The total variation flow in metric random walk spaces

Geometric And Functional Analysis, 1998

Advances in Calculus of Variations

In this paper we study the ( BV , L p ) {(\mathrm{BV},L^{p})} -decomposition, p = 1 , 2 {p=1,2} , of functions in metric random walk spaces, a general workspace that includes weighted graphs and nonlocal models used in image processing. We obtain the Euler-Lagrange equations of the corresponding variational problems and their gradient flows. In the case p = 1 {p=1} we also study the associated geometric problem and the thresholding parameters describing the behavior of its solutions.

Transactions of the American Mathematical Society, 1984

The difference Laplacian on a square lattice in Rn has been studied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many properties of the Laplacian in the continuous setting (e.g. the maximum principle, the Harnack inequality, and Cheeger's bound for the lowest eigenvalue) hold for this difference operator. The difference Laplacian governs the random walk on a graph, just as the Laplace operator governs the Brownian motion. As an application of the theory of the difference Laplacian, it is shown that the random walk on a class of graphs is transient.

Electronic Journal of Probability, 2014

We study a simple stochastic differential equation (SDE) driven by one Brownian motion on a general oriented metric graph whose solutions are stochastic flows of kernels. Under some condition, we describe the laws of all solutions. This work is a natural continuation of [16], [8] and [9] where some particular metric graphs are considered.

Potential Analysis, 2003

In this paper some isoperimetric problems are studied, particularly the extremal property of the mean exit time of the random walk from finite sets. This isoperimetric problem is inserted into the set of equivalent conditions of the diagonal upper estimate of transition probability of random walks on weighted graphs.

2002

We study diffusions, variational principles and associated boundary value problems on directed graphs with natural weightings. Using random walks and exit times, we associate to certain subgraphs (domains) a pair of sequences, each of which is invariant under the action of the automorphism group of the underlying graph. We prove that these invariants differ by an explicit combinatorial factor given

ESAIM: Control, Optimisation and Calculus of Variations

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.

Proceedings of the London Mathematical Society, 1992

We introduce an (r, /?)-net (0 < 2r < R) of a metric space M as a maximal graph whose vertices are elements in M of pairwise distance at least r such that any two vertices of distance at most R are adjacent. We show that, for a large class of metric spaces, including many Riemannian manifolds, the property of transience of a net and the property of the net carrying a non-constant harmonic function of bounded energy is independent of the choice of the net. We give a new necessary and sufficient condition for a graph with bounded degrees and satisfying an isoperimetric inequality to have no non-constant harmonic functions. For this purpose we develop equivalent analytic conditions for graphs satisfying an isoperimetric inequality. Some of these results have been discovered recently by others in more general settings, but our treatment here is specific and self-contained. We use graph transience to prove that Scherk's surface is hyperbolic, a problem posed by Osserman in 1965.

Annales de l'Institut Henri Poincare (B) Probability and Statistics, 2008

In this paper characterizations of graphs satisfying heat kernel estimates for a wide class of space-time scaling functions are given. The equivalence of the two-sided heat kernel estimate and the parabolic Harnack inequality is also shown via the equivalence of the upper (lower) heat kernel estimate to the parabolic mean value (and super mean value) inequality.

Journal of Statistical Physics, 2006

This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the volume doubling and time comparison principles. This and the other set of conditions provide the basic vwork for the study of (sub-) diffusive behavior of the random walks on weighted graphs.