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In this work, we show that for the martingale problem for a class of degenerate diffusions with bounded continuous drift and diffusion coefficients, the small noise limit of non-degenerate approximations leads to a unique Feller limit. The proof uses the theory of viscosity solutions applied to the associated backward Kolmogorov equations. Under appropriate conditions on drift and diffusion coefficients, we will establish a comparison principle and a one-one correspondence between Feller solutions to the martingale problem and continuous viscosity solutions of the associated Kolmogorov equation. This work can be considered as an extension to the work in Borkar and Kumar in (J Theor Probab 23(3): 729–747, 2010). [ABSTRACT FROM AUTHOR]

We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described by d X t ε = b (X t ε , Y t ε) d t + ε α d B t , d Y t ε = - 1 ε ∇ y U (X t ε , Y t ε) d t + s (ε) ε d W t , where B t , W t are independent Brownian motions on R d and R m respectively, b : R d × R m → R d , U : R d × R m → R and s : (0 , ∞) → (0 , ∞) . We impose regularity assumptions on b, U and let 0 < α < 1. When s (ε) goes to zero slower than a prescribed rate as ε → 0 , we characterize all weak limit points of X ε , as ε → 0 , as solutions to a differential equation driven by a measurable vector field. Under an additional assumption on the behaviour of U (x , ·) at its global minima we characterize all limit points as Filippov solutions to the differential equation. [ABSTRACT FROM AUTHOR]

We study the effect of additive Brownian noise on an ODE system that has a stable hyperbolic limit cycle, for initial data that are attracted to the limit cycle. The analysis is performed in the limit of small noise – that is, we modulate the noise by a factor ε ↘ 0 – and on a long time horizon. We prove explicit estimates on the proximity of the noisy trajectory and the limit cycle up to times exp ⁡ ( c ε − 2 ) , c > 0 , and we show both that on the time scale ε − 2 the dephasing (i.e., the difference between noiseless and noisy system measured in a natural coordinate system that involves a phase) is close to a Brownian motion with constant drift, and that on longer time scales the dephasing dynamics is dominated by the drift. The natural choice of coordinates, that reduces the dynamics in a neighborhood of the cycle to a rotation, plays a central role and makes the connection with the applied science literature in which noisy limit cycle dynamics are often reduced to a diffusion model for the phase of the limit cycle. [ABSTRACT FROM AUTHOR]

We consider the asymptotics of the invariant measure for the process of the empirical spatial distribution of N coupled Markov chains in the limit of a large number of chains. Each chain reflects the stochastic evolution of one particle. The chains are coupled through the dependence of the transition rates on this spatial distribution of particles in the various states. Our model is a caricature for medium access interactions in wireless local area networks. It is also applicable to the study of spread of epidemics in a network. The limiting process satisfies a deterministic ordinary differential equation called the McKean-Vlasov equation. When this differential equation has a unique globally asymptotically stable equilibrium, the spatial distribution asymptotically concentrates on this equilibrium.More generally, its limit points are supported on a subset of the ω-limit sets of the McKean-Vlasov equation. Using a control-theoretic approach, we examine the question of large deviations of the invariant measure from this limit.

We present explicit methods for simulating diffusions whose generator is self-adjoint with respect to a known (but possibly not normalizable) density. These methods exploit this property and combine an optimized Runge-Kutta algorithm with a Metropolis-Hastings Monte Carlo scheme. The resulting numerical integration scheme is shown to be weakly accurate at finite noise and to gain higher order accuracy in the small noise limit. It also permits the user to avoid computing explicitly certain terms in the equation, such as the divergence of the mobility tensor, which can be tedious to calculate. Finally, the scheme is shown to be ergodic with respect to the exact equilibrium probability distribution of the diffusion when it exists. These results are illustrated in several examples, including a Brownian dynamics simulation of DNA in a solvent. In this example, the proposed scheme is able to accurately compute dynamics at time step sizes that are an order of magnitude (or more) larger than those permitted with commonly used explicit predictor-corrector schemes. [ABSTRACT FROM AUTHOR]

A new selection principle is proposed for construction of a Feller solution to an ill-posed degenerate diffusion process. This is based on constructing the Feller transition kernel from the unique (under suitable conditions) continuous viscosity solutions to the backward Kolmogorov equation associated with the diffusion. The connection of this Feller process with the small-noise limits of the nondegenerate perturbations of the diffusion and several other related phenomena are also discussed. [ABSTRACT FROM AUTHOR]

Abstract: We consider multi-dimensional nondegenerate diffusions with invariant densities, with the diffusion matrix scaled by a small . The o.d.e. limit corresponding to is assumed to have the origin as its unique globally asymptotically stable equilibrium. Using control theoretic methods, we show that in the limit, the invariant density has the form , where the W is characterized as the optimal cost of a deterministic control problem. This generalizes an earlier work of Sheu. Extension to multiple equilibria is also given. [Copyright &y& Elsevier]

A dynamical system driven by non-Gaussian Lévy noises of small intensity is considered. The first exit time of solution orbits from a bounded neighborhood of an attracting equilibrium state is estimated. For a class of non-Gaussian Lévy noises, it is shown that the mean exit time is asymptotically faster than exponential (the well-known Gaussian Brownian noise case) but slower than polynomial (the stable Lévy noise case), in terms of the reciprocal of the small noise intensity. [ABSTRACT FROM AUTHOR]

It is known that Kalman–Bucy filter is stable with respect to initial conditions under the assumptions of uniform complete controllability and uniform complete observability. In this paper, we prove the stability of Kalman-Bucy filter for the case of noise free dynamical system, i.e., for deterministic processes. The earlier stability results cannot be applied for this case, as the system is not controllable. We further show that the optimal linear filter for a general class of non-Gaussian initial conditions is asymptotically proximal to Kalman–Bucy filter. It is also shown that the filter corresponding to non-zero system noise in the limit of small system noise approaches the filter corresponding to zero system noise in the case of Gaussian initial conditions. [ABSTRACT FROM AUTHOR]