Planar Brownian Motion and Complex Analysis

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

In the paper Dynkin construction for self-intersection local time of planar Wiener process is extended on Hilbert-valued weights.

Some infinite divisibility properties of the reciprocal of planar Brownian motion exit time from a cone

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.

Planar Brownian motion winds evenly along its trajectory

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.

Many proofs that $\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{π^2}{6}$ can be found in the conformal invariance of planar Brownian motion

A number of recent new proofs of Euler's celebrated identity are presented, all of which are probabilistic in nature and depend on the conformal invariance of planar Brownian motion.

On conformally invariant subsets of the planar Brownian curve

We define and study a family of generalized non-intersection exponents for planar Brownian motions that is indexed by subsets of the complex plane: For each $A\subset\CC$, we define an exponent $ξ(A)$ that describes the decay of certain non-intersection probabilities. To each of these exponents, we associate a conformally invariant subset of the planar Brownian path, of Hausdorff dimension $2-ξ(A)$. A consequence of this and continuity of $ξ(A)$ as a function of $A$ is the almost sure existence of pivoting points of any sufficiently small angle on a planar Brownian path.

Rescaled bipartite planar maps converge to the Brownian map

For every integer $n\geq 1$, we consider a random planar map $\mathcal{M}_n$ which is uniformly distributed over the class of all rooted bipartite planar maps with $n$ edges. We prove that the vertex set of $\mathcal{M}_n$ equipped with the graph distance rescaled by the factor $(2n)^{-1/4}$ converges in distribution, in the Gromov-Hausdorff sense, to the Brownian map. This complements several recent results giving the convergence of various classes of random planar maps to the Brownian map.

关于平面区域几何与平面 Brown 运动离开时间之间关系的结果合集，II

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。

On the expected diameter of planar Brownian motion

Known results show that the diameter $d_1$ of the trace of planar Brownian motion run for unit time satisfies $1.595 \leq \mathbb{E} d_1 \leq 2.507$. This note improves these bounds to $1.601 \leq \mathbb{E} d_1 \leq 2.355$. Simulations suggest that $\mathbb{E} d_1 \approx 1.99$.

Reflected planar Brownian motions, intertwining relations and crossing probabilities

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.

On the density of the winding number of planar Brownian motion

We obtain a formula for the density of the winding number of planar Brownian motion around the origin, and deduce from it asymptotic expansions in inverse powers of the logarithm of the squared time, explicit in the angular variable. In particular, we obtain the corrections of any order to the Spitzer's law, and also to a local limit theorem for the windings.

平面区域的 Brownian 对称化

Brownian symmetrization of planar domains

本文介绍源于平面 Skorokhod 嵌入问题的 Brownian 对称化，并通过其与 Steiner 对称化的性质比较，证明二者是不同且互不影响的变换。

Critical exponents, conformal invariance and planar Brownian motion

In this review paper, we first discuss some open problems related to two-dimensional self-avoiding paths and critical percolation. We then review some closely related results (joint work with Greg Lawler and Oded Schramm) on critical exponents for two-dimensional simple random walks, Brownian motions and other conformally invariant random objects.

Hitting distributions of planar Brownian motion

The purpose of the paper is to find the joint distribution of the hitting time and place of two-dimensional Brownian motion hitting the negative horizontal axis. We provide various formulas for Green functions as well as for the conditional distributions. In the end we treat an analogous problem for the case when vertical component has a negative drift.

Growth-fragmentation process embedded in a planar Brownian excursion

The aim of this paper is to present a self-similar growth-fragmentation process linked to a Brownian excursion in the upper half-plane $\mathbb{H}$, obtained by cutting the excursion at horizontal levels. We prove that the associated growth-fragmentation is related to one of the growth-fragmentation processes introduced by Bertoin, Budd, Curien and Kortchemski.

On the expected exit time of planar Brownian motion from simply connected domains

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…

平面域的布朗收敛与平面Skorokhod嵌入问题的稳定性

Brownian Convergence of Planar Domains and Stability of the Planar Skorokhod Embedding Problem

本文提出一个数值框架来近似平面Skorokhod嵌入问题中的μ-域，并证明在概率测度序列弱收敛下，相应的μ-域序列在适当意义下收敛到极限测度μ的域，同时给出实现策略、收敛速率估计和数值例子。

On the exit distribution of partially reflected Brownian motion in planar domains

We show that the dimension of the exit distribution of planar partially reflected Brownian motion can be arbitrarily close to 2.

Obliquely reflected Brownian motion in non-smooth planar domains

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…

Planar aggregation and the coalescing Brownian flow

We study a scaling limit associated to a model of planar aggregation. The model is obtained by composing certain independent random conformal maps. The evolution of harmonic measure on the boundary of the cluster is shown to converge to the coalescing Brownian flow.

