N. Dokuchaev
It is given an effective upper estimate of expectation of |T_1-T_2|, where T_1 and T_2 are the first exit times from a region for two vector diffusion processes.
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We use analytical methods to construct the two-parameter Feller semigroup associated with a Markov process on a line with a moving membrane such that at the points on both sides of the membrane it coincides with the ordinary diffusion processes given there and its behavior after reaching the membrane is described by a kind of nonlocal Feller-Wentzell conjugation condition.
For each prime $p$, a diffusion constant together with a positive exponent specify a Vladimirov operator and an associated $p$-adic diffusion equation. The fundamental solution of this pseudo-differential equation gives rise to a measure on the Skorokhod space of $p$-adic valued paths that is concentrated on the paths originating at the origin. We calculate the first exit probabilities of paths from balls and estimate these probabilities for the brownian bridges.
Capacity is an important quantity in potential theory and in the study of Markov processes. We give equivalent conditions between the capacity, the mean exit time, and the Green function for non-reversible diffusions.
Variational principles for the exit time of non-symmetric diffusions
该研究建立了非对称扩散过程离开区域时间的新变分原理，并据此得到不同扩散过程间的比较定理与单调性规律。
In this paper we study the fluctuations from the limiting behavior of small noise random perturbations of diffusions with multiple scales. The result is then applied to the exit problem for multiscale diffusions, deriving the limiting law of the joint distribution of the exit time and exit location. We apply our results to the first order Langevin equation in a rough potential, studying both fluctuations around the typical behavior and the conditional limiting exit law, conditional on the rare event of going against the underlying deterministic flow.
The objective of this dissertation is to prove a scaling limit for the exit of a domain problem of a small noise system with underlying hyperbolic dynamics. In this case, Large Deviation kind of estimates fail to provide a complete picture of the dynamics of the system under consideration. This is so because the rate function given by the Large Deviation Principle has several minimizing trajectories hence making them indistinguishable at the exponential level. We propose a pathwise approach based on the theory of normal forms combined with geometrical arguments. We prove a scaling limits and p…
In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm was already introduced in both the Brownian context and in the Ornstein-Uhlenbeck context. Here the aim is therefore to generalize this efficient numerical approach in order to obtain an approximation of both the exit time and position for either a general linear diffusion or a growth diffusion. The efficiency of the method is described with particular care through theoretical results and numerical examples.
The main result in this paper is a variational formula for the exit rate from a bounded domain for a diffusion process in terms of the stationary law of the diffusion constrained to remain in this domain forever. Related results on the geometric ergodicity of the controlled Q-process are also presented.
We provide Large Deviation estimates for the bridge of a $d$-dimensional general diffusion process as the conditioning time tends to $0$ and apply these results to the evaluation of the asymptotics of its exit time probabilities. We are motivated by applications to numerical simulation, especially in connection with stochastic volatility models.
A dynamical system perturbed by white noise in a neighborhood of an unstable fixed point is considered. We obtain the exit asymptotics in the limit of vanishing noise intensity. This is a refinement of a result by Kifer (1981).
Freidlin-Wentzell type exit-time estimates for time-inhomogeneous diffusions and their applications
本文研究非齐次扩散的逃逸时间，证明在漂移和扩散项一致接近时逃逸时间指数增长，并推广至McKean-Vlasov过程。
We study a class of time-inhomogeneous diffusion: the self-interacting one. We show a convergence result with a rate of convergence that does not depend on the diffusion coefficient. Finally, we establish a so-called Kramers' type law for the first exit-time of the process from domain of attractions when the landscapes are uniformly convex.
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 1-d diffusions conditioned on rare events.
