Two-dimensional Brownian motion and harmonic functions

Brownian Motion and Harmonic Functions

The Lie group Sol(p, q) is the semidirect product induced by the action of R on R which is given by (x, y) 7→ (ex, e−qzy), z ∈ R. Viewing Sol(p, q) as a 3-dimensional manifold, it carries a natural Riemannian metric and Laplace-Beltrami operator. We add a linear drift term in the z-variable to the latter, and study the associated Brownian motion with drift. We derive a central limit theorem and...

Harmonic functions of subordinate killed Brownian motion

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D: We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded L...

Harmonic Functions and Brownian Motion in Several Dimensions

Definition 1. A standard d−dimensional Brownian motion is an Rd−valued continuous-time stochastic process {Wt}t≥0 (i.e., a family of d−dimensional random vectors Wt indexed by the set of nonnegative real numbers t) with the following properties. (A)’ W0 = 0. (B)’ With probability 1, the function t→Wt is continuous in t. (C)’ The process {Wt}t≥0 has stationary, independent increments. (D)’ The i...

Brownian motion and Harmonic functions on Sol(p,q)

The Lie group Sol(p, q) is the semidirect product induced by the action of R on R which is given by (x, y) 7→ (ex, e−qzy), z ∈ R. Viewing Sol(p, q) as a 3-dimensional manifold, it carries a natural Riemannian metric and Laplace-Beltrami operator. We add a linear drift term in the z-variable to the latter, and study the associated Brownian motion with drift. We derive a central limit theorem and...

Brownian Motion, Harmonic Functions and Hyperbolicity for Euclidean Complexes