On the most visited sites of planar Brownian motion

Let (B_t : t > 0) be a planar Brownian motion and define gauge functions $ϕ_α(s)=log(1/s)^{-α}$ for $α>0$. If $α<1$ we show that almost surely there exists a point x in the plane such that $H^{ϕ_α}({t > 0 : B_t=x})>0$, but if $α>1$ almost surely $H^{ϕ_α} ({t > 0 : B_t=x})=0$ simultaneously for all $x\in R^2$. This resolves a longstanding open problem posed by S.,J. Taylor in 1986.

Some Estimates for Planar Random Walk and Brownian Motion

The purpose of this note is to collect in one place a few results about simple random walk and Brownian motion which are often useful. These include standard results such as Beurling estimates, large deviation estimates, and a method for coupling the two processes, as well as solutions to the discrete Dirichlet problem in various domains which, to the author's knowledge, have not been published anywhere. The main focus is on the two-dimensional processes.

平面布朗凸包的覆盖概率

Coverage probability of the planar Brownian convex hull

本文精确计算了平面布朗运动凸包包含给定点的概率，通过径向函数与支撑函数的关系及高斯椭圆比较，并推广至布朗桥，给出渐近结果。

On the planar Brownian Green's function for stopping times

It has been known for some time that the Green's function of a planar domain can be defined in terms of the exit time of Brownian motion, and this definition has been extended to stopping times more general than exit times. In this paper, we extend the notion of conformal invariance of Green's function to analytic functions which are not injective, and use this extension to calculate the Green's function for a stopping time defined by the winding of Brownian motion. These considerations lead to a new proof of the Riemann mapping theorem. We also show how this invariance can be used to deduce s…

平面布朗运动h函数反问题的一种求解方法

A method of solution for the inverse problem for $h$-functions of planar Brownian motion

本文研究平面布朗运动h函数的反问题，通过将区域h函数推广到停时并利用共形不变性，给出了判定给定函数是否为某区域h函数的方法，并构造了一大类对应停时为击中时间的例子。

Random cubic planar graphs converge to the Brownian sphere

In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere. The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances. Then, Whitney's theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit…

Convex hull of n planar Brownian paths: an exact formula for the average number of edges

We establish an exact formula for the average number of edges appearing on the boundary of the global convex hull of n independent Brownian paths in the plane. This requires the introduction of a counting criterion which amounts to "cutting off" edges that are, in a specific sense, small. The main argument consists in a mapping between planar Brownian convex hulls and configurations of constrained, independent linear Brownian motions. This new formula is confirmed by retrieving an existing exact result on the average perimeter of the boundary of Brownian convex hulls in the plane.

Hitting distribution of a correlated planar Brownian motion in a disk

In this paper we study the hitting probability of a circumference $C_R$ for a correlated Brownian motion $\underline{B}(t)=\left(B_1(t), B_2(t)\right)$, $ρ$ being the correlation coefficient. The analysis starts by first mapping the circle $C_R$ into an ellipse $E$ with semiaxes depending on $ρ$ and transforming the differential operator governing the hitting distribution into the classical Laplace operator. By means of two different approaches (one obtained by applying elliptic coordinates) we obtain the desired distribution as a series of Poisson kernels.

Self-intersection local time of planar Brownian motion based on a strong approximation by random walks

The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka--Rosen--Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-i…

布朗运动与布朗桥的凸包

Convex hull of Brownian motion and Brownian bridge

本文研究平面布朗运动与独立布朗桥轨迹并集的凸包，给出其周长与面积期望的精确值及联合最大值时刻密度，并推广到多个独立过程。

平面布朗运动凸包尺寸及相关逆过程的界限

Bounds on the size of the convex hull of planar Brownian motion and related inverse processes

本文研究了标准平面布朗运动凸包的外接圆半径与内切圆半径期望值上下界，并给出相关逆过程界限以揭示凸包尺寸增长速度。

The critical 2D delta-Bose gas as mixed-order asymptotics of planar Brownian motion

We consider the 2D delta-Bose gas by a smooth mollification of the delta potential, where the coupling constant is in the critical window. The main result proves that for two particles, the approximate semigroups on $\mathscr B_b(\Bbb R^2)$ for the Schrödinger operator with singular interaction at the origin converge pointwise in the initial condition. This convergence extends earlier functional analytic results for the convergence in the $L_2$-norm resolvent sense. The central methods introduced here apply the excursion theory of the 2D Bessel process and the ergodicity of the winding number …

