The Optimal Sink and the Best Source in a Markov Chain
@article{Bakhtin2011TheOS, title={The Optimal Sink and the Best Source in a Markov Chain}, author={Yuri Bakhtin and Leonid A. Bunimovich}, journal={Journal of Statistical Physics}, year={2011}, volume={143}, pages={943-954} }
It is well known that the distributions of hitting times in Markov chains are quite irregular, unless the limit as time tends to infinity is considered. We show that nevertheless for a typical finite irreducible Markov chain and for nondegenerate initial distributions the tails of the distributions of the hitting times for the states of a Markov chain can be ordered, i.e., they do not overlap after a certain finite moment of time. If one considers instead each state of a Markov chain as a…
11 Citations
Where and When Orbits of Strongly Chaotic Systems Prefer to Go
- Physics
- 2017
We prove that transport in the phase space of the "most strongly chaotic" dynamical systems has three different stages. Consider a finite Markov partition (coarse graining) $\xi$ of the phase space…
Faster than expected escape for a class of fully chaotic maps.
- MathematicsChaos
- 2012
An exact periodic orbit formula is derived for finite size Markov holes which differs from other periodic expansions in the literature and can account for additional distortion to maps with piecewise constant expansion rate.
Short- and long-term forecast for chaotic and random systems (50 years after Lorenz's paper)
- Physics
- 2014
We briefly review a history of the impact of the famous 1963 paper by E Lorenz on hydrodynamics, physics and mathematics communities on both sides of the iron curtain. This paper was an attempt to…
Leaking chaotic systems
- Physics
- 2013
There are numerous physical situations in which a hole or leak is introduced in an otherwise closed chaotic system. The leak can have a natural origin, it can mimic measurement devices, and it can…
Why escape is faster than expected
- Mathematics
- 2020
We consider chaotic (hyperbolic) dynamical systems which have a generating Markov partition. Then, open dynamical systems are built by making one element of a Markov partition a hole through which…
System Reliability at the Crossroads
- Computer Science
- 2012
It is argued that while greater understanding of the physics of failure has led to significant progress at the component level, there are significant challenges remaining at the system level and system reliability, a field of applied mathematics that addresses the latter challenges, is at a juncture where fundamental changes are likely.
IMPROVED ESTIMATES OF SURVIVAL PROBABILITIES VIA ISOSPECTRAL TRANSFORMATIONS
- Mathematics
- 2014
We consider open systems generated from one-dimensional maps that admit a finite Markov partition and use the recently developed theory of isospectral graph transformations to estimate a system’s…
Some new surprises in chaos.
- PhysicsChaos
- 2015
Numerical results demonstrate that some parts of the phase space of chaotic systems are more likely to be visited earlier than other parts and prove a statement that could be naturally considered as a dual one to the Poincaré theorem on recurrences.
References
SHOWING 1-10 OF 16 REFERENCES
Hitting and return times in ergodic dynamical systems
- Mathematics
- 2004
Given an ergodic dynamical system (X, T,μ), and U C X measurable with μ(U) > 0, let μ(U)τ U (x) denote the normalized hitting time of x ∈ X to U. We prove that given a sequence (U n ) with μ(U n ) →…
Where to place a hole to achieve a maximal escape rate
- Physics
- 2008
A natural question of how the survival probability depends upon a position of a hole was seemingly never addressed in the theory of open dynamical systems. We found that this dependency could be very…
ASYMPTOTICS FOR HITTING TIMES
- Mathematics
- 2005
(essentially decreasing sequences of balls in ametric space X) and identified to be the distribution function of the posi-tive exponential law with parameter 1, in many classes of mixing systems,in…
The limiting distribution and error terms for return times of dynamical systems
- Mathematics
- 2004
We develop a new framework that allows to prove that
the limiting distribution of return times for a large class of
mixing dynamical systems are Poisson
distributed.
We demonstrate our technique…
Possible limit laws for entrance times of an ergodic aperiodic dynamical system
- Mathematics
- 2002
LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ0∞G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and…
Sharp error terms and neccessary conditions for exponential hitting times in mixing processes
- Mathematics
- 2004
We prove an upper bound for the error in the exponential approximation of the hitting time law of a rare event in α-mixing processes with exponential decay, ϕ-mixing processes with a summable…
On the notion of recurrence in discrete stochastic processes
- Mathematics
- 1947
Received by the editors March 3, 1947. 1 John Simon Guggenheim Memorial Fellow. 2 That the theory of stationary stochastic processes is mathematically equivalent with an "ergodic" theory (to which…
Entropy estimation and fluctuations of Hitting and Recurrence Times for Gibbsian sources
- Computer Science
- 2004
This work provides a comprehensive analysis of hitting times of cylinder sets in the setting of Gibbsian sources and proves two strong approximation results from which it can be deduced pointwise convergence to entropy, lognormal fluctuations, precise large deviation estimates and an explicit formula for the hitting-time multifractal spectrum.
Which hole is leaking the most: a topological approach to study open systems
- Mathematics
- 2010
We study the process of escape of orbits through a hole in the phase space of a dynamical system generated by a chaotic map of an interval. If this hole is an element of Markov partition we are able…
An Introduction To Probability Theory And Its Applications
- Mathematics
- 1950
A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics.