Athanasios Batakis, Viet Hung Nguyen
We show that the dimension of the exit distribution of planar partially reflected Brownian motion can be arbitrarily close to 2.
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For each prime $p$, a diffusion constant together with a positive exponent specify a Vladimirov operator and an associated $p$-adic diffusion equation. The fundamental solution of this pseudo-differential equation gives rise to a measure on the Skorokhod space of $p$-adic valued paths that is concentrated on the paths originating at the origin. We calculate the first exit probabilities of paths from balls and estimate these probabilities for the brownian bridges.
We supplement a very recent paper of G. Markowsky concerned with the expected exit times of Brownian motion from planar domains. Besides the use of conformal mapping, we apply results from potential theory. We treat the case of a wedge-shaped region exactly, subsuming an example of Markowsky. A number of other results are presented, including for a half disc, $n$-grams, and a variety of other regions.
We introduce a class of interesting stochastic processes based on Brownian-time processes. These are obtained by taking Markov processes and replacing the time parameter with the modulus of Brownian motion. They generalize the iterated Brownian motion (IBM) of Burdzy and the Markov snake of Le Gall, and they introduce new interesting examples. After defining Brownian-time processes, we relate them to fourth order parabolic PDEs. We then study their exit problem as they exit nice domains in $\Rd$, and connect it to elliptic PDEs. We show that these processes have the peculiar property that they…
We consider the tail probabilities for Brownian exit time from a class of perturbed multi-strips in Euclidean plane. Under some assumptions we prove that the long stays in a perturbed multi-strip are more likely than those in a strip of the same width. This effect is related to the existence of the trapped modes in waveguides.
We study the distribution of the exit place of iterated Brownian motion in a cone, obtaining information about the chance of the exit place having large magnitude. Along the way, we determine the joint distribution of the exit time and exit place of Brownian motion in a cone. This yields information on large values of the exit place (harmonic measure) for Brownian motion. The harmonic measure for cones has been studied by many authors for many years. Our results are sharper than any previously obtained
With the help of the Gauss-Laplace transform for the exit time from a cone of planar Brownian motion, we obtain some infinite divisibility properties for the reciprocal of this exit time.
We study the first exit time of a multi-dimensional fractional Brownian motion from unbounded domains. In particular, we are interested in the upper tail of the corresponding distribution when the domain is parabola-shaped.
This paper is devoted to studying the properties of the exit times of stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs). In particular, we prove that the exit times of $G$-SDEs has the quasi-continuity property. As an application, we give a probabilistic representation for a large class of fully nonlinear elliptic equations with Dirichlet boundary.
We consider the one-sided exit problem for fractional Brownian motion (FBM), which is equivalent to the question of the distribution of the lower tail of the maximum of FBM on the unit interval. We improve the bounds given by Molchan (1999) and shed some light on the relation to the quantity $I$ studied there.
Exit Times for Brownian Motion and Location Detection
该研究针对二维等多种平面区域，利用布朗运动样本路径的首出时间矩，结合泊松层级工具，实现了流形上的位置检测，不同区域所需矩序列长度不同。
We provide a new methodology to simulate the first exit times of a vector of Brownian motions from an orthant. This new approach can be used to simulate the first exit times of dimension higher than two. When at least one Brownian motion has non-zero drift, the joint density function of the first exit times in N dimensions needs to be known, or approximated. However, when the drifts are all zero, a simpler simulation algorithm is obtained without using the joint density function.
For a Brownian bridge from $0$ to $y$ we prove that the mean of the first exit time from interval $(-h,h), \,\, h>0,$ behaves as $O(h^2)$ when $h \downarrow 0.$ Similar behavior is seen to hold also for the 3-dimensional Bessel bridge. For Brownian bridge and 3-dimensional Bessel bridge this mean of the first exit time has a puzzling representation in terms of the Kolmogorov distribution. The result regarding the Brownian bridge is applied to prove in detail an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.
Estimates for exit time from an interval of length 2r before a prescribed time T are derived for solutions of a class of stochastic partial differential equations used to characterize two population models: super-Brownian motion and Fleming-Viot Process. These types of estimates are then derived for the two population models. The corresponding large deviation results are also applied for the acquired bounds.
