Exit asymptotics for small diffusion about an unstable equilibrium

Atypical exit events near a repelling equilibrium

We consider exit problems for small white noise perturbations of a dynamical system generated by a vector field, and a domain containing a critical point with all positive eigenvalues of linearization. We prove that, in the vanishing noise limit, the probability of exit through a generic set on the boundary is asymptotically polynomial in the noise strength, with exponent depending on the mutual position of the set and the flag of the invariant manifolds associated with the top eigenvalues. Furthermore, we compute the limiting exit distributions conditioned on atypical exit events of polynomia…

Weak noise and non-hyperbolic unstable fixed points: Sharp estimates on transit and exit times

We consider certain one dimensional ordinary stochastic differential equations driven by additive Brownian motion of variance $\varepsilon ^2$. When $\varepsilon =0$ such equations have an unstable non-hyperbolic fixed point and the drift near such a point has a power law behavior. For $\varepsilon >0$ small, the fixed point property disappears, but it is replaced by a random escape or transit time which diverges as $\varepsilon \searrow0$. We show that this random time, under suitable (easily guessed) rescaling, converges to a limit random variable that essentially depends only on the power e…

An intermediate regime for exit phenomena driven by non-Gaussian Levy noises

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.

Exit time asymptotics for small noise stochastic delay differential equations

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 o…

Metastability and exit problems for systems of stochastic reaction-diffusion equations

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…

具有私人信息的退出博弈

Exit game with private information

针对竞争与不确定下的战略退出决策难题，研究了参与者不知对手退出价值的随机退出博弈，基于一般一维扩散收益流构建了含状态变量与后验信念的显式均衡策略，并证明其在对称贝叶斯均衡子类中唯一。

First-exit-time probability density tails for a local height of a non-equilibrium Gaussian interface

We study the long-time behavior of the probability density Q_t of the first exit time from a bounded interval [-L,L] for a stochastic non-Markovian process h(t) describing fluctuations at a given point of a two-dimensional, infinite in both directions Gaussian interface. We show that Q_t decays when t \to \infty as a power-law $^{-1 - α}, where αis non-universal and proportional to the ratio of the thermal energy and the elastic energy of a fluctuation of size L. The fact that αappears to be dependent on L, which is rather unusual, implies that the number of existing moments of Q_t depends on …

On stochastic perturbations of slowly changing dynamical systems

In this paper we consider a diffusion process obtained as a small random perturbation of a dynamical system attracted to a stable equilibrium point. The drift and the diffusive perturbation are assumed to evolve slowly in time. We describe the asymptotics of the time it takes the process to exit a given domain and the limiting distribution of the exit point.

On the noise-induced passage through an unstable periodic orbit I: Two-level model

We consider the problem of stochastic exit from a planar domain, whose boundary is an unstable periodic orbit, and which contains a stable periodic orbit. This problem arises when investigating the distribution of noise-induced phase slips between synchronized oscillators, or when studying stochastic resonance far from the adiabatic limit. We introduce a simple, piecewise linear model equation, for which the distribution of first-passage times can be precisely computed. In particular, we obtain a quantitative description of the phenomenon of cycling: The distribution of first-passage times rot…

连续时间和离散时间随机过程向不稳定平衡的非收敛性

Non-convergence to unstable equilibriums for continuous-time and discrete-time stochastic processes

该研究证明连续与离散时间随机过程向不稳定平衡的非收敛定理，拓展修正了过往离散时间相关结论，并将其应用于两类顶点强化随机游走的陷阱非收敛性证明。

Quasipotentials with more than two variables: new evaluation at equilibrium points of the drift

The relevant quasipotential near an equilibrium point is determined by a new linear matrix equation, with less unknowns than an existing (possibly nonlinear) one. This also assures the asymptotic fulfillment of the Fokker-Planck equation, even globally due to the second term in the noise strength. An auxiliary result for the exit problem is derived as well.

Exit time and principal eigenvalue of non-reversible elliptic diffusions

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 …

Decision making times in mean-field dynamic Ising model

We consider a dynamic mean-field ferromagnetic model in the low-temperature regime in the neighborhood of the zero magnetization state. We study the random time it takes for the system to make a decision, i.e., to exit the neighborhood of the unstable equilibrium and approach one of the two stable equilibrium points. We prove a limit theorem for the distribution of this random time in the thermodynamic limit.

Exit time tails from pairwise decorrelation in hidden Markov chains, with applications to dynamical percolation

Consider a Markov process ω_t at equilibrium and some event C (a subset of the state-space of the process). A natural measure of correlations in the process is the pairwise correlation \Pr[ω_0,ω_t \in C] - \Pr[ω_0 \in C]^2. A second natural measure is the probability of the continual occurrence event \{ω_s \in C, \forall s\in [0,t]\}. We show that for reversible Markov chains, and any event C, pairwise decorrelation of the event C implies a decay of the probability of the continual occurrence event \{ω_s \in C, \forall s \in [0,t]\} as t\to\infty. We provide examples showing that our results a…

随机反应扩散方程在中等偏差区域的重要性采样

Importance sampling for stochastic reaction-diffusion equations in the moderate deviation regime

本文提出一种可证明高效的重要性采样方案，用于估计小噪声随机反应扩散方程的解从稳定平衡点标定邻域退出的概率，通过中等偏差缩放和线性化近似，并结合随机控制与变分方法证明了其在零噪声极限和预渐近情况下的良好表现。