相关论文: Brownian symmetrization of planar domains
We present a numerical framework for approximating the $\mu$-domain in the planar Skorokhod embedding problem PSEP, recently introduced in \cite{gross2019}. We show that under weak convergence of a sequence of probability measures…
The Skorokhod Embedding problem is well understood when the underlying process is a Brownian motion. We examine the problem when the underlying is the simple symmetric random walk and when no external randomisation is allowed. We prove that…
Start a planar Brownian motion and let it run until it hits some given barrier. We show that the barrier may be crafted so that the x coordinate at the hitting time has any prescribed centered distribution with finite variance. This…
We establish a new symmetrization procedure for the isoperimetric problem in symmetric spaces of noncompact type. This symmetrization generalizes the well known Steiner symmetrization in euclidean space. In contrast to the classical…
Under continuity and recurrence assumptions, we prove that the iteration of successive partial symmetrizations that form a time-homogeneous Markov process, converges to a symmetrization. We cover several settings, including the…
We revisit the planar Skorokhod embedding problem introduced by Gross and developed further by Boudabra-Markowsky, and we place it in a fully variational framework. For a centered probability measure $\mu$ with finite second moment, we show…
This work contributes a systematic survey and complementary insights of reflecting Brownian motion and its properties. Extension of the Skorohod problem's solution to more general cases is investigated, based on which a discussion is…
In this paper we consider a connection between the famous Skorohod embedding problem and the Shiryaev inverse problem for the first hitting time distribution of a Brownian motion: given a probability distribution, $F$, find a boundary such…
Steiner symmetrization is well known for its rounding and general convergence properties. We identify a whole family of symmetrizations sharing analogue behaviors: In fact we prove that all these symmetrizations share the same converging…
The concept of an $i$-symmetrization is introduced, which provides a convenient framework for most of the familiar symmetrization processes on convex sets. Various properties of $i$-symmetrizations are introduced and the relations between…
The invariance properties of Brownian motion are investigated and revisited within a recent Lie symmetry approach to stochastic differential equations. Some notable properties of the process can be recovered by a related integration by…
We introduce a technique to merge two biased Brownian motions into a single regular process. The outcome follows a stochastic differential equation with a constant diffusion coefficient and a non-linear drift. The emerging stochastic…
We discuss a one-parameter family of transformations which changes sets and functions continuously into their (k,n)-Steiner symmetrizations. Our construction consists of two stages. First, we employ a continuous symmetrization introduced by…
We consider the model of the Brownian plane, which is a pointed non-compact random metric space with the topology of the complex plane. The Brownian plane can be obtained as the scaling limit in distribution of the uniform infinite planar…
We consider the area of spheres centered at the distinguished point in the Brownian plane. As a function of the radius, the resulting process has continuously differentiable sample paths. Furthermore, the pair consisting of the process and…
In this paper we consider the Skorokhod embedding problem in Brownian motion. In particular, we give a solution based on the local time at zero of a variably skewed Brownian motion related to the underlying Brownian motion. Special cases of…
Consider an n-fold integrated Brownian motion. We show that a simple change in time and scale transforms it into a stationary Gaussian process. The collection of stationary processes so constructed not only constitutes an interesting family…
A new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations (SDEs). The correspondence between an algebra of weak symmetries for a given SDE and an…
We consider the Skorokhod problem in a time-varying interval. We prove existence and uniqueness for the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. We…
We establish a new spatial Markov property of the Brownian half-plane. According to this property, if one removes a hull centered at a boundary point, the remaining space equipped with an intrinsic metric is still a Brownian half-plane,…