Conditioned Brownian motion in planar domains

Summary We give an upper bound for the Green functions of conditioned Brownian motion in planar domains. A corollary is the conditional gauge theorem in bounded planar domains.Supported in part by NSF grant DMS-9100244 and an AMS Centennial Fellowship     

Lifetime of conditioned Brownian motion in Lipschitz domains

Summary If D- ^d , d3, is bounded and has Lipschitz boundary then the expected lifetime of any Brownian h-path process in D is finite.     

Distoriton of area and conditioned Brownian motion

Summary In a simply connected planar domainD the expected lifetime of conditioned Brownian motion may be viewed as a function on the set of hyperbolic geodesics for the domain. We show that each hyperbolic geodesic induces a decomposition ofD into disjoint subregions and that the subregions are obtained in a natural way using Euclidean geometric quantities relating toD. The lifetime associated with on each _j is then shown to be bounded by the product of the diameter of the smallest ball containing _j and the diameter of the largest ball in _j . Because this quantity is never larger than, and in general is much smaller than, the area of the largest ball in _j it leads to finite lifetime estimates in a variety of domains of infinite area.Research of the first author was supported in part by NSF Grant DMS-9100811Research of the second author was supported in part by NSF Grant DMS-9105407     

On the lifetime of a conditioned Brownian motion in the ball

Consider the Brownian motion conditioned to start in x, to converge to y, with , and to be killed at the boundary ∂Ω. Here Ω is a bounded domain in Rn. For which x and y is the lifetime of this Brownian motion maximal? One would guess for x and y being opposite boundary points and we will show that this holds true for balls in Rn. As a consequence we find the best constant for the positivity preserving property of some elliptic systems and an identity between this constant and a sum of inverse Dirichlet eigenvalues.     

On the lifetime of a conditioned Brownian motion on a fish bowl

It has been conjectured that for 2-dimensional planar domains the lifetime of a conditioned Brownian motion is maximized for two boundary points. The present example shows that such a claim is in general not true on manifolds. Received: 3 April 2007     

Brownian motion in twisted domains

The tail behavior of a Brownian motion's exit time from an unbounded domain depends upon the growth of the ``inner radius' of the domain. In this article we quantify this idea by introducing the notion of a twisted domain in the plane. Roughly speaking, such a domain is generated by a planar curve as follows. As a traveler proceeds out along the curve, the boundary curves of the domain are obtained by moving out units along the unit normal to the curve when the traveler is units away from the origin. The function is called the growth radius. Such domains can be highly nonconvex and asymmetric. We give a detailed account of the case , . When , a twisted domain can reasonably be interpreted as a ``twisted cone.'

     

Brownian motion in infinite dimensional manifolds

We discuss the extension to infinite dimensional Riemannian—Wiener manifolds of the transport approximation to Brownian motion, which was formulated by M. Pinsky for finite dimensional manifolds. A global representation is given for the Laplace—Beltrami operator in terms of the Riemannian spray and a homogenizing operator based upon the central hitting measure of the surface of the unit ball with respect to the Brownian motion on the model space.Research supported by NSF grant MCS8202319.     

Sojourns and future infima of planar Brownian motion

Summary Let {W(t); 0t1} be a two-dimensional Wiener process starting from 0. We are interested in the almost sure asymptotic behaviour, asr tends to 0, of the processesX(r) andY(r), whereX(r) denotes the total time spent byW in the ball centered at 0 with radiusr andY(r) the distance between 0 and the curve {W(t);rt1}. While a characterization of the lower functions ofY was previously established by Spitzer [S], we characterize via integral tests its upper functions as well as the upper and lower functions ofX.     

On the conditioned exit measures of super Brownian motion

In this paper we present a martingale related to the exit measures of super Brownian motion. By changing measure with this martingale in the canonical way we have a new process associated with the conditioned exit measure. This measure is shown to be identical to a measure generated by a non-homogeneous branching particle system with immigration of mass. An application is given to the problem of conditioning the exit measure to hit a number of specified points on the boundary of a domain. The results are similar in flavor to the “immortal particle” picture of conditioned super Brownian motion but more general, as the change of measure is given by a martingale which need not arise from a single harmonic function. Received: 27 August 1998 / Revised version: 8 January 1999     

Hilbert-valued self-intersection local times for planar Brownian motion

Dynkin's construction for self-intersection local time of a planar Wiener process is extended to Hilbert-valued weights. In Dynkin's construction, the weight is bounded and measurable. Since the weight function describes the properties of the medium in which the Brownian motion moves, relative to the external medium's properties, the weight function can be random and unbounded. In this article, we discuss the possibility to consider Hilbert-space-valued weights. It appears that the existence of Hilbert-valued Dynkin-renormalized self-intersection local time is equivalent to the embedding of the values of Hilbert-valued weight into a Hilbert–Schmidt brick. Using Dorogovtsev's sufficient condition for the embedding of compact sets into a Hilbert–Schmidt brick in terms of an isonormal process, we prove the existence of Hilbert-valued Dynkin-renormalized self-intersection local time. Also using Dynkin's construction we construct the self-intersection local time for the deterministic image of the planar Wiener process.     

Pathwise uniqueness for reflecting Brownian motion in Euclidean domains

For a bounded C ^1,α domain in ℝ ^d we show that there exists a strong solution to the multidimensional Skorokhod equation and that weak uniqueness holds for this equation. These results imply that pathwise uniqueness and strong uniqueness hold for the Skorokhod equation. Received: 3 February 1999 / Revised version: 2 September 1999 /?Published online: 11 April 2000     

Tanaka's formula for multiple intersections of planar Brownian motion

We establish a Tanaka-like formula relating the local times of r and r + 1 fold self-intersections of a Brownian path in the plane.     

On the variances of occupation times of conditioned Brownian motion

We extend some bounds on the variance of the lifetime of two--dimensional Brownian motion, conditioned to exit a planar domain at a given point, to certain domains in higher dimensions. We also give a short ``analytic' proof of some existing results.

     

Brownian motion characterization of some Besov-Lipschitz spaces on domains

We characterize the Besov-Lipschitz spaces with zero boundary conditions on bounded smooth domains. We prove that the appropriate first and second difference norms are equivalent to the norm given in terms of the transition kernel of the Brownian motion killed upon exit from the domain.