一维扩散过程的小随机扰动

本文考虑一维扩散过程的小随机扰动．我们应用随机分析方法，给出了当扰动趋于零时，平均越出时间的渐近估计和越出时间的概率估计．     

The Poincaré Map of Randomly Perturbed Periodic Motion

A system of autonomous differential equations with a stable limit cycle and perturbed by small white noise is analyzed in this work. In the vicinity of the limit cycle of the unperturbed deterministic system, we define, construct, and analyze the Poincaré map of the randomly perturbed periodic motion. We show that the time of the first exit from a small neighborhood of the fixed point of the map, which corresponds to the unperturbed periodic orbit, is well approximated by the geometric distribution. The parameter of the geometric distribution tends to zero together with the noise intensity. Therefore, our result can be interpreted as an estimate of the stability of periodic motion to random perturbations. In addition, we show that the geometric distribution of the first exit times translates into statistical properties of solutions of important differential equation models in applications. To this end, we demonstrate three distinct examples from mathematical neuroscience featuring complex oscillatory patterns characterized by the geometric distribution. We show that in each of these models the statistical properties of emerging oscillations are fully explained by the general properties of randomly perturbed periodic motions identified in this paper.     

Scaling limit for the diffusion exit problem in the Levinson case

The exit problem for small perturbations of a dynamical system in a domain is considered. It is assumed that the unperturbed dynamical system and the domain satisfy the Levinson conditions. We assume that the random perturbation affects the driving vector field and the initial condition, and each of the components of the perturbation follows a scaling limit. We derive the joint scaling limit for the random exit time and exit point. We use this result to study the asymptotics of the exit time for 1D diffusions conditioned on rare events.     

Global well-posedness for the KP-II equation on the background of a non-localized solution

Motivated by transverse stability issues, we address the time evolution under the KP-II flow of perturbations of a solution which does not decay in all directions, for instance the KdV-line soliton. We study two different types of perturbations: perturbations that are square integrable in R×T and perturbations that are square integrable in R². In both cases we prove the global well-posedness of the Cauchy problem associated with such initial data.     

Exit times for multivariate autoregressive processes

We study exit times from a set for a family of multivariate autoregressive processes with normally distributed noise. By using the large deviation principle, and other methods, we show that the asymptotic behavior of the exit time depends only on the set itself and on the covariance matrix of the stationary distribution of the process. The results are extended to exit times from intervals for the univariate autoregressive process of order n$n$, where the exit time is of the same order of magnitude as the exponential of the inverse of the variance of the stationary distribution.     

Multiscale analysis of exit distributions for random walks in random environments

We present a multiscale analysis for the exit measures from large balls in , of random walks in certain i.i.d. random environments which are small perturbations of the fixed environment corresponding to simple random walk. Our main assumption is an isotropy assumption on the law of the environment, introduced by Bricmont and Kupiainen. Under this assumption, we prove that the exit measure of the random walk in a random environment from a large ball, approaches the exit measure of a simple random walk from the same ball, in the sense that the variational distance between smoothed versions of these measures converges to zero. We also prove the transience of the random walk in random environment. The analysis is based on propagating estimates on the variational distance between the exit measure of the random walk in random environment and that of simple random walk, in addition to estimates on the variational distance between smoothed versions of these quantities. Partially supported by NSF grant DMS-0503775.     

Moderate deviations for randomly perturbed dynamical systems

A Moderate Deviation Principle is established for random processes arising as small random perturbations of one-dimensional dynamical systems of the form X_n=f(X_n−1). Unlike in the Large Deviations Theory the resulting rate function is independent of the underlying noise distribution, and is always quadratic. This allows one to obtain explicit formulae for the asymptotics of probabilities of the process staying in a small tube around the deterministic system. Using these, explicit formulae for the asymptotics of exit times are obtained. Results are specified for the case when the dynamical system is periodic, and imply stability of such systems. Finally, results are applied to the model of density-dependent branching processes.     

Estimates for first exit times of non-Markovian Itô processes

First exit times and their path-wise dependence on trajectories are studied for non-Markovian Itô processes. Estimates of distances between two exit times are obtained. In particular, it follows that first exit times of two Itô processes are close if their trajectories are close.     

