Brownian Motion and Harmonic Functions on Manifolds of Negative Curvature

1976 ◽  
Vol 21 (1) ◽  
pp. 81-95 ◽  
Author(s):  
Yu. I. Kifer
Keyword(s):  
Brownian Motion ◽  
Harmonic Functions ◽  
Negative Curvature

scholarly journals Brownian Motion and Harmonic Analysis on Sierpinski Carpets

1999 ◽  
Vol 51 (4) ◽  
pp. 673-744 ◽  
Author(s):  
Martin T. Barlow ◽  
Richard F. Bass
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Heat Kernel ◽  
Lower Bounds ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Upper And Lower Bounds ◽  
Isotropic Diffusion ◽  
Sierpinski Carpets

AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

scholarly journals A diffusion model for Bookstein triangle shape

1996 ◽  
Vol 28 (02) ◽  
pp. 334-335
Author(s):  
Wilfrid S. Kendall
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Hyperbolic Plane ◽  
Negative Curvature ◽  
General Linear Group ◽  
Shape Space ◽  
Shape Theory ◽  
Independent Brownian Motion ◽  
Invertible Matrices

This reports on work in progress, developing a dynamical context for Bookstein's shape theory. It shows how Bookstein's shape space for triangles arises when the landmarks are moved around by a particular Brownian motion on the general linear group of (2 × 2) invertible matrices. Indeed, suppose that the random process G(t) ∈ GL(2, ℝ) solves the Stratonovich stochastic differential equation dsG = (dsB)G for a Brownian matrix B (independent Brownian motion entries). If {x1 x2, x3 } is a fixed (non-degenerate) triple of planar points then Xi(t) = G(t)xi ; determines a triple {X1 X2, X3 } whose shape performs a diffusion which can be shown to be Brownian motion on the hyperbolic plane of negative curvature − 2.

scholarly journals Harmonic functions on Walsh’s Brownian motion

2014 ◽  
Vol 124 (6) ◽  
pp. 2228-2248 ◽  
Author(s):  
Patrick J. Fitzsimmons ◽  
Kristin E. Kuter
Keyword(s):  
Brownian Motion ◽  
Harmonic Functions

scholarly journals Brownian Motion and Harmonic Functions on Sol(p,q)†

2011 ◽  
Vol 2012 (22) ◽  
pp. 5182-5218 ◽  
Author(s):  
Sara Brofferio ◽  
Maura Salvatori ◽  
Wolfgang Woess
Keyword(s):  
Brownian Motion ◽  
Harmonic Functions

scholarly journals Brownian motion on stationary random manifolds

2016 ◽  
Vol 16 (02) ◽  
pp. 1660001
Author(s):  
Pablo Lessa
Keyword(s):  
Brownian Motion ◽  
Harmonic Functions ◽  
Dimensional Space ◽  
Volume Growth ◽  
Upper And Lower Bounds ◽  
Entropy Theory ◽  
Liouville Property ◽  
Average Volume ◽  
Infinite Dimensional ◽  
Bounded Harmonic Functions

We introduce the notion of a stationary random manifold and develop the basic entropy theory for it. Examples include manifolds admitting a compact quotient under isometries and generic leaves of a compact foliation. We prove that the entropy of an ergodic stationary random manifold is zero if and only if the manifold satisfies the Liouville property almost surely, and is positive if and only if it admits an infinite dimensional space of bounded harmonic functions almost surely. Upper and lower bounds for the entropy are provided in terms of the linear drift of Brownian motion and average volume growth of the manifold. Other almost sure properties of these random manifolds are also studied.

Sign in / Sign up

Export Citation Format

Share Document