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Register a new userFirst hitting and strong metastability times
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Added by
Elisabetta Scoppola
Publication date
2018
fields
Physics
and research's language is
English
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In the setting of non-reversible Markov chains on finite or countable state space, exact results on the distribution of the first hitting time to a given set $G$ are obtained. A new notion of strong metastability time is introduced to describe the local relaxation time. This time is defined via a generalization of the strong stationary time to a conditionally strong quasi-stationary time(CSQST). Rarity of the target set $G$ is not required and the initial distribution can be completely general. The results clarify the the role played by the initial distribution on the exponential law; they are used to give a general notion of metastability and to discuss the relation between the exponential distribution of the first hitting time and metastability.
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We generalize the notion of strong stationary time and we give a representation formula for the hitting time to a target set in the general case of non-reversible Markov processes.
For the last ten years, almost every theoretical result concerning the expected run time of a randomized search heuristic used drift theory, making it the arguably most important tool in this domain. Its success is due to its ease of use and its powerful result: drift theory allows the user to derive bounds on the expected first-hitting time of a random process by bounding expected local changes of the process -- the drift. This is usually far easier than bounding the expected first-hitting time directly. Due to the widespread use of drift theory, it is of utmost importance to have the best drift theorems possible. We improve the fundamental additive, multiplicative, and variable drift theorems by stating them in a form as general as possible and providing examples of why the restrictions we keep are still necessary. Our additive drift theorem for upper bounds only requires the process to be nonnegative, that is, we remove unnecessary restrictions like a finite, discrete, or bounded search space. As corollaries, the same is true for our upper bounds in the case of variable and multiplicative drift.
This paper is concerned with the analysis of blow-ups for two McKean-Vlasov equations involving hitting times. Let $(B(t); , t ge 0)$ be standard Brownian motion, and $tau:= inf{t ge 0: X(t) le 0}$ be the hitting time to zero of a given process $X$. The first equation is $X(t) = X(0) + B(t) - alpha mathbb{P}(tau le t)$. We provide a simple condition on $alpha$ and the distribution of $X(0)$ such that the corresponding Fokker-Planck equation has no blow-up, and thus the McKean-Vlasov dynamics is well-defined for all time $t ge 0$. Our approach relies on a connection between the McKean-Vlasov equation and the supercooled Stefan problem, as well as several comparison principles. The second equation is $X(t) = X(0) + beta t + B(t) + alpha log mathbb{P}(tau > t)$, whose Fokker-Planck equation is non-local. We prove that for $beta > 0$ sufficiently large and $alpha$ no greater than a sufficiently small positive constant, there is no blow-up and the McKean-Vlasov dynamics is well-defined for all time $t ge 0$. The argument is based on a new transform, which removes the non-local term, followed by a relative entropy analysis.
Let 0<alpha<1/2. We show that the mixing time of a continuous-time reversible Markov chain on a finite state space is about as large as the largest expected hitting time of a subset of stationary measure at least alpha of the state space. Suitably modified results hold in discrete time and/or without the reversibility assumption. The key technical tool is a construction of a random set A such that the hitting time of A is both light-tailed and a stationary time for the chain. We note that essentially the same results were obtained independently by Peres and Sousi [arXiv:1108.0133].
In noisy environments such as the cell, many processes involve target sites that are often hidden or inactive, and thus not always available for reaction with diffusing entities. To understand reaction kinetics in these situations, we study the first hitting time statistics of a Brownian particle searching for a target site that switches stochastically between visible and hidden phases. At high crypticity, an unexpected rate limited power-law regime emerges for the first hitting time density, which markedly differs from the classic $t^{-3/2}$ scaling for steady targets. Our problem admits an asymptotic mapping onto a mixed, or Robin, boundary condition. Similar results are obtained with non-Markov targets and particles diffusing anomalously.
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