平面布朗运动的骨干概率

Backbone probability of planar Brownian motion

研究平面布朗运动中两个不相交子路径连接微观与宏观距离的概率，揭示其渐近行为为C(log|logε|)^{-1}。

On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol's identity

Some identities in law in terms of planar complex valued Ornstein-Uhlenbeck processes $(Z_{t}=X_{t}+iY_{t},t\geq0)$ including planar Brownian motion are established and shown to be equivalent to the well known Bougerol identity for linear Brownian motion:$(β_{t},t\geq0)$: for any fixed $u>0$: \sinh(β_{u}) \stackrel{(law)}{=} \hatβ_{(\int^{u}_{0}ds\exp(2β_{s}))}. These identities in law for 2-dimensional processes allow to study the distributions of hitting times $T^θ_{c}\equiv\inf\{t:θ_{t} =c \}, (c>0)$, $T^θ_{-d,c}\equiv\inf\{t:θ_{t}\notin(-d,c) \}, (c,d>0)$ and more specifically of $T^θ_{-c,…

On occupation times of the first and third quadrants for planar Brownian motion

An open problem of interest, first infused into the applied probability community in the work of Bingham and Doney in 1988, (see \cite{Bingham}) is stated as follows: find the distribution of the quadrant occupation time of planar Brownian motion. In this short communication, we study an alternate formulation of this longstanding open problem: let $X(t), Y(t), t \geq 0$ be standard Brownian motions starting at $x,y$ respectively. Find the distribution of the total time $T=Leb\{t \in [0,1]: X(t) \times Y(t) >0\}$, when $x=y=0$, i.e., the occupation time of the union of the first and third quadr…

Trapping planar Brownian motion in a non circular trap

Brownian motion in the plane in the presence of a "trap" at which motion is stopped is studied. If the trap $T$ is a connected compact set, it is shown that the probability for planar Brownian motion to hit this set before a given time $t$ is well approximated even at short times by the probability that Brownian motion hits a disk of radius $r_T$ equal to the conformal radius of the trap $T$.

Planar Brownian motion and Gaussian multiplicative chaos

We construct the analogue of Gaussian multiplicative chaos measures for the local times of planar Brownian motion by exponentiating the square root of the local times of small circles. We also consider a flat measure supported on points whose local time is within a constant of the desired thickness level and show a simple relation between the two objects. Our results extend those of Bass, Burdzy and Khoshnevisan and in particular cover the entire $L^1$-phase or subcritical regime. These results allow us to obtain a nondegenerate limit for the appropriately rescaled size of thick points, thereb…

Expected exit times of Brownian motion from planar domains: Complements to a paper of Markowsky

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.

Geodesics in large planar maps and in the Brownian map

We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the root by more than one geodesic. The set S is dense in the Brownian map and homeomorphic to a non-compact real tree. Furthermore, for every x in S, the number of distinct geodesics from x to the root is equal to the number of connected components of the complement of {x} in S. In particular, points …

Integrability properties and limit theorems for the exit time from a cone of planar Brownian motion

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.

布朗运动和布朗循环汤中几种随机分形的不存在性

Non-existence of several random fractals in Brownian motion and Brownian loop soup

本文通过统一方法证明了三种随机分形的不存在性，包括平面布朗运动的先驱三重点、平面和三维布朗运动的先驱双断点，以及临界强度下平面布朗循环汤簇边界上的双点，解决了相关开放问题。

The Dimension of the Planar Brownian Frontier is 4/3

In a series of recent preprints, we have proven that with probability one the Hausdorff dimension on the outer boundary of planar Brownian motion is 4/3, confirming a conjecture by Mandelbrot. It is also shown that the Hausdorff dimension of the set of cut points is a.s. 3/4. The present paper is an expository outline of the arguments involved in the proof of these and related results.

平面布朗运动与桥的星形包的期望面积

Expected area of the star hull of planar Brownian motion and bridge

研究平面布朗运动与桥的星形包，证明其期望面积分别为3π/8和π/4，通过分析半直线首达时与位置的联合分布得到简洁的拉普拉斯变换。

具有随机高斯端点的平面布朗桥的凸包

On the convex hull of a planar Brownian bridge with a random Gaussian endpoint

研究一类各向同性平面高斯过程，计算其凸包的期望周长和面积，得到关于参数σ的闭式公式，并推广到多个独立过程的情形。

Brownian geometry

We present different continuous models of random geometry that have been introduced and studied in the recent years. In particular, we consider the Brownian map, which is the universal scaling limit of large planar maps in the Gromov-Hausdorff sense, and the Brownian disk, which appears as the scaling limit of planar maps with a boundary. We discuss the construction of these models, and we emphasize the role played by Brownian motion indexed by the Brownian tree.

The kinetic limit of a system of coagulating planar Brownian particles

We study a model of mass-bearing coagulating planar Brownian particles. Coagulation is prone to occur when two particles become within a distance of order $ε$. We assume that the initial number of particles is of the order of $| \log ε|. Under suitable assumptions on the initial distribution of particles and the microscopic coagulation propensities, we show that the macroscopic particle densities satisfy a Smoluchowski-type equation.