On the dimension of the subspace of Liouvillian solutions of a Fuchsian system

The effect of small constantly acting random perturbations of white noise type on a dynamical system with locally stable fixed point is studied. The perturbed system is considered in the form of Itô stochastic differential equations, and it is assumed that the perturbation does not vanish at a fixed point. In this case, the trajectories of the stochastic system issuing from points near the stable fixed point exit from the neighborhood of equilibrium with probability 1. Classes of perturbations such that the equilibrium of a deterministic system is stable in probability on an asymptotically large time interval are described.     

An entry-exit model involving construction and abandonment periods

We analyze, using the optimal stopping theory, the entry-exit decision on a project, which takes time to be constructed and abandoned. We obtain the closed-form expressions of optimal start time of entry, optimal start time of exit, and the maximal expected present value of the project. In addition, we examine the effects of construction and abandonment periods on the optimal start times of entry and exit.     

Property of “correct exit” and one limit theorem for semi-Markov processes

A necessary and sufficient condition on weak convergence of a sequence or probability measures in the space D_{[o, ∞]}(X) is formulated in terms of first exit times. The proof of necessity is based on continuity of first exit times and first exit points with respect to the Stone-Skorokhod metric on the set of functions that “correctly exit” from an open set Δ?X. A limit theorem for semi-Markov processes is proved as an application.     

The Weighted Transience and Recurrence of Markov Processes

Transience and recurrence are among the most important concepts in Markov processes. In this paper, we study the transience and recurrence for right processes with a given weight function, and characterize them by potentials, excessive functions, first hitting times and last exit times of the process. We also study the properties of recurrent states.     

Exit Times from Equilateral Triangles

In this paper we obtain a closed form expression of the expected exit time of a Brownian motion from equilateral triangles. We consider first the analogous problem for a symmetric random walk on the triangular lattice and show that it is equivalent to the ruin problem of an appropriate three player game. A suitable scaling of this random walk allows us to exhibit explicitly the relation between the respective exit times. This gives us the solution of the related Poisson equation.     

Exit Times from Equilateral Triangles

In this paper we obtain a closed form expression of the expected exit time of a Brownian motion from equilateral triangles. We consider first the analogous problem for a symmetric random walk on the triangular lattice and show that it is equivalent to the ruin problem of an appropriate three player game. A suitable scaling of this random walk allows us to exhibit explicitly the relation between the respective exit times. This gives us the solution of the related Poisson equation.     

Long waves in ferromagnetic media, Khokhlov-Zabolotskaya equation

We are interested here in small perturbations of electromagnetic waves in a saturated ferromagnetic media. By means of an asymptotic expansion we prove that the solution remains close on long times of the one of the Khokhlov-Zabolotskaya equation.     

Exit identities for diusion processes observed at Poisson arrival times

For diffusion processes, we extend various two-sided exit identities to the situation when the process is only observed at arrival times of an independent Poisson process. The results are expressed in terms of solutions to the differential equations associated with the diffusions generators.     

Robustness for Stable Impulsive Equations via Quadratic Lyapunov Functions

For a linear impulsive differential equation, we give a complete characterization of the existence of a nonuniform exponential contraction in terms of quadratic Lyapunov functions and of the operators defining them. This corresponds to consider a nonuniform exponential stability of the dynamics, which is typical for example in the context of ergodic theory. As an application, we use this characterization to establish in a very simple manner the robustness property of a nonuniform exponential contraction under sufficiently small linear perturbations. In addition, we obtain versions of the robustness property for perturbations of the jumping times and of a strong nonuniform exponential contraction. The latter corresponds to consider not only an upper bound for the dynamics but also a lower bound.     

Exit problems associated with finite reflection groups

We obtain a formula for the distribution of the first exit time of Brownian motion from a fundamental region associated with a finite reflection group. In the type A case it is closely related to a formula of de Bruijn and the exit probability is expressed as a Pfaffian. Our formula yields a generalisation of de Bruijn’s. We derive large and small time asymptotics, and formulas for expected first exit times. The results extend to other Markov processes. By considering discrete random walks in the type A case we recover known formulas for the number of standard Young tableaux with bounded height.Mathematics Subject Classification (2000): 20F55, 60J65     

A priori bounds for elliptic equations

In this paper we study the relationship existing between the maximum principle and the principal eigenvalue of second order elliptic operators and the expected exit times of the corresponding diffusions. A probabilistic approach is discussed that provides a good understanding of classical results established analytically. We also obtain an Alexandrov-Bakelman-Pucci type of estimate using similar methods.