Limit theorems for the Green function of the lattice Laplacian under large deviations of the random walk

2012 ◽  
Vol 76 (6) ◽  
pp. 1190-1217 ◽  
Author(s):  
Stanislav A Molchanov ◽  
E B Yarovaya
Keyword(s):  
Random Walk ◽  
Green Function ◽  
Large Deviations ◽  
Limit Theorems ◽  
The Green Function

A global random walk on grid algorithm for second order elliptic equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Karl K. Sabelfeld ◽  
Dmitrii Smirnov
Keyword(s):  
Differential Equation ◽  
Random Walk ◽  
Green Function ◽  
Elliptic Equations ◽  
Symmetry Property ◽  
Variable Coefficients ◽  
Second Order ◽  
Grid Algorithm ◽  
The Green Function ◽  
Second Order Elliptic Equations

Abstract We suggest in this paper a global random walk on grid (GRWG) method for solving second order elliptic equations. The equation may have constant or variable coefficients. The GRWS method calculates the solution in any desired family of m prescribed points of the gird in contrast to the classical stochastic differential equation based Feynman–Kac formula, and the conventional random walk on spheres (RWS) algorithm as well. The method uses only N trajectories instead of mN trajectories in the RWS algorithm and the Feynman–Kac formula. The idea is based on the symmetry property of the Green function and a double randomization approach.

The fundamental matrix for a certain random walk

1999 ◽  
Vol 36 (02) ◽  
pp. 320-333
Author(s):  
Howard M. Taylor
Keyword(s):  
Random Walk ◽  
Green Function ◽  
Explicit Formula ◽  
Fundamental Matrix ◽  
Simple Random Walk ◽  
Limiting Behavior ◽  
Point Of Entry ◽  
Discrete Lattices ◽  
The Green Function ◽  
The Right

Consider the random walk {Sn} whose summands have the distributionP(X=0) = 1-(2/π), andP(X= ±n) = 2/[π(4n2−1)], forn≥ 1. This random walk arises when a simple random walk in the integer plane is observed only at those instants at which the two coordinates are equal. We derive the fundamental matrix, or Green function, for the process on the integral [0,N] = {0,1,…,N}, and from this, an explicit formula for the mean timexkfor the random walk starting fromS0=kto exit the interval. The explicit formula yields the limiting behavior ofxkasN→ ∞ withkfixed. For the random walk starting from zero, the probability of exiting the interval on the right is obtained. By lettingN→ ∞ in the fundamental matrix, the Green function on the interval [0,∞) is found, and a simple and explicit formula for the probability distribution of the point of entry into the interval (−∞,0) for the random walk starting fromk= 0 results. The distributions for some related random variables are also discovered.Applications to stress concentration calculations in discrete lattices are briefly reviewed.

The fundamental matrix for a certain random walk

1999 ◽  
Vol 36 (2) ◽  
pp. 320-333 ◽  
Author(s):  
Howard M. Taylor
Keyword(s):  
Random Walk ◽  
Green Function ◽  
Explicit Formula ◽  
Fundamental Matrix ◽  
Simple Random Walk ◽  
Limiting Behavior ◽  
Point Of Entry ◽  
Discrete Lattices ◽  
The Green Function ◽  
The Right

Consider the random walk {Sn} whose summands have the distribution P(X=0) = 1-(2/π), and P(X = ± n) = 2/[π(4n2−1)], for n ≥ 1. This random walk arises when a simple random walk in the integer plane is observed only at those instants at which the two coordinates are equal. We derive the fundamental matrix, or Green function, for the process on the integral [0,N] = {0,1,…,N}, and from this, an explicit formula for the mean time xk for the random walk starting from S0 = k to exit the interval. The explicit formula yields the limiting behavior of xk as N → ∞ with k fixed. For the random walk starting from zero, the probability of exiting the interval on the right is obtained. By letting N → ∞ in the fundamental matrix, the Green function on the interval [0,∞) is found, and a simple and explicit formula for the probability distribution of the point of entry into the interval (−∞,0) for the random walk starting from k = 0 results. The distributions for some related random variables are also discovered.Applications to stress concentration calculations in discrete lattices are briefly reviewed.

