Joint asymptotic behavior of local and occupation times of random walk in higher dimension

2007 ◽  
Vol 44 (4) ◽  
pp. 535-563 ◽  
Author(s):  
Endre Csáki ◽  
Antónia Földes ◽  
Pál Révész
Keyword(s):  
Asymptotic Behavior ◽  
Random Walk ◽  
Unit Ball ◽  
Local Time ◽  
Occupation Time ◽  
Local Times ◽  
Higher Dimension ◽  
Occupation Times ◽  
Symmetric Random Walk

Considering a simple symmetric random walk in dimension d ≧ 3, we study the almost sure joint asymptotic behavior of two objects: first the local times of a pair of neighboring points, then the local time of a point and the occupation time of the surface of the unit ball around it.

scholarly journals Optimal Bounds for the Variance of Self-Intersection Local Times

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
George Deligiannidis ◽  
Sergey Utev
Keyword(s):  
Random Walk ◽  
Local Time ◽  
Simple Random Walk ◽  
The Self ◽  
The Other ◽  
Local Times ◽  
Intersection Local Time ◽  
Moment Conditions ◽  
Symmetric Random Walk ◽  
Optimal Bounds

For a Zd-valued random walk (Sn)n∈N0, let l(n,x) be its local time at the site x∈Zd. For α∈N, define the α-fold self-intersection local time as Ln(α)≔∑xl(n,x)α. Also let LnSRW(α) be the corresponding quantities for the simple random walk in Zd. Without imposing any moment conditions, we show that the variance of the self-intersection local time of any genuinely d-dimensional random walk is bounded above by the corresponding quantity for the simple symmetric random walk; that is, var(Ln(α))=O(var⁡(LnSRW(α))). In particular, for any genuinely d-dimensional random walk, with d≥4, we have var⁡(Ln(α))=O(n). On the other hand, in dimensions d≤3 we show that if the behaviour resembles that of simple random walk, in the sense that lim infn→∞var⁡Lnα/var⁡(LnSRW(α))>0, then the increments of the random walk must have zero mean and finite second moment.

scholarly journals Asymptotic Behavior of the Local Time of a Recurrent Random Walk

1984 ◽  
Vol 12 (1) ◽  
pp. 64-85 ◽  
Author(s):  
Naresh C. Jain ◽  
William E. Pruitt
Keyword(s):  
Asymptotic Behavior ◽  
Random Walk ◽  
Local Time ◽  
Recurrent Random Walk

A random walk analogue of Lévy’s Theorem

2008 ◽  
Vol 45 (2) ◽  
pp. 223-233 ◽  
Author(s):  
Takahiko Fujita
Keyword(s):  
Random Walk ◽  
Local Time ◽  
Stochastic Analysis ◽  
Type Theorem ◽  
Symmetric Random Walk ◽  
Absolute Value ◽  
Lévy’S Theorem ◽  
Definition Of ◽  
Skorokhod Reflection ◽  
Tanaka Formula

In this paper we will give a simple symmetric random walk analogue of Lévy’s Theorem. We will give a new definition of a local time of the simple symmetric random walk. We apply a discrete Itô formula to some absolute value like function to obtain a discrete Tanaka formula. Results in this paper rely upon a discrete Skorokhod reflection argument. This random walk analogue of Lévy’s theorem was already obtained by G. Simons ([14]) but it is still worth noting because we will use a discrete stochastic analysis to obtain it and this method is applicable to other research. We note some connection with previous results by Csáki, Révész, Csörgő and Szabados. Finally we observe that the discrete Lévy transformation in the present version is not ergodic. Lastly we give a Lévy-type theorem for simple nonsymmetric random walk using a discrete bang-bang process.

scholarly journals On a Generalization of One-Dimensional Kinetics

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1264
Author(s):  
Vladimir V. Uchaikin ◽  
Renat T. Sibatov ◽  
Dmitry N. Bezbatko
Keyword(s):  
Exact Solution ◽  
Asymptotic Behavior ◽  
Constant Velocity ◽  
Telegraph Equation ◽  
Material Derivative ◽  
Symmetric Random Walk ◽  
One Dimensional ◽  
Special Cases ◽  
Fractional Telegraph Equation ◽  
Asymmetric Case

One-dimensional random walks with a constant velocity between scattering are considered. The exact solution is expressed in terms of multiple convolutions of path-distributions assumed to be different for positive and negative directions of the walk axis. Several special cases are considered when the convolutions are expressed in explicit form. As a particular case, the solution of A. S. Monin for a symmetric random walk with exponential path distribution and its generalization to the asymmetric case are obtained. Solution of fractional telegraph equation with the fractional material derivative is presented. Asymptotic behavior of its solution for an asymmetric case is provided.

scholarly journals Representations of hermite processes using local time of intersecting stationary stable regenerative sets

2020 ◽  
Vol 57 (4) ◽  
pp. 1234-1251
Author(s):  
Shuyang Bai
Keyword(s):  
Long Range ◽  
Local Time ◽  
Limit Theorems ◽  
Local Times ◽  
Long Range Dependence ◽  
Random Interval ◽  
Stationary Increments ◽  
Self Similar ◽  
Regenerative Sets

AbstractHermite processes are a class of self-similar processes with stationary increments. They often arise in limit theorems under long-range dependence. We derive new representations of Hermite processes with multiple Wiener–Itô integrals, whose integrands involve the local time of intersecting stationary stable regenerative sets. The proof relies on an approximation of regenerative sets and local times based on a scheme of random interval covering.

scholarly journals The exterior Dirichlet problems of Monge–Ampère equations in dimension two

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Unit Ball ◽  
Bounded Domain ◽  
Viscosity Solutions ◽  
Sufficient Conditions ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Necessary And Sufficient ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

AbstractIn this paper, we study the Monge–Ampère equations $\det D^{2}u=f$ det D 2 u = f in dimension two with f being a perturbation of $f_{0}$ f 0 at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.

Self-avoiding random walk: A Brownian motion model with local time drift

1987 ◽  
Vol 74 (2) ◽  
pp. 271-287 ◽  
Author(s):  
J. R. Norris ◽  
L. C. G. Rogers ◽  
David Williams
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Local Time ◽  
Motion Model ◽  
Time Drift ◽  
Brownian Motion Model
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