scholarly journals On the scaling limit of loop-erased random walk excursion

2012 ◽  
Vol 50 (2) ◽  
pp. 331-357
Author(s):  
Fredrik Johansson Viklund
Keyword(s):  
Random Walk ◽  
Scaling Limit

scholarly journals Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling

2018 ◽  
Vol 24 (3) ◽  
pp. 1075-1105
Author(s):  
Andrei Agrachev ◽  
Ugo Boscain ◽  
Robert Neel ◽  
Luca Rizzi
Keyword(s):  
Random Walk ◽  
Riemannian Manifolds ◽  
Riemannian Geometry ◽  
Stochastic Analysis ◽  
Scaling Limit ◽  
Primary Motivation ◽  
Infinitesimal Generators ◽  
Integral Curves ◽  
Parabolic Scaling ◽  
Riemannian Case

We relate some constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular attention to walks where the choice of step is influenced by a volume on the manifold. A primary motivation is to explore how one can pass, in the parabolic scaling limit, from geodesics, orthonormal frames, and/or volumes to diffusions, and hence their infinitesimal generators, on sub-Riemannian manifolds, which is interesting in light of the fact that there is no completely canonical notion of sub-Laplacian on a general sub-Riemannian manifold. However, even in the Riemannian case, this random walk approach illuminates the geometric significance of Ito and Stratonovich stochastic differential equations as well as the role played by the volume.

scholarly journals Scaling limit of the loop-erased random walk Green’s function

2015 ◽  
Vol 166 (1-2) ◽  
pp. 271-319 ◽  
Author(s):  
Christian Beneš ◽  
Gregory F. Lawler ◽  
Fredrik Viklund
Keyword(s):  
Random Walk ◽  
Green’S Function ◽  
Green's Function ◽  
Scaling Limit

Limit theorems for continuous-time random walks with infinite mean waiting times

2004 ◽  
Vol 41 (03) ◽  
pp. 623-638 ◽  
Author(s):  
Mark M. Meerschaert ◽  
Hans-Peter Scheffler
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Continuous Time ◽  
Waiting Times ◽  
Scaling Limit ◽  
Domain Of Attraction ◽  
Simple Random Walk ◽  
Lévy Motion ◽  
Hamiltonian Chaos ◽  
Continuous Time Random Walks

A continuous-time random walk is a simple random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper we show that, when the time between renewals has infinite mean, the scaling limit is an operator Lévy motion subordinated to the hitting time process of a classical stable subordinator. Density functions for the limit process solve a fractional Cauchy problem, the generalization of a fractional partial differential equation for Hamiltonian chaos. We also establish a functional limit theorem for random walks with jumps in the strict generalized domain of attraction of a full operator stable law, which is of some independent interest.

scholarly journals Optimal scaling of the random walk Metropolis algorithm under Lp mean differentiability

2017 ◽  
Vol 54 (4) ◽  
pp. 1233-1260 ◽  
Author(s):  
Alain Durmus ◽  
Sylvain Le Corff ◽  
Eric Moulines ◽  
Gareth O. Roberts
Keyword(s):  
Random Walk ◽  
Scaling Limit ◽  
Optimal Scaling ◽  
High Dimensional ◽  
Random Walk Metropolis ◽  
Langevin Diffusion ◽  
Random Walk Metropolis Algorithm ◽  
The One ◽  
Original Derivation ◽  
Practical Implications

Abstract In this paper we consider the optimal scaling of high-dimensional random walk Metropolis algorithms for densities differentiable in the Lp mean but which may be irregular at some points (such as the Laplace density, for example) and/or supported on an interval. Our main result is the weak convergence of the Markov chain (appropriately rescaled in time and space) to a Langevin diffusion process as the dimension d goes to ∞. As the log-density might be nondifferentiable, the limiting diffusion could be singular. The scaling limit is established under assumptions which are much weaker than the one used in the original derivation of Roberts et al. (1997). This result has important practical implications for the use of random walk Metropolis algorithms in Bayesian frameworks based on sparsity inducing priors.

scholarly journals On the csáki-vincze transformation

2013 ◽  
Vol 50 (2) ◽  
pp. 266-279
Author(s):  
Hatem Hajri
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Asymptotic Properties ◽  
Scaling Limit ◽  
Simple Random Walk ◽  
Precise Description ◽  
Discrete Transformation

Csáki and Vincze have defined in 1961 a discrete transformation T which applies to simple random walks and is measure preserving. In this paper, we are interested in ergodic and asymptotic properties of T. We prove that T is exact: ∩k≧1σ(Tk(S)) is trivial for each simple random walk S and give a precise description of the lost information at each step k. We then show that, in a suitable scaling limit, all iterations of T “converge” to the corresponding iterations of the continuous Lévy transform of Brownian motion.

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