In this paper, we study the asymptotic of exit problem for controlled Markov diffusion processes with random jumps and vanishing diffusion terms, where the random jumps are introduced in order to modify the evolution of the controlled diffusions by switching from one mode of dynamics to another. That is, depending on the state-position and state-transition information, the dynamics of the controlled diffusions randomly switches between the different drift and diffusion terms. Here, we specifically investigate the asymptotic exit problem concerning such controlled Markov diffusion processes in …
Integration by parts formula for exit times of one dimensional diffusions
本文为一维扩散过程的首个退出时间建立了分部积分公式，采用基于时间变量的微积分技术，而非传统的空间马尔可夫链微分方法。
We study the exit-time from a domain of a self-interacting diffusion, where the Brownian motion is replaced by $σB_t$ for a constant $σ$. The first part of this work consists in showing that the rate of convergence (of the occupation measure of the self-interacting process toward some explicit Gibbs measure) previously obtained in \cite{kk-ejp} for a convex confinment potential $V$ and a convex interaction potential can be bounded uniformly with respect to $σ$. Then, we prove an Arrhenius-type law for the first exit-time from a domain (satisfying classical hypotheses of Freidlin-Wentzell theor…
In this work we prove a Kramers' type law for the low-temperature behavior of the exit-times from a metastable state for a class of self-interacting nonlinear diffusion processes. Contrary to previous works, the interaction is not assumed to be convex, which means that this result covers cases where the exit-time for the interacting process is smaller than the exit-time for the associated non-interacting process. The technique of the proof is based on the fact that, under an appropriate contraction condition, the interacting process is conveniently coupled with a non-interacting (linear) Marko…
In this article we derive moment estimates, exponential integrability, concentration inequalities and exit times estimates for canonical diffusions in two settings each beyond the scope of Riemannian geometry. Firstly, we consider sub-Riemannian limits of Riemannian foliations. Secondly, we consider the non-smooth setting of $\mathrm{RCD}^*(K,N)$ spaces. In each case the necessary ingredients are an Itô formula and a comparison theorem for the Laplacian, for which we refer to the recent literature. As an application, we derive pointwise Carmona-type estimates on eigenfunctions of Schrödinger o…
In order to describe or estimate different quantities related to a specific random variable, it is of prime interest to numerically generate such a variate. In specific situations, the exact generation of random variables might be either momentarily unavailable or too expensive in terms of computation time. It therefore needs to be replaced by an approximation procedure. As was previously the case, the ambitious exact simulation of exit times for diffusion processes was unreachable though it concerns many applications in different fields like mathematical finance, neuroscience or reliability. …
The goal of this paper is to supplement the large deviation principle of the Freidlin--Wentzell theory on exit problems for diffusion processes with results of classical central limit theorem kind. We describe a class of situations where conditioning on exit through unlikely locations leads to a Gaussian scaling limit for the exit distribution. Our results are based on Doob's h-transform and new asymptotic convergence gradient estimates for elliptic nonlinear equations that allow to reduce the problem to the Levinson case. We devote a separate section to a rigorous and general discussion of h-…
In this paper we develop a metastability theory for a class of stochastic reaction-diffusion equations exposed to small multiplicative noise. We consider the case where the unperturbed reaction-diffusion equation features multiple asymptotically stable equilibria. When the system is exposed to small stochastic perturbations, it is likely to stay near one equilibrium for a long period of time, but will eventually transition to the neighborhood of another equilibrium. We are interested in studying the exit time from the full domain of attraction (in a function space) surrounding an equilibrium a…
This paper studies, in dimensions greater than two, stationary diffusion processes in random environment which are small, isotropic perturbations of Brownian motion satisfying a finite range dependence. Such processes were first considered in the continuous setting by Sznitman and Zeitouni [20]. Building upon their work, it is shown by analyzing the associated elliptic boundary-value problem that, almost surely, the smoothed (in the sense that the boundary data is continuous) exit law of the diffusion from large domains converges, as the domain's scale approaches infinity, to that of a Brownia…
Exact simulation of first exit times for one-dimensional diffusion processes
本文提出了一种新的算法，用于精确模拟一维扩散过程的首次退出时间，利用Girsanov变换、收敛级数方法和经典拒绝采样技术，提高了模拟的准确性。
Measure-dependent non-linear diffusions with superlinear drifts: asymptotic behaviour of the first exit-times
本文在无凸性假设的可逆 McKean-Vlasov SDE 中，利用大偏差原理和改进耦合方法证明了首次退出时间的 Kramers 型指数渐近律，显著推广了已有结果。
The purpose of this paper is to consider the exit-time problem for a finite-range Markov jump process, i.e, the distance the particle can jump is bounded independent of its location. Such jump diffusions are expedient models for anomalous transport exhibiting super-diffusion or nonstandard normal diffusion. We refer to the associated deterministic equation as a volume-constrained nonlocal diffusion equation. The volume constraint is the nonlocal analogue of a boundary condition necessary to demonstrate that the nonlocal diffusion equation is well-posed and is consistent with the jump process. …
We explain the connection between the Gumbel limit for diffusion exit times and the theory of extreme values.
Stochastic homogenization is achieved for a class of elliptic and parabolic equations describing the lifetime, in large domains, of stationary diffusion processes in random environment which are small, statistically isotropic perturbations of Brownian motion in dimension at least three. Furthermore, the homogenization is shown to occur with an algebraic rate. Such processes were first considered in the continuous setting by Sznitman and Zeitouni [21], upon whose results the present work relies strongly, and more recently their smoothed exit distributions from large domains were shown to conver…
We investigate exit times from domains of attraction for the motion of a self-stabilized particle traveling in a geometric (potential type) landscape and perturbed by Brownian noise of small amplitude. Self-stabilization is the effect of including an ensemble-average attraction in addition to the usual state-dependent drift, where the particle is supposed to be suspended in a large population of identical ones. A Kramers' type law for the particle's exit from the potential's domains of attraction and a large deviations principle for the self-stabilizing diffusion are proved. It turns out that …
We consider a finite dimensional deterministic dynamical system with a global attractor A with a unique ergodic measure P concentrated on it, which is uniformly parametrized by the mean of the trajectories in a bounded set D containing A. We perturbe this dynamical system by a multiplicative heavy tailed Lévy noise of small intensity ε>0 and solve the asymptotic first exit time and location problem from a bounded domain D around the attractor A in the limit of ε to 0. In contrast to the case of Gaussian perturbations, the exit time has the asymptotically algebraic exit rate as a function of ε,…
First Passage and First Exit Times for diffusion processes related to a general growth curve
针对包含多种经典生长方程的一般生长曲线，提出加性、乘性噪声下的两种随机扩散模型，推导了时变阈值下首达时间与首出时间概率密度的闭式解。
We study reaction diffusion equations with a deterministic reaction term as well as two random reaction terms, one that acts on the interior of the domain, and another that acts only on the boundary of the domain. We are interested in the regime where the relative sizes of the diffusion and reaction terms are different. Specifically, we consider the case where the diffusion rate is much larger than the rate of reaction, and the deterministic rate of reaction is much larger than either of the random rate of reactions.
In this work, we analyse the metastability of non-reversible diffusion processes $$dX_t=\boldsymbol{b}(X_t)dt+\sqrt h\,dB_t$$ on a bounded domain $Ω$ when $\mathbf{b}$ admits the decomposition $\mathbf{b}=-(\nabla f+\mathbf{\ell})$ and $\nabla f \cdot \mathbf{\ell}=0$. In this setting, we first show that, when $h\to 0$, the principal eigenvalue of the generator of $(X_t)_{t\ge 0}$ with Dirichlet boundary conditions on the boundary $\partialΩ$ of $Ω$ is exponentially close to the inverse of the mean exit time from $Ω$, uniformly in the initial conditions $X_0=x$ within the compacts of $Ω$. The …
Let $Ω\subset\mathbb R^n$ be a bounded domain and for $x\inΩ$ let $τ(x)$ be the expected exit time from $Ω$ of a diffusing particle starting at $x$ and advected by an incompressible flow $u$. We are interested in the question which flows maximize $\|τ\|_{L^\infty(Ω)}$, that is, they are most efficient in the creation of hotspots inside $Ω$. Surprisingly, among all simply connected domains in two dimensions, the discs are the only ones for which the zero flow $u\equiv 0$ maximises $\|τ\|_{L^\infty(Ω)}$. We also show that in any dimension, among all domains with a fixed volume and all incompress…
First exit times from regions and their dependence on variations of boundaries are discussed for diffusion processes. The paper presents an estimate of $L_1$-distance between exit times from two regions via expectations of exit times.
