A simple random walk and a Brownian motion are considered on a spider that is a collection of half lines (we call them legs) joined in the origin. We give a strong approximation of these two objects and their local times. For fixed number of legs we establish limit theorems on local and occupation times in n steps.
We consider the Brownian ``spider process'', also known as Walsh Brownian motion, first introduced in the epilogue of Walsh 1978. The paper provides the best constant $C_n$ for the inequality $$ E D_τ\leq C_n \sqrt{E τ},$$ where $τ$ is the class of all adapted and integrable stopping times and $D$ denotes the diameter of the spider process measured in terms of the British rail metric. The proof relies on the explicit identification of the value function for the associated optimal stopping problem.
Sticky Brownian motion on the real line can be obtained as a weak solution of a system of stochastic differential equations. We find the conditional distribution of the process given the driving Brownian motion, both at an independent exponential time and at a fixed time t>0. As a classical problem, we find the distribution of the occupation times of a half-line, and at 0, which is the sticky point for the process.
A diffusion spider is a strong Markov process with continuous paths taking values on a graph with one vertex and a finite number of edges (of infinite length). An example is Walsh's Brownian spider where the process on each edge behaves as Brownian motion. We calculate the density of the resolvent kernel in terms of the characteristics of the underlying diffusion. Excessive functions are studied via the Martin boundary theory. The main result is an explicit expression for the representing measure of a given excessive function. These results are used to solve optimal stopping problems for diffu…
Corrections for "Occupation and local times for skew Brownian motion with applications to dispersion across an interface"
我们正在进行更正，并感谢同事指出我们最近发表在Annals of Applied Probability（2011）21（1）183-214上的题为“Occupation and local times for skew布朗运动with applications to dispersion across an interface”的论文中的错误。具体的更正是：1。不需要定理1.3中将$γ$限制为非负值。但只有当$γ$为非负时才有概率解释。2.说明推论3.3中的正确公式，因为它们是原始公式中的计算错误。我们感谢Pierre Etoiré和Miguel Martinez指出这些错误。
Stochastic processes time-changed by an inverse subordinator have been suggested as a way to model the price of assets in illiquid markets, where the jumps of the subordinator correspond to periods of time where one is unable to sell an asset. We develop an excursion theory for time-changed reflected Brownian motion and use this to express the price of certain European options with Parisian barrier condition in terms of solutions of a time-fractional PDE. We provide a general description of the occupation measures of time-changed processes and use this to prove a Ray-Knight theorem for the occ…
This note concerns distributions of Skew Brownian motion with dry friction and its occupation time. These distributions were obtained in [2] by using the Laplace transform and joint characteristic functions. We provide an alternative approach to deriving these distributions. Our approach is based on using the results for Skew Brownian motion obtained in [3].
Let $D_N$ be the set of points around which a planar Brownian motion winds at least $N$ times. We prove that the random measure on the plane with density $2 πN 1_{D_N}$ with respect to the Lebesgue measure converges almost surely weakly, as $N$ tends to infinity, towards the occupation measure of the Brownian motion.
Starting with a Brownian motion, we define and study a novel diffusion process by combining stickiness and oscillation properties. The associated stochastic differential equation, resolvent and semigroup are provided. Also the trivariate density of position, local time and occupation time of this diffusion is obtained explicitly. Furthermore, we give a construction of two Brownian motions with drift and scaling whose difference is an oscillating sticky Brownian motion, up to a multiplicative constant.
We show that the Brydges-Fröhlich-Spencer-Dynkin and the Le Jan's isomorphisms between the Gaussian free fields and the occupation times of symmetric Markov processes generalize to the $β$-Dyson's Brownian motion. For $β\in\{1,2,4\}$ this is a consequence of the Gaussian case, however the relation holds for general $β$. We further raise the question whether there is an analogue of $β$-Dyson's Brownian motion on general electrical networks, interpolating and extrapolating the fields of eigenvalues in matrix-valued Gaussian free fields. In the case $n=2$ we give a simple construction.
The binary contact path process (BCPP) introduced in Griffeath (1983) describes the spread of an epidemic on a graph and is an auxiliary model in the study of improving upper bounds of the critical value of the contact process. In this paper, we are concerned with the central limit theorem of the occupation time of a normalized version of the BCPP (NBCPP) on a lattice. We show that the centred occupation time process of the NBCPP converges in finite dimensional distributions to a Brownian motion when the dimension of the lattice and the infection rate of the model are sufficiently large and th…
The strong $L^2$-approximation of occupation time functionals is studied with respect to discrete observations of a $d$-dimensional càdlàg process. Upper bounds on the error are obtained under weak assumptions, generalizing previous results in the literature considerably. The approach relies on regularity for the marginals of the process and applies also to non-Markovian processes, such as fractional Brownian motion. The results are used to approximate occupation times and local times. For Brownian motion, the upper bounds are shown to be sharp up to a log-factor.
We condition a Brownian motion with arbitrary starting point $y \in \mathbb{R}$ on spending at most $1$ time unit below $0$ and provide an explicit description of the resulting process. In particular, we provide explicit formulas for the distributions of its last zero $g=g^y$ and of its occupation time $Γ=Γ^y$ below $0$ as functions of $y$. This generalizes a result of Benjamini and Berestycki from 2011, which covers the special case $y=0$. Additionally, we study the behavior of the distributions of $g^y$ and $Γ^y$, respectively, for $y \to \pm\infty$.
On the sojourn time of a Generalized Brownian meander
本文研究了以$ u $为起始点的漂移布朗运动在满足$ \min_{ 0\leq z \leq l} B(z)> v $，有$ u > v $的条件下，在正半直线上的逗留时间.这一过程是一个漂移的布朗曲流到时间$ l $，然后演变为一个自由的布朗运动。我们还考虑了一类桥式过程的逗留时间，在此我们增加了一个附加条件，使其在时间间隔的末端返回到初始水平。我们分析了占据泛函$ u \downarrow v $的弱极限。我们得到明确的分布结果时，障碍被放置在零级，也在特殊情况下，漂移为零。
