Control of the mean exiting time for the correlated random walk via initial position setting

Automatica ◽  
2019 ◽  
Vol 103 ◽  
pp. 239-242
Author(s):  
Jianliang Tang ◽  
Mingqing Xiao
Keyword(s):  
Random Walk ◽  
Initial Position ◽  
Correlated Random Walk ◽  
The Mean

scholarly journals Can Mean Helicity Fluctuations Explain the Variability of the Solar Cycle ?

1993 ◽  
Vol 157 ◽  
pp. 71-75
Author(s):  
P. Hoyng
Keyword(s):  
Random Walk ◽  
Solar Cycle ◽  
Phase Shift ◽  
Correlated Random Walk ◽  
The Mean ◽  
Dynamo Wave

I consider the effect of rapid fluctuations in the mean helicity on a plane dynamo wave in the αω-approximation and in the weak forcing limit. The phase shift and the logarithmic amplitude of the wave exhibit a correlated random walk, so that weaker (stronger) cycles last longer (shorter). The solar cycle data follow this prediction rather well. Mean helicity fluctuations are concluded to be an important source of solar cycle variability.

Mortal random walks on a family of treelike regular fractals with a deep trap

2020 ◽  
Vol 34 (06) ◽  
pp. 2050031
Author(s):  
Zikai Wu ◽  
Guangyao Xu
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Network Models ◽  
Deep Trap ◽  
Initial Position ◽  
Single Step ◽  
The Mean ◽  
Walk Dimension ◽  
Mean Time ◽  
Temporal Factor

Due to the ubiquitous occurrence of evanescence in many physical, chemical and biological scenarios, mortal random walks that incorporate evanescence explicitly have drawn more and more attention. It has been a hot topic to study mortal random walks on distinct network models. In this paper, we study mortal random walks on T fractal and a family of treelike regular fractals with a trap located at central node (i.e., innermost node). First, with self-similar setting composed of T fractal, initial position of the walker and location of trap, the total trapping probability of the mortal walker reduces to a function of walker’s single-step survival parameter [Formula: see text]. In more detail, the total trapping probability is expressed by the [Formula: see text]th iteration of map (scaling function) of [Formula: see text]. Based on the map, the analytical expression of total trapping probability’s dominant behavior, the mean time to trapping (MFPT) and temporal factor are obtained, which are related to random walk dimension. Last, we extend the analysis to a family of treelike regular fractals. On them, the total trapping probability is still expressed as the [Formula: see text]th iteration of the map scaling [Formula: see text]. Accordingly, dominant behavior of total trapping probability, MFPT and temporal factor are determined analytically. Both analytical results obtained on T fractal and more general treelike regular fractals show that the mean time to trapping and desired random walk dimension can be obtained by tuning the survival probability parameter [Formula: see text]. In summary, the work advances the understanding of mortal random walks on more general deterministic networks.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

A compass without a map: tortuosity and orientation of eastern painted turtles (Chrysemys picta picta) released in unfamiliar territory

2006 ◽  
Vol 84 (8) ◽  
pp. 1129-1137 ◽  
Author(s):  
I.R. Caldwell ◽  
V.O. Nams
Keyword(s):  
Fractal Dimension ◽  
Random Walk ◽  
Spatial Scales ◽  
Chrysemys Picta ◽  
Reference Stimulus ◽  
Correlated Random Walk ◽  
Specific Mechanism ◽  
Minimal Time ◽  
Painted Turtles

Orientation mechanisms allow animals to spend minimal time in hostile areas while reaching needed resources. Identification of the specific mechanism used by an animal can be difficult, but examining an animal's path in familiar and unfamiliar areas can provide clues to the type of mechanism in use. Semiaquatic turtles are known to use a homing mechanism in familiar territory to locate their home lake while on land, but little is known about their ability to locate habitat in unfamiliar territory. We tested the tortuosity and orientation of 60 eastern painted turtles ( Chrysemys picta picta (Schneider, 1783)). We released turtles at 20 release points located at five distances and in two directions from two unfamiliar lakes. Turtle trails were quite straight (fractal dimension between 1.1 and 1.025) but were not oriented towards water from any distance (V-test; u < 0.72; P > 0.1). Turtles maintained their initially chosen direction but either could not detect water or were not motivated to reach it. Furthermore, paths were straighter at larger spatial scales than at smaller spatial scales, which could not have occurred if the turtles had been using a correlated random walk. Turtles must therefore be using a reference stimulus for navigation even in unfamiliar areas.

scholarly journals Approximation and Analytical Studies of Inter-clustering Performances of Space-Filling Curves

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Ho-Kwok Dai ◽  
Hung-Chi Su
Keyword(s):  
Random Walk ◽  
Dimensional Space ◽  
Hilbert Curve ◽  
Space Filling ◽  
Space Filling Curve ◽  
Space Filling Curves ◽  
Order Curve ◽  
Cluster Distance ◽  
The Mean ◽  
Curve Family

International audience A discrete space-filling curve provides a linear traversal/indexing of a multi-dimensional grid space.This paper presents an application of random walk to the study of inter-clustering of space-filling curves and an analytical study on the inter-clustering performances of 2-dimensional Hilbert and z-order curve families.Two underlying measures are employed: the mean inter-cluster distance over all inter-cluster gaps and the mean total inter-cluster distance over all subgrids.We show how approximating the mean inter-cluster distance statistics of continuous multi-dimensional space-filling curves fits into the formalism of random walk, and derive the exact formulas for the two statistics for both curve families.The excellent agreement in the approximate and true mean inter-cluster distance statistics suggests that the random walk may furnish an effective model to develop approximations to clustering and locality statistics for space-filling curves.Based upon the analytical results, the asymptotic comparisons indicate that z-order curve family performs better than Hilbert curve family with respect to both statistics.

Optimal stopping rules for correlated random walks with a discount

2004 ◽  
Vol 41 (2) ◽  
pp. 483-496 ◽  
Author(s):  
Pieter Allaart
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Optimal Stopping ◽  
Constant Factor ◽  
Stopping Rules ◽  
Correlated Random Walk ◽  
Numerical Examples ◽  
Optimal Rule ◽  
Correlated Random Walks ◽  
Optimal Stopping Rules

Optimal stopping rules are developed for the correlated random walk when future returns are discounted by a constant factor per unit time. The optimal rule is shown to be of dual threshold form: one threshold for stopping after an up-step, and another for stopping after a down-step. Precise expressions for the thresholds are given for both the positively and the negatively correlated cases. The optimal rule is illustrated by several numerical examples.

Series expansions for the all-time maximum of α-stable random walks

2016 ◽  
Vol 48 (3) ◽  
pp. 744-767
Author(s):  
Clifford Hurvich ◽  
Josh Reed
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Queueing Theory ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Series Expansions ◽  
Stable Random Variables ◽  
The Mean ◽  
The Stability

AbstractWe study random walks whose increments are α-stable distributions with shape parameter 1<α<2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter β=-1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.

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