Isoperimetric inequalities in the Brownian plane

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 triangulation or the uniform infinite planar quadrangulation and is conjectured to be the universal scaling limit of many others random planar lattices. We establish sharp bounds on the probability of having a short cycle separating the ball of radius $r$ centered at the distinguished point from infinity. Then we prove a strong version of the spatial Markov p…

A Weak Limit Shape Theorem For Planar Isotropic Brownian Flows

It has been shown by various authors under different assumptions that the diameter of a bounded non-trivial set $γ$ under the action of a stochastic flow grows linearly in time. We show that the asymptotic linear expansion speed if properly defined is deterministic i.e. we show for a $2$-dimensional isotropic Brownian flow $Φ$ with a positive Lyapunov exponent that there exists a non-random set $\mathcal{B}$ such that we have for $ε>0$, arbitrary connected $γ\subset\subset\R^2$ consisting of at least two different points and arbitrarily large times $T$ that $$(1-ε)T\mathcal{B}\subset \cup_{0\l…

Homotopy and holonomy of the planar Brownian motion in a Poisson punctured plane

We define a family of diffeomorphism-invariant models of random connections on principal $G$-bundles over the plane, whose curvatures are concentrated on singular points. In a limit when the number of point grows whilst the singular curvature on each point diminishes, the model converges in some sense towards a Yang--Mills field. We study another regime for which we prove that the holonomy along a Brownian trajectory converges towards an explicit limit.

The Mean First Rotation Time of a planar polymer

We estimate the mean first time, called the mean rotation time (MRT), for a planar random polymer to wind around a point. This polymer is modeled as a collection of n rods, each of them being parameterized by a Brownian angle. We are led to study the sum of i.i.d. imaginary exponentials with one dimensional Brownian motions as arguments. We find that the free end of the polymer satisfies a novel stochastic equation with a nonlinear time function. Finally, we obtain an asymptotic formula for the MRT, whose leading order term depends on the square root of n and, interestingly, depends weakly on …

How round are the complementary components of planar Brownian motion?

Consider a Brownian motion $W$ in ${\bf C}$ started from $0$ and run for time 1. Let $A(1),A(2),\dots$ denote the bounded connected components of ${\bf C}-W([0,1])$. Let $R(i)$ (resp. $r(i)$) denote the out-radius (resp. in-radius) of $A(i)$ for $i\in\bf N$. Our main result is that ${\bf E}[\sum_i R(i)^2|\log R(i)|^θ]<\infty$ for any $θ<1$. We also prove that $\sum_i r(i)^2|\log r(i)|=\infty$ almost surely. These results have the interpretation that most of the components $A(i)$ have a rather regular or round shape.

Conformal invariance, universality, and the dimension of the Brownian frontier

This paper describes joint work with Oded Schramm and Wendelin Werner establishing the values of the planar Brownian intersection exponents from which one derives the Hausdorff dimension of certain exceptional sets of planar Brownian motion. In particular, we proof a conjecture of Mandelbrot that the dimension of the frontier is 4/3. The proof uses a universality principle for conformally invariant measures and a new process, the stochastic Loewner evolution ($SLE$), introduced by Schramm. These ideas can be used to study other planar lattice models from statistical physics at criticality. I d…

Asymptotic support theorem for planar isotropic Brownian flows

It has been shown by various authors that the diameter of a given nontrivial bounded connected set $\mathcal{X}$ grows linearly in time under the action of an isotropic Brownian flow (IBF), which has a nonnegative top-Lyapunov exponent. In case of a planar IBF with a positive top-Lyapunov exponent, the precise deterministic linear growth rate K of the diameter is known to exist. In this paper we will extend this result to an asymptotic support theorem for the time-scaled trajectories of a planar IBF $φ$, which has a positive top-Lyapunov exponent, starting in a nontrivial compact connected set…

平面布朗乘法混沌的特征化

Characterisation of planar Brownian multiplicative chaos

本文通过特征化平面布朗乘法混沌测度，揭示了随机游走厚点集的标度极限，并得出了归一化后厚点数量的收敛性结果。

The expected area of the filled planar Brownian loop is Pi/5

Let B_t be a planar Brownian loop of time duration 1 (a Brownian motion conditioned so that B_0 = B_1). We consider the compact hull obtained by filling in all the holes, i.e. the complement of the unique unbounded component of R^2\B[0,1]. We show that the expected area of this hull is Pi/5. The proof uses, perhaps not surprisingly, the Schramm Loewner Evolution (SLE). Also, using the result of Yor about the law of the index of a Brownian loop, we show that the expected areas of the regions of non-zero index n equal 1/(2 Pi n^2). As a consequence, we find that the expected area of the region o…