant measure of the process as , and the second is the distribution of the position of a trajectory at the first time of its exit from a compact domain. An important role is played in these problems by an estimate of the probability for a trajectory of not to deviate from a smooth function by more than during the time . It turns out that the main term of this probability for small and has the form , where is a certain non-negative functional of . A function , the minimum of over the set of all functions connecting and , is involved in the answers to both the problems. By means of we introduce an independent of perturbations relation of equivalence in the phase-space. We show, under certain assumption, at what point of the phase-space the invariant measure concentrates in the limit. In both the problems we approximate the process in question by a certain Markov chain; the answers depend on the behaviour of on graphs that are associated with this chain. Let us remark that the second problem is closely related to the behaviour of the solution of a Dirichlet problem with a small parameter at the highest derivatives. We consider the Markov diffusion process ξ∈(t), transforming when ɛ=0 into the solution of an ordinary differential equation with a turning point ℴ of the hyperbolic type. The asymptotic behevior as ɛ→0 of the exit time, of its expectation of the probability distribution of exit points for the process ξ∈(t) is studied. These indicate also the asymptotic behavior of solutions of the corresponding singularly perturbed elliptic boundary value problems. Following classical work by Freidlin [Trans. Amer. Math. Soc. (1988) 305 665--657] and subsequent works by Sowers [Ann. Probab. (1992) 20 504--537] and Peszat [Probab. Theory Related Fields (1994) 98 113--136], we prove large deviation estimates for the small noise limit of systems of stochastic reaction--diffusion equations with globally Lipschitz but unbounded diffusion coefficients, however, assuming the reaction terms to be only locally Lipschitz with polynomial growth. This generalizes results of the above mentioned authors. Our results apply, in particular, to systems of stochastic Ginzburg--Landau equations with multiplicative noise. Abstract We consider the exit problem for an asymptotically small random perturbation of a stable dynamical system in a region D. We show the standard large deviations results for the exit distribution and mean exit time, as obtained by Wentzell and Freidlin under the assumption of nontangential drift 〈b, n〉