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; similarly probabilistic proofs of some recent results of Karafyllia on harmonic measure on starlike domains; examples of domains and their complements which are simultaneously large when measured by the moments of exit time of Brownian motion, and examples of domains and their complements which are simultaneously small; and proofs of several identities involving the…
In this paper we address the question of finding the point which maximizes the $p$-th moment of the exit time of planar Brownian motion from a given domain. We present a geometrical method of excluding parts of the domain from consideration which makes use of a coupling argument and the conformal invariance of Brownian motion. In many cases the maximizing point can be localized to a relatively small region. Several illustrative examples are presented.
We consider the system of one-sided reflected Brownian motions which is in variational duality with Brownian last passage percolation. We show that it has integrable transition probabilities, expressed in terms of Hermite polynomials and hitting times of exponential random walks, and that it converges in the 1:2:3 scaling limit to the KPZ fixed point, the scaling invariant Markov process defined in [MQR17] and believed to govern the long time large scale fluctuations for all models in the KPZ universality class. Brownian last passage percolation was shown recently in [DOV18] to converge to the…
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 …
Let $U$ be a domain, convex in $x$ and symmetric about the y-axis, which is contained in a centered and oriented rectangle $R$. \linebreak If $τ_A$ is the first exit time of Brownian motion from $A$ and $A^+=A\cap \{(x,y):x>0\}$, it is proved that $P^z(τ_{U^+}>s\mid τ_{R^+}>t)\leq P^z(τ_{U}>s\mid τ_{R}>t)$ for every $s,t>0$ and every $z\in U^+$.
We access the edge of Gaussian beta ensembles with one spike by analyzing high powers of the associated tridiagonal matrix models. In the classical cases beta=1, 2, 4, this corresponds to studying the fluctuations of the largest eigenvalues of additive rank one perturbations of the GOE/GUE/GSE random matrices. In the infinite-dimensional limit, we arrive at a one-parameter family of random Feynman-Kac type semigroups, which features the stochastic Airy semigroup of Gorin and Shkolnikov [13] as an extreme case. Our analysis also provides Feynman-Kac formulas for the spiked stochastic Airy opera…
(i) Uncountably many synchronized reflected Brownian motions can hit the boundary of a $C^2$ domain at the same time. (ii) Measures associated to local times of two synchronized reflected Brownian motions are mutually singular until the time when the normal vectors at the reflection locations become identical. (iii) Mirror coupled reflected Brownian motions can simultaneously hit opposite sides of a wedge at different distances from the origin.
In this paper we prove matching upper and lower bounds for the transition density function of the subordinate reflected Brownian motion on fractals.
We use a renormalization of the total mass of the exit measure from the complement of a small ball centered at $x\in \mathbb{R}^d$ for $d\leq 3$ to give a new construction of the total local time $L^x$ of super-Brownian motion at $x$. In \cite{Hong20} a more singular renormalization of the total mass of the exit measure concentrating on $x$, where the exit measure is positive but unusually small, is used to build a boundary local time supported on the topological boundary of the range of super-Brownian motion. Our exit measure construction of $L^x$ motivates this renormalization. We give an im…
Stochastic Domination of Exit Times for Random Walks and Brownian Motion with Drift
本文利用Girsanov变换证明对称区间上有偏随机游走和带漂移布朗运动的退出时间关于漂移参数随机单调，并为Peköz和Righter的2024年结果提供新证明。
We prove that a stochastic flow of reflected Brownian motions in a smooth multidimensional domain is differentiable with respect to its initial position. The derivative is a linear map represented by a multiplicative functional for reflected Brownian motion. The method of proof is based on excursion theory and analysis of the deterministic Skorokhod equation.
This paper presents some results on the expected exit time of Brownian motion from simply connected domains in $\CC$. We indicate a way in which Brownian motion sees the identity function and the Koebe function as the smallest and largest analytic functions, respectively, in the Schlicht class. We also give a sharpening of a result of McConnell's concerning the moments of exit times of Schlicht domains. We then show how a simple formula for expected exit time can be applied in a series of examples. Included in the examples given are the expected exit times from given points of a cardioid and r…
We obtain some integrability properties and some limit theorems for the exit time from a cone of a planar Brownian motion, and we check that our computations are correct via Bougerol's identity.
