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.
We present a random walk approximation to fractional Brownian motion where the increments of the fractional random walk are defined as a weighted sum of the past increments of a Bernoulli random walk.
We establish the Brownian bridge asymptotics for a scaled self-avoiding walk conditioned on arriving to a far away point $n \vec{a}$ for $\vec{a}$ in $Z^d$, as $n$ increases to infinity.
We present a duality relation between two systems of coalescing random walks and an analogous duality relation between two systems of coalescing Brownian motions. Our results extends previous work in the literature and we apply it to the study of a system of coalescing Brownian motions with Poisson immigration.
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.
In this article, we show a result of approximation in law to subfractional Brownian motion, with $H>\frac{1}{2}$, in the Skorohod topology. The construction of these approximations is based on a sequence of I.I.D random variables
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.
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.
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.
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.
Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam and van Moerbeke that the process of eigenvalues of two consecutive minors of an Hermitian Brownian motion is a Markov process, whereas if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion.
We derive asymptotics for the probability of the origin to be an extremal point of a random walk in R^n. We show that in order for the probability to be roughly 1/2, the number of steps of the random walk should be between e^{c n / log n}$ and e^{C n log n}. As a result, we attain a bound for the ?pi/2-covering time of a spherical brownian motion.
Let $ξ(k,n)$ be the local time of a simple symmetric random walk on the line. We give a strong approximation of the centered local time process $ξ(k,n)-ξ(0,n)$ in terms of a Wiener sheet and an independent Wiener process, time changed by an independent Brownian local time. Some related results and consequences are also established.
We introduce a Multifractal Random Walk (MRW) defined as a stochastic integral of an infinitely divisible noise with respect to a dependent fractional Brownian motion. Using the techniques of the Malliavin calculus, we study the existence of this object and its properties. We then propose a continuous time model in finance that captures the main properties observed in the empirical data, including the leverage effect. We illustrate our result by numerical simulations.
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 $\mathbb{E} d_1 \approx 1.99$.
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.
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.
In this paper we define Brownian local time as the almost sure limit of the local times of a nested sequence of simple, symmetric random walks. The limit is jointly continuous in $(t,x)$. The rate of convergence is $n^{\frac14} (\log n)^{\frac34}$ that is close to the best possible. The tools we apply are almost exclusively from elementary probability theory.
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.
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.
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.
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.
In this article it is shown that the Brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete $n$-vertex ordered graph trees whose search-depth functions converge to the Brownian excursion as $n\to\infty$. We prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of our main result. The assumptions of the article cover the important example of simple random walks on the trees generated by the Galton-Watson branching process, conditioned on the total…
We focus on the existence and its characterization of limit for a certain critical branching random walks in time-space random environment in 1 dimension which was introduced by Birkner et.al. Each particle performs simple random walk on $\mathbb{Z}$ and branching mechanism depends on the time-space site. The weak limit points of this measure valued processes are characterized as a solution of the non-trivial martingale problem and called super-Brownian motions in random environment by Mytnik.
We analyze the Brownian Motion limit of a prototypical unit step reinforced random-walk on the half line. A reinforced random walk is one which changes the weight of any edge (or vertex) visited to increase the frequency of return visits. The generating function for the discrete case is first derived for the joint probability distribution of $S_N$ (the location of the walker at the $N$^{th}$ step) and $A_N$ the maximum location the walker achieved in $N$ steps. Then the bulk of the analysis concerns the statistics of the limiting Brownian walker, and of its "environment", both parametrized by …
We focus on the existence and characterization of the limit for a certain critical branching random walks in time-space random environment in one dimension which was introduced by M. Birnkenr et.al. Each particle performs simple random walk on $\mathbb{Z}$ and branching mechanism depends on the time-space site. The weak limit of this measure valued processes is characterized as a solution to the non-trivial martingale problem and called super-Brownian motions in random environment by L. Mytnik. Moreover, we will show the weak uniqueness of the solutions with some initial condition.
We consider the continuous time symmetric random walk with a slow bond on $\mathbb Z$, which rates are equal to $1/2$ for all bonds, except for the bond of vertices $\{-1,0\}$, which associated rate is given by $αn^{-β}/2$, where $α\geq 0$ and $β\in [0,\infty]$ are the parameters of the model. We prove here a functional central limit theorem for the random walk with a slow bond: if $β<1$, then it converges to the usual Brownian motion. If $β\in (1,\infty]$, then it converges to the reflected Brownian motion. And at the critical value $β=1$, it converges to the snapping out Brownian motion (SNO…
In this note, we prove an $L^p$ uniform approximation of the fractional Brownian motion with Hurst exponent $0 < H < \frac{1}{2}$ by means of a family of continuous-time random walks imbedded on a given Brownian motion. The approximation is constructed via a pathwise representation of the fractional Brownian motion in terms of a standard Brownian motion. For an arbitrary choice $ε_k$ for the size of the jumps of the family of random walks, the rate of convergence of the approximation scheme is $O(ε_k^{p(1-2λ)+ 2(δ-1)})$ whenever $\max\{0,1-\frac{pH}{2}\}< δ< 1$, $λ\in \big(\frac{1-H}{2}, \frac…
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 study the convergence in rough path topology of a certain class of discrete processes, the hidden Markov walks, to a Brownian motion with an area anomaly. This area anomaly, which is a new object, keeps track of the time-correlation of the discrete models and brings into light the question of embeddings of discrete processes into continuous time. We also identify an underlying combinatorial structure in the hidden Markov walks, which turns out to be a generalization of the occupation time from the classical ergodic theorem in the spirit of rough paths.
