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Register a new userTight estimates of exit and containment probabilities for nonlinear stochastic systems
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Tight estimates of exit/containment probabilities are of particular importance in many control problems. Yet, estimating the exit/containment probabilities is non-trivial: even for linear systems (Ornstein-Uhlenbeck processes), the containment probability can be computed exactly for only some particular values of the system parameters. In this paper, we derive tight bounds on the containment probability for a class of nonlinear stochastic systems. The core idea is to compare the pull strength (how hard the deterministic part of the system dynamics pulls towards the origin) experienced by the nonlinear system at hand with that of a well-chosen process for which tight estimates of the containment probability are known or can be numerically obtained (e.g. an Ornstein-Uhlenbeck process). Specifically, the main technical contribution of this paper is to define a suitable dominance relationship between the pull strengths of two systems and to prove that this dominance relationship implies an order relationship between their containment probabilities. We also discuss the link with contraction theory and highlight some examples of applications.
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Freidlin-Wentzell theory of large deviations can be used to compute the likelihood of extreme or rare events in stochastic dynamical systems via the solution of an optimization problem. The approach gives exponential estimates that often need to be refined via calculation of a prefactor. Here it is shown how to perform these computations in practice. Specifically, sharp asymptotic estimates are derived for expectations, probabilities, and mean first passage times in a form that is geared towards numerical purposes: they require solving well-posed matrix Riccati equations involving the minimizer of the Freidlin-Wentzell action as input, either forward or backward in time with appropriate initial or final conditions tailored to the estimate at hand. The usefulness of our approach is illustrated on several examples. In particular, invariant measure probabilities and mean first passage times are calculated in models involving stochastic partial differential equations of reaction-advection-diffusion type.
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 and therefore do not assume that the domain of attraction features uniform attraction to the equilibrium. This means that the boundary of the domain of attraction is allowed to contain saddles and limit cycles. Our method of proof is purely infinite dimensional, i.e., we do not go through finite dimensional approximations. In addition, we address the multiplicative noise case and we do not impose gradient type of assumptions on the nonlinearity. We prove large deviations logarithmic asymptotics for the exit time and for the exit shape, also characterizing the most probable set of shapes of solutions at the time of exit from the domain of attraction.
Let $X$ be the constrained random walk on ${mathbb Z}_+^d$ representing the queue lengths of a stable Jackson network and $x$ its initial position. Let $tau_n$ be the first time the sum of the components of $X$ equals $n$. $p_n doteq P_x(tau_n < tau_0)$ is a key performance measure for the queueing system represented by $X$, stability implies $p_nrightarrow 0$ exponentially. Currently the only analytic method available to approximate $p_n$ is large deviations analysis, which gives the exponential decay rate of $p_n$. Finer results are available via rare event simulation. The present article develops a new method to approximate $p_n$ and related expectations. The method has two steps: 1) with an affine transformation, move the origin onto the exit boundary of $tau_n$, take limits to remove some of the constraints on the dynamics, this yields a limit unstable constrained walk $Y$ 2) Construct a basis of harmonic functions of $Y$ and use them to apply the classical superposition principle of linear analysis. The basis functions are linear combinations of $log$-linear functions and come from solutions of harmonic systems, which are graphs whose vertices represent points on the characteristic surface of $Y$, the edges between the vertices represent conjugacy relations between the points, the loops represent membership in the boundary characteristic surfaces. Using our method we derive explicit, simple and almost exact formulas for $P_x(tau_n < tau_0)$ for $d$-tandem queues, similar to the product form formulas for the stationary distribution of $X$. The same method allows us to approximate the Balayage operator mapping $f$ to $x rightarrow {mathbb E}_x left[ f(X_{tau_n}) 1_{{tau_n < tau_0}} right]$ for a range of stable constrained random walks in $2$ dimensions. We indicate how the ideas of the paper relate to more general processes and exit boundaries.
Let $mathbb{hat{E}}$ be the upper expectation of a weakly compact but non-dominated family $mathcal{P}$ of probability measures. Assume that $Y$ is a $d$-dimensional $mathcal{P}$-semimartingale under $mathbb{hat{E}}$. Given an open set $Qsubsetmathbb{R}^{d}$, the exit time of $Y$ from $Q$ is defined by [ {tau}_{Q}:=inf{tgeq0:Y_{t}in Q^{c}}. ] The main objective of this paper is to study the quasi-continuity properties of ${tau}_{Q}$ under the nonlinear expectation $mathbb{hat{E}}$. Under some additional assumptions on the growth and regularity of $Y$, we prove that ${tau}_{Q}wedge t$ is quasi-continuous if $Q$ satisfies the exterior ball condition. We also give the characterization of quasi-continuous processes and related properties on stopped processes. In particular, we get the quasi-continuity of exit times for multi-dimensional $G$-martingales, which nontrivially generalizes the previous one-dimensional result of Song.
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.
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