The fundamental solution of a pseudo-differential equation for functions defined on the $d$-fold product of the $p$-adic numbers, $\mathbb{Q}_p$, induces an analogue of the Wiener process in $\mathbb{Q}_p^d$. As in the real setting, the components are $1$-dimensional $p$-adic Brownian motions with the same diffusion constant and exponent as the original process. Asymptotic analysis of the conditional probabilities shows that the vector components are dependent for all time. Exit time probabilities for the higher dimensional processes reveal a concrete effect of the component dependency.
We introduce several martingale changes of measure of the law of the exit measure of super Brownian motion. These changes of measure include and generalize one arising by conditioning the exit measures to charge a point on the boun dary of a 2-dimensional domain. In the case we discuss this is a non-degenerate conditioning. We give characterizations of the new processes in terms of "immortal particle" branching processes with immigration of mass, and give application s to the study of solutions to Lu = cu^2 in D. The representations are related to those in an earlier paper, which treated the c…
Transport phenomena are ubiquitous in nature and known to be important for various scientific domains. Examples can be found in physics, electrochemistry, heterogeneous catalysis, physiology, etc. To obtain new information about diffusive or Laplacian transport towards a semi-permeable or resistive interface, one can study the random trajectories of diffusing particles modeled, in a first approximation, by the partially reflected Brownian motion. This stochastic process turns out to be a convenient mathematical foundation for discrete, semi-continuous and continuous theoretical descriptions of…
In Gambler's Ruin when both players start with the same amount of money, we show the playing time stochastically increases when the games are made more fair. We give two different arguments for this fact that extend results from \cite{Pek2021}. We then use this to show that the exit time from a symmetric interval for Brownian motion with drift stochastically increases as the drift moves closer to zero; this result is not easily obtainable from available explicit formulas for the density.
In this paper we introduce a new method for the simulation of the exit time and position of a $δ$-dimensional Brownian motion from a domain. The main interest of our method is that it avoids splitting time schemes as well as inversion of complicated series. The idea is to use the connexion between the $δ$-dimensional Bessel process and the $δ$-dimensional Brownian motion thanks to an explicit Bessel hitting time distribution associated with a particular curved boundary. This allows to build a fast and accurate numerical scheme for approximating the hitting time. Numerical comparisons with exis…
We investigate the tail distribution of the first exit time of Brownian motion with drift from a cone and find its exact asymptotics for a large class of cones. Our results show in particular that its exponential decreasing rate is a function of the distance between the drift and the cone, whereas the polynomial part in the asymptotics depends on the position of the drift with respect to the cone and its polar cone, and reflects the local geometry of the cone at the point where the drift is orthogonally projected.
A comb domain is defined to be the entire complex plain with a collection of vertical slits, symmetric over the real axis, removed. In this paper, we consider the question of determining whether the exit time of planar Brownian motion from such a domain has finite $p$-th moment. This question has been addressed before in relation to starlike domains, but these previous results do not apply to comb domains. Our main result is a sufficient condition on the location of the slits which ensures that the $p$-th moment of the exit time is finite. Several auxiliary results are also presented, includin…
In this paper, we study the reflected backward stochastic differential equation driven by G-Brownian motion (reflected G-BSDE for short) with an upper obstacle. The existence is proved by approximation via penalization. By using a variant comparison theorem, we show that the solution we constructed is the largest one.
In this paper we consider a (reflected) Brownian motion with broken drift hitting a random boundary. Some dedicated calculations allow us to obtain the formula on the joint Laplace transform of the hitting time and hitting position. These develop the research on first rendezvous times of (reflected) Brownian motion and compound Poisson-type processes by Perry et al. (2004).
In this note, we study the asymptotical frontier behavior of a branching reflected Brownian motion. There is essentially no difference in maximal displacement between a branching Brownian motion and its reflected counterpart. We provide two proofs of this fact, one via a soft argument on the dependance of two-sided extremal particles in a branching Brownian motion and the other based on direct computations as in Roberts. The asymptotics of minimal displacement is also given.
