scholarly journals Geodesic Random Walks, Diffusion Processes and Brownian Motion on Finsler Manifolds

2021 ◽  
Author(s):  
Tianyu Ma ◽  
Vladimir S. Matveev ◽  
Ilya Pavlyukevich
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Random Walks ◽  
Diffusion Processes ◽  
Riemannian Metric ◽  
Finsler Manifold ◽  
Finsler Manifolds ◽  
Bounded Geometry ◽  
And Brownian Motion

AbstractWe show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.

SMOLYANOV–WEIZSÄCKER SURFACE MEASURES GENERATED BY DIFFUSIONS ON THE SET OF TRAJECTORIES IN RIEMANNIAN MANIFOLDS

2008 ◽  
Vol 11 (01) ◽  
pp. 21-31 ◽  
Author(s):  
ILYA V. TELYATNIKOV
Keyword(s):  
Brownian Motion ◽  
Euclidean Space ◽  
Diffusion Process ◽  
Riemannian Manifolds ◽  
Marginal Distribution ◽  
Diffusion Processes ◽  
Ambient Space ◽  
Time Interval ◽  
Standard Brownian Motion ◽  
Limit Surface

We consider surface measures on the set of trajectories in a smooth compact Riemannian submanifold of Euclidean space generated by diffusion processes in the ambient space. A construction of surface measures on the path space of a smooth compact Riemannian submanifold of Euclidean space was introduced by Smolyanov and Weizsäcker for the case of the standard Brownian motion. The result presented in this paper extends the result of Smolyanov and Weizsäcker to the case when we consider measures generated by diffusion processes in the ambient space with nonidentical correlation operators. For every partition of the time interval, we consider the marginal distribution of the diffusion process in the ambient space under the condition that it visits the manifold at all times of the partition, when the mesh of the partition tends to zero. We prove the existence of some limit surface measures and the equivalence of the above measures to the distribution of some diffusion process on the manifold.

The ruin problem and cover times of asymmetric random walks and Brownian motions

2000 ◽  
Vol 32 (01) ◽  
pp. 177-192 ◽  
Author(s):  
K. S. Chong ◽  
Richard Cowan ◽  
Lars Holst
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Cover Time ◽  
Brownian Motions ◽  
Brownian Motion With Drift ◽  
And Brownian Motion ◽  
Cover Times ◽  
Ruin Problem

A simple asymmetric random walk on the integers is stopped when its range is of a given length. When and where is it stopped? Analogous questions can be stated for a Brownian motion. Such problems are studied using results for the classical ruin problem, yielding results for the cover time and the range, both for asymmetric random walks and Brownian motion with drift.

scholarly journals Limit Theorems for Local and Occupation Times of Random Walks and Brownian Motion on a Spider

2017 ◽  
Vol 32 (1) ◽  
pp. 330-352
Author(s):  
Endre Csáki ◽  
Miklós Csörgő ◽  
Antónia Földes ◽  
Pál Révész
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Limit Theorems ◽  
Occupation Times ◽  
And Brownian Motion
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