Metastability of diffusion processes in narrow tubes
研究狭窄管域中扩散过程的亚稳态行为，通过分析特征时间尺度和退出分布，揭示了其渐近性质及瓶颈效应下的逃逸机制。
We give some relationships between the first Dirichlet eigenvalues and the exit time moments for the general symmetric Markov processes. As applications, we present some examples, including symmetric diffusions and $α$-stable processes, and provide the estimates of their first Dirichlet eigenvalues and the exit time moments.
In this review, an outline of the so called Freidlin-Wentzell theory and its recent extensions is given. Broadly, this theory studies the exponential rate at which the probabilities of rare events related to random perturbation of ODE decays. The typical situation is when an ODE has several stable equilibria, in which case, the theory predicts the most likely paths in which the randomly perturbed system goes from one equilibria to another. In recent developments I will outline how recent approaches allows to distinguish between paths that are otherwise exponentially equivalent and provide an o…
We study quantitative asymptotics of planar random walks that are spatially non-homogeneous but whose mean drifts have some regularity. Specifically, we study the first exit time $τ_α$ from a wedge with apex at the origin and interior half-angle $α$ by a non-homogeneous random walk on the square lattice with mean drift at $x$ of magnitude $O(1/|x|)$ as $|x| \to \infty$. This is the critical regime for the asymptotic behaviour: under mild conditions, a previous result of the authors (see arXiv:0910.1772) stated that $τ_α< \infty$ a.s. for any $α$ (while for a stronger drift field $τ_α$ is infin…
We consider the exit problem for small white noise perturbation of a smooth dynamical system on the plane in the neighborhood of a hyperbolic critical point. We show that if the distribution of the initial condition has a scaling limit then the exit distribution and exit time also have a joint scaling limit as the noise intensity goes to zero. The limiting law is computed explicitly. The result completes the theory of noisy heteroclinic networks in two dimensions. The analysis is based on normal forms theory.
This article studies the dynamics of a nonlinear dissipative reaction-diffusion equation with well-separated stable states which is perturbed by infinite-dimensional multiplicative Lévy noise with a regularly varying component at intensity $ε>0$. The main results establish the precise asymptotics of the first exit times and locus of the solution $X^ε$ from the domain of attraction of a deterministic stable state, in the limit as $ε\rightarrow 0$. In contrast to the exponential growth for respective Gaussian perturbations the exit times grow essentially as a power function of the noise intensit…
Let X be some homogeneous additive functional of a skew Bessel process Y. In this note, we compute the asymptotics of the first passage time of X to some fixed level b, and study the position of Y when X exits a bounded interval [a, b]. As a by-product, we obtain the probability that X reaches the level b before the level a. Our results extend some previous works on additive functionals of Brownian motion by Isozaki and Kotani for the persistence problem, and by Lachal for the exit time problem.
Freidlin and Wentzell characterized the logarithmic asymptotics of the exit time from a basin of attraction for a finite dimensional diffusion with small noise. After that, several authors studied the same properties for exit problems associated to specific infinite dimensional system. In this paper, we present a general method, based a control theoretic approach, to establish exit time and exit place results for a large class of stochastic equations in Banach spaces.
In his seminal work from the 1950s, William Feller classified all one-dimensional diffusions on $-\infty\leq a<b\leq \infty$ in terms of their ability to access the boundary (Feller's test for explosions) and to enter the interior from the boundary. Feller's technique is restricted to diffusion processes as the corresponding differential generators allow explicit computations and the use of Hille-Yosida theory. In the present article we study exit and entrance from infinity for the most natural generalization, that is, jump diffusions of the form \[ dZ_t=σ(Z_{t-})\,dX_t, \] driven by stable Lé…
We consider a dynamical system described by the differential equation $\dot{Y}_t=-U'(Y_t)$ with a unique stable point at the origin. We perturb the system by the Lévy noise of intensity $\varepsilon$ to obtain the stochastic differential equation $dX^{\varepsilon}_t=-U'(X^{\varepsilon}_{t-}) dt+\varepsilon dL_t.$ The process $L$ is a symmetric Lévy process whose jump measure $ν$ has exponentially light tails, $ν([u,\infty))\sim\exp(-u^α)$, $α>0$, $u\to \infty$. We study the first exit problem for the trajectories of the solutions of the stochastic differential equation from the interval $(-1,1…
We consider exit problems for general Lévy processes, where the first passage over a threshold is detected either immediately or at an epoch of an independent homogeneous Poisson process. It is shown that the two corresponding one-sided problems are related through a surprisingly simple identity. Moreover, we identify a simple link between two-sided exit problems with one continuous and one Poisson exit. Finally, Poisson exit of a reflected process is connected to the continuous exit of a process reflected at Poisson epochs, and a link between some Parisian type exit problems is established. W…
Physical notions of stochastic resonance for potential diffusions in periodically changing double-well potentials such as the spectral power amplification have proved to be defective. They are not robust for the passage to their effective dynamics: continuous-time finite-state Markov chains describing the rough features of transitions between different domains of attraction of metastable points. In the framework of one-dimensional diffusions moving in periodically changing double-well potentials we design a new notion of stochastic resonance which refines Freidlin's concept of quasi-periodic m…
In this paper, we consider a diffusion process pertaining to a chain of distributed control systems with small random perturbation. The distributed control system is formed by n subsystems that satisfy an appropriate Hormander condition, i.e., the second subsystem assumes the random perturbation entered into the first subsystem, the third subsystem assumes the random perturbation entered into the first subsystem then was transmitted to the second subsystem and so on, such that the random perturbation propagates through the entire distributed control system. Note that the random perturbation en…
This paper develops a new technique for the path approximation of one-dimensional stochastic processes, more precisely the Brownian motion and families of stochastic differential equations sharply linked to the Brownian motion (usually known as L and G-classes). We are interested here in the $ε$-strong approximation. We propose an explicit and easy to implement procedure that constructs jointly, the sequences of exit times and corresponding exit positions of some well chosen domains. The main results control the number of steps to cover a fixed time interval and the convergence theorems for ou…
In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm so-called Walk on Moving Spheres was already introduced in the Brownian context. The aim is therefore to generalize this numerical approach to the Ornstein-Uhlenbeck process and to describe the efficiency of the method.
We establish general moment estimates for the discrete and continuous exit times of a general Itô process in terms of the distance to the boundary. These estimates serve as intermediate steps to obtain strong convergence results for the approximation of a continuous exit time by a discrete counterpart, computed on a grid. In particular, we prove that the discrete exit time of the Euler scheme of a diffusion converges in the L1 norm with an order 1/2 with respect to the mesh size.
Exit-problem for a class of non-Markov processes with path dependency
本文研究了一类具有路径依赖性的自相互作用扩散过程在开域中的退出时间问题，在较弱的势函数假设下建立了大偏差原理，并证明了退出时间满足 Kramers 型定律，最后给出了扩散退出位置的结果。
We prove the kernel estimates related to subordinated semigroups on homogeneous trees. We study the long time propagation problem. We exploit this to show exit time estimates for (large) balls. We use an abstract setting of metric measure spaces. This enables us to give these results for trees end hyperbolic spaces as well. Finally, we show some estimates for the Poisson kernel of a ball.
