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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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〉 In this paper we show how to accelerate randomized coordinate descent methods and achieve faster convergence rates without paying per-iteration costs in asymptotic running time. In particular, we show how to generalize and efficiently implement a method proposed by Nesterov, giving faster asymptotic running times for various algorithms that use standard coordinate descent as a black box. In addition to providing a proof of convergence for this new general method, we show that it is numerically stable, efficiently implementable, and in certain regimes, asymptotically optimal. To highlight the computational power of this algorithm, we show how it can used to create faster linear system solvers in several regimes: - We show how this method achieves a faster asymptotic runtime than conjugate gradient for solving a broad class of symmetric positive definite systems of equations. - We improve the best known asymptotic convergence guarantees for Kaczmarz methods, a popular technique for image reconstruction and solving overdetermined systems of equations, by accelerating a randomized algorithm of Strohmer and Vershynin. - We achieve the best known running time for solving Symmetric Diagonally Dominant (SDD) system of equations in the unit-cost RAM model, obtaining an O(m log^ 3 2 n (log log n)^ 1 2 log (log n eps)) asymptotic running time by accelerating a recent solver by Beyond the independent interest of these solvers, we believe they highlight the versatility of the approach of this paper and we hope that they will open the door for further algorithmic improvements in the future. In this paper we show how to accelerate randomized coordinate descent methods and achieve faster convergence rates without paying per-iteration costs in asymptotic running time. In particular, we show how to generalize and efficiently implement a method proposed by Nesterov, giving faster asymptotic running times for various algorithms that use standard coordinate descent as a black box. In addition to providing a proof of convergence for this new general method, we show that it is numerically stable, efficiently implementable, and in certain regimes, asymptotically optimal. To highlight the power of this algorithm, we show how it can used to create faster linear system solvers in several regimes: - We show how this method achieves a faster asymptotic runtime than conjugate gradient for solving a broad class of symmetric positive definite systems of equations. - We improve the convergence guarantees for Kaczmarz methods, a popular technique for image reconstruction and solving over determined systems of equations, by accelerating an algorithm of Strohmer and Vershynin. - We achieve the best known running time for solving Symmetric Diagonally Dominant (SDD) system of equations in the unit-cost RAM model, obtaining a running time of O(m log3 2n (log log n)1 2 log((log n) eps)) by accelerating a recent solver by Beyond the independent interest of these solvers, we believe they highlight the versatility of the approach of this paper and we hope that they will open the door for further algorithmic improvements in the future. 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.