scholarly journals Local limit theorems in relatively hyperbolic groups I: rough estimates

2021 ◽  
pp. 1-41
Author(s):  
MATTHIEU DUSSAULE
Keyword(s):  
Green Function ◽  
Random Walks ◽  
Limit Theorems ◽  
Local Limit ◽  
Hyperbolic Groups ◽  
Relatively Hyperbolic Groups ◽  
The Green Function ◽  
Positive Recurrent ◽  
Relatively Hyperbolic ◽  
Local Limit Theorems

Abstract This is the first of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this first paper, we prove rough estimates for the Green function. Along the way, we introduce the notion of relative automaticity which will be useful in both papers and we show that relatively hyperbolic groups are relatively automatic. We also define the notion of spectral positive recurrence for random walks on relatively hyperbolic groups. We then use our estimates for the Green function to prove that $p_n\asymp R^{-n}n^{-3/2}$ for spectrally positive-recurrent random walks, where $p_n$ is the probability of going back to the origin at time n and where R is the inverse of the spectral radius of the random walk.

scholarly journals Gillis' Random Walks on Graphs

2005 ◽  
Vol 42 (1) ◽  
pp. 295-301 ◽  
Author(s):  
Nadine Guillotin-Plantard
Keyword(s):  
Random Walk ◽  
Green Function ◽  
Random Walks ◽  
Regular Graph ◽  
Simple Random Walk ◽  
Square Lattice ◽  
Random Walker ◽  
Unit Step ◽  
Random Walks On Graphs ◽  
The Green Function

We consider a random walker on a d-regular graph. Starting from a fixed vertex, the first step is a unit step in any one of the d directions, with common probability 1/d for each one. At any later step, the random walker moves in any one of the directions, with probability q for a reversal of direction and probability p for any other direction. This model was introduced and first studied by Gillis (1955), in the case when the graph is a d-dimensional square lattice. We prove that the Gillis random walk on a d-regular graph is recurrent if and only if the simple random walk on the graph is recurrent. The Green function of the Gillis random walk will be also given, in terms of that of the simple random walk.

scholarly journals Gillis' Random Walks on Graphs

2005 ◽  
Vol 42 (01) ◽  
pp. 295-301 ◽  
Author(s):  
Nadine Guillotin-Plantard
Keyword(s):  
Random Walk ◽  
Green Function ◽  
Random Walks ◽  
Regular Graph ◽  
Simple Random Walk ◽  
Square Lattice ◽  
Random Walker ◽  
Unit Step ◽  
Random Walks On Graphs ◽  
The Green Function

We consider a random walker on ad-regular graph. Starting from a fixed vertex, the first step is a unit step in any one of theddirections, with common probability 1/dfor each one. At any later step, the random walker moves in any one of the directions, with probabilityqfor a reversal of direction and probabilitypfor any other direction. This model was introduced and first studied by Gillis (1955), in the case when the graph is ad-dimensional square lattice. We prove that the Gillis random walk on ad-regular graph is recurrent if and only if the simple random walk on the graph is recurrent. The Green function of the Gillis random walk will be also given, in terms of that of the simple random walk.

Exponentially Convergent Approximation to the Elliptic Solution Operator

2006 ◽  
Vol 6 (4) ◽  
pp. 386-404 ◽  
Author(s):  
Ivan. P. Gavrilyuk ◽  
V.L. Makarov ◽  
V.B. Vasylyk
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Solution Operator ◽  
Hyperbolic Operator ◽  
Elliptic Solution ◽  
Elliptic Boundary Value ◽  
Sine Family ◽  
The Green Function

AbstractWe develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

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