Wiener Process with Reflection in Non-Smooth Narrow Tubes

Wiener process with instantaneous reflection in narrow tubes of width {\epsilon}<<1 around axis x is considered in this paper. The tube is assumed to be (asymptotically) non-smooth in the following sense. Let $V^{\epsilon}(x)$ be the volume of the cross-section of the tube. We assume that $V^{\epsilon}(x)/{\epsilon}$ converges in an appropriate sense to a non-smooth function as {\epsilon}->0. This limiting function can be composed by smooth functions, step functions and also the Dirac delta distribution. Under this assumption we prove that the x-component of the Wiener process converges weakly to a Markov process that behaves like a standard diffusion process away from the points of discontinuity and has to satisfy certain gluing conditions at the points of discontinuity.Comment: 28 pages, 1 figur

Rare event simulation for multiscale diffusions in random environments

We consider systems of stochastic differential equations with multiple scales and small noise and assume that the coefficients of the equations are ergodic and stationary random fields. Our goal is to construct provably-efficient importance sampling Monte Carlo methods that allow efficient computation of rare event probabilities or expectations of functionals that can be associated with rare events. Standard Monte Carlo algorithms perform poorly in the small noise limit and hence fast simulations algorithms become relevant. The presence of multiple scales complicates the design and the analysis of efficient importance sampling schemes. An additional complication is the randomness of the environment. We construct explicit changes of measures that are proven to be logarithmic asymptotically efficient with probability one with respect to the random environment (i.e., in the quenched sense). Numerical simulations support the theoretical results.Comment: Final version, paper to appear in SIAM Journal Multiscale Modelling and Simulatio

Fluctuation analysis and short time asymptotics for multiple scales diffusion processes

We consider the limiting behavior of fluctuations of small noise diffusions with multiple scales around their homogenized deterministic limit. We allow full dependence of the coefficients on the slow and fast motion. These processes arise naturally when one is interested in short time asymptotics of multiple scale diffusions. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. Depending on the order of interaction between the fast scale and the size of the noise we get different behavior. In certain cases additional drift terms arise in the limiting process, which are explicitly characterized. These results provide a better approximation to the limiting behavior of such processes when compared to the law of large numbers homogenization limit

Large Deviations and Importance Sampling for Systems of Slow-Fast Motion

In this paper we develop the large deviations principle and a rigorous mathematical framework for asymptotically efficient importance sampling schemes for general, fully dependent systems of stochastic differential equations of slow and fast motion with small noise in the slow component. We assume periodicity with respect to the fast component. Depending on the interaction of the fast scale with the smallness of the noise, we get different behavior. We examine how one range of interaction differs from the other one both for the large deviations and for the importance sampling. We use the large deviations results to identify asymptotically optimal importance sampling schemes in each case. Standard Monte Carlo schemes perform poorly in the small noise limit. In the presence of multiscale aspects one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. It turns out that one has to consider the so called cell problem from the homogenization theory for Hamilton-Jacobi-Bellman equations in order to guarantee asymptotic optimality. We use stochastic control arguments.Comment: More detailed proofs. Differences from the published version are editorial and typographica