Let $T^D$ denote the first exit time of a planar Brownian motion from a domain $D$. Given two simply connected planar domains $U,W \neq \SC$ containing $0$, we investigate the cases in which we are more likely to have fast exits (meaning for instance ${\bf P}(T^U<t) > {\bf P}(T^W<t)$ for $t$ small) from $U$ than from $W$, or long stays (meaning ${\bf P}(T^U>t) > {\bf P}(T^W>t)$ for $t$ large). We prove several results on these questions. In particular, we show that the primary factor in the probability of fast exits is the proximity of the boundary to the origin, while for long stays an import…
A limit theorem for certain Feller semigroups, and the distribution of the local time for Brownian motion when it exits an interval
本文从半群理论视角研究一维 Laplace 算子的局部奇异摄动，证明其收敛至零点具 Robin 型边界条件的热半群，并借此提出计算 Brownian motion 离开区间时局部时分布的方法。
In this paper, we prove that there exists at least one solution for the reflected forward-backward stochastic differential equation driven by G-Brownian motion satisfying the obstacle constraint with monotone coefficients.
This paper studies the regularity properties of the density of the exit measure for super-Brownian motion with (1+β)-stable branching mechanism. It establishes the continuity of the density in dimension d=2 and the unboundedness of the density in all other dimensions where the density exists. An alternative description of the exit measure and its density is also given via a stochastic integral representation. Results are applied to the probabilistic representation of nonnegative solutions of the partial differential equation Δu=u^{1+β}.
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 function on that domain is either bounded by its supremum on the boundary or else goes to infinity along some sequence more rapidly than $e^{|z|^{2p}}$. We also present a method for constructing domains whose exit time has finite p-th moment, thereby giving a general Phragmen-Lindelof principle for spiral-like and star-like domains, and in the process giving a …
The perturbed GUE corners ensemble is the joint distribution of eigenvalues of all principal submatrices of a matrix $G+\mathrm{diag}(\mathbf{a})$, where $G$ is the random matrix from the Gaussian Unitary Ensemble (GUE), and $\mathrm{diag}(\mathbf{a})$ is a fixed diagonal matrix. We introduce Markov transitions based on exponential jumps of eigenvalues, and show that their successive application is equivalent in distribution to a deterministic shift of the matrix. This result also leads to a new distributional symmetry for a family of reflected Brownian motions with drifts coming from an arith…
We study the total mass of a d-dimensional super-Brownian motion as it first exits an increasing sequence of balls. The process of the total mass is a time-inhomogeneous continuous-state branching process, where the increasing radii of the balls are taken as the time parameter. We are able to characterise its time-dependent branching mechanism and show that it converges, as time goes to infinity, towards the branching mechanism of the total mass of a one-dimensional super-Brownian motion as it first crosses above an increasing sequence of levels. Our results allow us to identify the compact su…
We prove that a sequence of semi-discrete approximations converges to a multiplicative functional for reflected Brownian motion, which intuitively represents the Lyapunov exponent for the corresponding stochastic flow. The method of proof is based on a study of the deterministic version of the problem and the excursion theory.
We study a class of reflected McKean-Vlasov diffusions over a convex domain with self-stabilizing coefficients. This includes coefficients that do not satisfy the classical Wasserstein Lipschitz condition. Further, the process is constrained to a (not necessarily bounded) convex domain by a local time on the boundary. These equations include the subclass of reflected self-stabilizing diffusions that drift towards their mean via a convolution of the solution law with a stabilizing potential. Firstly, we establish existence and uniqueness results for this class and address the propagation of cha…
In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion (RGBSDE for short). The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected GBSDEs, we apply a "martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization.