Moment estimates, exponential integrability, concentration inequalities and exit times estimates on evolving manifolds
该研究在带C¹族完备黎曼度量与C^{1,∞}族向量场的非紧光滑演化流形上，证明了扩散过程的矩估计、指数可积性、集中不等式及首离时估计，拓展了演化流形上扩散过程的分析结果。
For a stopped diffusion process in a multidimensional time-dependent domain $\D$, we propose and analyse a new procedure consisting in simulating the process with an Euler scheme with step size $Δ$ and stopping it at discrete times $(iΔ)_{i\in\N^*}$ in a modified domain, whose boundary has been appropriately shifted. The shift is locally in the direction of the inward normal $n(t,x)$ at any point $(t,x)$ on the parabolic boundary of $\D$, and its amplitude is equal to $0.5826 (...) |n^*σ|(t,x)\sqrt Δ$ where $σ$ stands for the diffusion coefficient of the process. The procedure is thus extremel…
The fundamental solution of a pseudo-differential equation for functions defined on the $d$-fold product of the $p$-adic numbers, $\mathbb{Q}_p$, induces an analogue of the Wiener process in $\mathbb{Q}_p^d$. As in the real setting, the components are $1$-dimensional $p$-adic Brownian motions with the same diffusion constant and exponent as the original process. Asymptotic analysis of the conditional probabilities shows that the vector components are dependent for all time. Exit time probabilities for the higher dimensional processes reveal a concrete effect of the component dependency.
We provide sharp Large Deviation estimates for the probability of exit from a domain for the bridge of a $d$-dimensional general diffusion process $X$, as the conditioning time tends to $0$. This kind of results is motivated by applications to numerical simulation. In particular we investigate the influence of the drift $b$ of $X$. It turns out that the sharp asymptotics for the exit time probability are independent of the drift, provided $b$ enjoyes a simple condition that is always satisfied in dimension $1$. On the other hand, we show that the drift can be influential if this assumption is …
We study a class of reflected McKean-Vlasov diffusions over a convex domain with self-stabilizing coefficients. This includes coefficients that do not satisfy the classical Wasserstein Lipschitz condition. Further, the process is constrained to a (not necessarily bounded) convex domain by a local time on the boundary. These equations include the subclass of reflected self-stabilizing diffusions that drift towards their mean via a convolution of the solution law with a stabilizing potential. Firstly, we establish existence and uniqueness results for this class and address the propagation of cha…
We report some additional examples of explicit solutions to an inverse first-passage place problem for one-dimensional diffusions with jumps, introduced in a previous paper. If $X(t)$ is a one-dimensional diffusion with jumps, starting from a random position $η\in [a,b],$ let be $τ_{a,b}$ the time at which $X(t)$ first exits the interval $(a,b),$ and $π_a = P(X(τ_{a,b}) \le a)$ the probability of exit from the left of $(a,b).$ Given a probability $q \in (0,1),$ the problem consists in finding the density $g$ of $η$ (if it exists) such that $π_a = q;$ it can be seen as a problem of optimization…
Let $D\subset R^d$ be a bounded domain and denote by $\mathcal P(D)$ the space of probability measures on $D$. Let \begin{equation*} L=\frac12\nabla\cdot a\nabla +b\nabla \end{equation*} be a second order elliptic operator. Let $μ\in\mathcal P(D)$ and $δ>0$. Consider a Markov process $X(t)$ in $D$ which performs diffusion in $D$ generated by the operator $δL$ and is stopped at the boundary, and which while running, jumps instantaneously, according to an exponential clock with spatially dependent intensity $V>0$, to a new point, according to the distribution $μ$. The Markov process is generated…
We establish the singularity with respect to Lebesgue measure as a function of time of the conditional probability that the sum of two one-dimensional Brownian motions will exit from the unit interval before time $t$, given the trajectory of the second Brownian motion up to the same time. On the way of doing so we show that if one solves the one-dimensional heat equation with zero condition on a trajectory of a one-dimensional Brownian motion, which is the lateral boundary, then for each moment of time with probability one the normal derivative of the solution is zero, provided that the diffus…
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