Related papers: On the planar Brownian Green's function for stoppi…
A method is given of deriving the distribution of planar Brownian motion evaluated at certain stopping times using analytic functions. This method relies upon a generalization of the standard conformal invariance of harmonic measure. A…
We prove a number of results relating exit times of planar Brownian with the geometric properties of the domains in question. Included are proofs of the conformal invariance of moduli of rectangles and annuli using Brownian motion;…
It is well known that Brownian motion enjoys several distributional invariances such as the scaling property and the time reversal. In this paper, we prove another invariance of Brownian motion that is compatible with the time reversal. The…
We construct Green's functions for divergence form, second order parabolic systems in non-smooth time-varying domains whose boundaries are locally represented as graph of functions that are Lipschitz continuous in the spatial variables and…
The conformal invariance of Brownian motion is used to give a short proof of the Open Mapping Theorem for analytic functions.
We study semi-martingale obliquely reflected Brownian motion with drift in the first quadrant of the plane in the transient case. Our main result determines a general explicit integral expression for the moment generating function of…
Given a random time, we characterize the set of martingales for which the stopping theorems still hold. We also investigate how the stopping theorems are modified when we consider arbitrary random times. To this end, we introduce some…
Given a planar domain $D$, the harmonic measure distribution function $h_D(r)$, with base point $z$, is the harmonic measure with pole at $z$ of the parts of the boundary which are within a distance $r$ of $z$. Equivalently it is the…
In this note, we prove a version of the Phragmen-Lindelof principle using probabilistic techniques. In particular, we will show that if the p-th moment of the exit time of Brownian motion from a planar domain is finite, then an analytic…
We consider in the plane the problem of reconstructing a domain from the normal derivative of its Green's function with pole at a fixed point in the domain. By means of the theory of conformal mappings, we obtain existence, uniqueness,…
We study several quantities associated to the Green's function of a multiply connected domain in the complex plane. Among them are some intrinsic properties such as geodesics, curvature, and $L^2$-cohomology of the capacity metric and…
This paper is the sequel to another with the same name (Buttigieg et al., Comput. Methods Funct. Theory, 2023), and is concerned with results of the same type. We deduce a result on the moments of the exit time of Brownian motion from…
In this paper we are interested in obtaining the exact expression and the study of the constant sign of the Green's function related to a second order perturbed periodic problem coupled with integral boundary conditions at the extremes of…
In this paper we will deduce several properties of the Green's functions related to the Hill's equation coupled to various boundary value conditions. In particular, the idea is to study the Green's functions of the second order differential…
This paper is the first part of our survey on various results about the distribution of exponential type Brownian functionals defined as an integral over time of geometric Brownian motion. Several related topics are also mentioned.
It has been discovered previously that the topological order parameter could be identified from the topological data of the Green's function, namely the (generalized) TKNN invariant in general dimensions, for both non-interacting and…
We establish a variational formula for the exponential decay rate of the Green function of Brownian motion evolving in a random stationary and ergodic nonnegative potential. Such a variational formula is established by Schroeder in 'Green's…
Motivated by a common Mathematical Finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of an associated Brownian motion. We prove a conjecture…
This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional…
The classical Green's function associated to a simply connected domain in the complex plane is easily expressed in terms of a Riemann mapping function. The purpose of this paper is to express the Green's function of a finitely connected…