Space-time transport of Brownian exit laws
本文系统研究p-布朗收敛，将其纳入最优输运理论框架，并得到位似域情形下的渐近行为。
Reflected Stochastic Differential Equations Driven by G-Brownian Motion with Nonlinear Constraints
本文研究由G-布朗运动驱动的带有两个非线性约束的反射随机微分方程，通过非线性约束的Skorokhod问题研究了双反射G-布朗运动，并通过先验估计和皮卡迭代法证明了反射G-SDE的唯一性和存在性，给出了解与约束过程的比较定理。
The purpose of this note is to give details for an argument of Sullivan to construct eigenfunctions of the Laplacian on a Riemannian manifold using exit times of Brownian motion \cite{sullivanpos}. Let $X$ be a complete, simply connected Riemannian manifold of pinched negative sectional curvature. Let $λ_1 = λ_1(X) < 0$ be the supremum of the spectrum of the Laplacian on $L^2(X)$, and let $D \subset X$ be a bounded domain in $X$ with smooth boundary. Let $(B_t)_{t \geq 0}$ be Brownian motion on $X$ and let $τ= τ_D$ be the first exit time of Brownian motion from $D$. For each $λ\in \mathbb{C}$ …
We extend earlier results on conditioning of super-Brownian motion to general branching rules. We obtain representations of the conditioned process, both as an $h$-transform, and as an unconditioned superprocess with immigration along a branching tree. Unlike the finite-variance branching setting, these trees are no longer binary, and strictly positive mass can be created at branch points. This construction is singular in the case of stable branching. We analyze this singularity first by approaching the stable branching function via analytic approximations. In this context the singularity of t…
In this paper, we first review the penalization method for solving deterministic Skorokhod problems in non-convex domains and establish estimates for problems with $α$-Hölder continuous functions. With the help of these results obtained previously for deterministic problems, we pathwisely define the reflected $G$-Brownian motion and prove its existence and uniqueness in a Banach space. Finally, multi-dimensional reflected stochastic differential equations driven by $G$-Brownian motion are investigated via a fixed-point argument.
Prompted by an example arising in critical percolation, we study some reflected Brownian motions in symmetric planar domains and show that they are intertwined with one-dimensional diffusions. In the case of a wedge, the reflected Brownian motion is intertwined with the 3-dimensional Bessel process. This implies some simple hitting distributions and sheds some light on the formula proposed by Watts for double-crossing probabilities in critical percolation.
A collection of results relating the geometry of plane domains and the exit time of planar Brownian motion, II
该续作研究平面区域几何与 Brown 运动离开时间矩的关系，分析虚线边界和周期铺砌区域，并构造类楔形区域以解答相关推测。
Expected perimeter and area of the convex hull of planar Brownian motion stopped upon exiting the unit disk
研究平面布朗运动在退出单位盘时凸包的周长期望，通过计算最大水平位移的期望值，利用截断盘的调和测度得出精确表达式，并讨论凸包面积期望的非平凡界及计算难度。
In this paper, we consider the reflected backward stochastic differential equations driven by G-Brownian motion (reflected G-BSDEs) whose coefficients satisfy the beta-order Mao's condition. The uniqueness is obtained by some a priori estimates and the existence can be proved by two different methods. The first one is Picard iteration and the second one is approximation via penalization. The latter construction is useful to get the comparison theorem.
Let $B_H(\cdot)$ be a fractional Brownian motion with Hurst parameter $H\in(0,1]$. Motivated by applications to maximal inequalities for fractional Brownian motion, in this note we derive bounds for K_T(H,γ):=E[\sup_{t\in[0,T]}|B_H(t)|]^γ, with $γ, T>0$.
We give necessary and sufficient conditions for the stationary density of semimartingale reflected Brownian motion in a wedge to be written as a finite sum of terms of exponential product form. Relying on geometric ideas reminiscent of the reflection principle, we give an explicit formula for the density in such cases.
We construct obliquely reflected Brownian motions in all bounded simply connected planar domains, including non-smooth domains, with general reflection vector fields on the boundary. Conformal mappings and excursion theory are our main technical tools. A key intermediate step, which may be of independent interest, is an alternative characterization of reflected Brownian motions in smooth bounded planar domains with a given field of angles of oblique reflection on the boundary in terms of a pair of quantities, namely an integrable positive harmonic function, which represents the stationary dist…
We establish the singularity with respect to Lebesgue measure as a function of time of the conditional probability that the sum of two one-dimensional Brownian motions will exit from the unit interval before time $t$, given the trajectory of the second Brownian motion up to the same time. On the way of doing so we show that if one solves the one-dimensional heat equation with zero condition on a trajectory of a one-dimensional Brownian motion, which is the lateral boundary, then for each moment of time with probability one the normal derivative of the solution is zero, provided that the diffus…
We supply two different descriptions of the pushing process driving the reflected Brownian motion in Weyl chambers, when the latter domains are simplexes. The first one shows that a simple root lies in one and only one orbit if and only if the pushing process in the direction of that simple root increases as the sum of all the Brownian local times in the directions of the orbit's positive elements. The last one shows that the pushing process may be written as the sum of an inward normal vector at the chamber's boundary and an inward normal vector at the origin, yielding a kind of a multivoque …
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