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.
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…
We consider the backbone of the infinite cluster generated by supercritical oriented site percolation in dimension 1 +1. A directed random walk on this backbone can be seen as an "ancestral line" of an individual sampled in the stationary discrete-time contact process. Such ancestral lineages were investigated in [BCDG13] where a central limit theorem for a single walker was proved. Here, we consider infinitely many coalescing walkers on the same backbone starting at each space-time point. We show that, after diffusive rescaling, the collection of paths converges in distribution to the Brownia…
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…
We study the persistence exponent for the first passage time of a random walk below the trajectory of another random walk. More precisely, let $\{B_n\}$ and $\{W_n\}$ be two centered, weakly dependent random walks. We establish that $\mathbb{P}(\forall_{n\leq N} B_n \geq W_n|W) = N^{-γ+ o(1)}$ for a non-random $γ\geq 1/2$. In the classical setting, $W_n \equiv 0$, it is well-known that $γ= 1/2$. We prove that for any non-trivial $W$ one has $γ>1/2$ and the exponent $γ$ depends only on $\text{Var}(B_1)/\text{Var}(W_1)$. Our result holds also in the continuous setting, when $B$ and $W$ are indep…
We introduce oscillatory analogues of fractional Brownian motion, sub-fractional Brownian motion and other related long range dependent Gaussian processes, we discuss their properties, and we show how they arise from particle systems with or without branching and with different types of initial conditions, where the individual particle motion is the so-called c-random walk on a hierarchical group. The oscillations are caused by the discrete and ultrametric structure of the hierarchical group, and they become slower as time tends to infinity and faster as time approaches zero. We also give othe…
We ask if it is possible to find some particular continuous paths of unit length in linear Brownian motion. Beginning with a discrete version of the problem, we derive the asymptotics of the expected waiting time for several interesting patterns. These suggest corresponding results on the existence/non-existence of continuous paths embedded in Brownian motion. With further effort we are able to prove some of these existence and non-existence results by various stochastic analysis arguments. A list of open problems is presented.
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.
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…
The Brownian Web (BW) is a family of coalescing Brownian motions starting from every point in space and time $\R\times\R$. It was first introduced by Arratia, and later analyzed in detail by Tóth and Werner. More recently, Fontes, Isopi, Newman and Ravishankar gave a characterization of the BW, and general convergence criteria allowing either crossing or noncrossing paths, which they verified for coalescing simple random walks. Later Ferrari, Fontes, and Wu verified these criteria for a two dimensional Poisson Tree. In both cases, the paths are noncrossing. In this thesis, we formulate new con…
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 …
In this paper we investigate the argmin process of Brownian motion $B$ defined by $α_t:=\sup\left\{s \in [0,1]: B_{t+s}=\min_{u \in [0,1]}B_{t+u} \right\}$ for $t \geq 0$. The argmin process $α$ is stationary,with invariant measure which is arcsine distributed. We prove that $(α_t; t \geq 0)$ is a Markov process with the Feller property, and provide its transition kernel $Q_t(x,\cdot)$ for $t>0$ and $x \in [0,1]$. Similar results for the argmin process of random walks and Lévy processes are derived. We also consider Brownian extrema of a given length. We prove that these extrema form a delayed…
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,…
We study the simple random walk on stochastic hyperbolic half planar triangulations constructed in Angel and Ray [3]. We show that almost surely the walker escapes the boundary of the map in positive speed and that the return probability to the starting point after n steps scales like $\exp(-cn^{1/3})$.
In this paper we study a random walk on an affine building of type $\tilde{A}_r$, whose radial part, when suitably normalized, converges to the Brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane \cite{Bia2}. This extends also the link at the probabilistic level between Riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously discovered by Bougerol and Jeulin in rank one \cite{BJ}. The main ingredients of the proof are a combinatorial formula on the building and the est…
We examine escape and entrance times, Green's functions, local times, and hitting distributions of discs and annuli of a symmetric random walk on $\Z^2$ projected onto the periodic lattice $\Z^2_K$. This extends a framework for the simple planar random walk in Dembo, et al. (2006) to the large class of planar random walks in Bass, Rosen (2007). The approach uses comparisons between $\Z^2$ and $\Z^2_K$ hitting times and distributions on annuli, and uses only random walk methods.
The fractional Brownian motion is a generalization of ordinary Brownian motion, used particularly when long-range dependence is required. Its explicit introduction is due to B.B. Mandelbrot and J.W. van Ness (1968) as a self-similar Gaussian process $\WH (t)$ with stationary increments. Here self-similarity means that $(a^{-H}\WH(at): t \ge 0) \stackrel{d}{=} (\WH(t): t \ge 0)$, where $H\in (0, 1)$ is the Hurst parameter of fractional Brownian motion. F.B. Knight gave a construction of ordinary Brownian motion as a limit of simple random walks in 1961. Later his method was simplified by P. Rév…
We prove that the local time process of a planar simple random walk, when time is scaled logarithmically, converges to a non-degenerate pure jump process. The convergence takes place in the Skorokhod space with respect to the $M1$ topology and fails to hold in the $J1$ topology.
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…
