Davar Khoshnevisan, David A. Levin, Zhan Shi
We present an extreme-value analysis of the classical law of the iterated logarithm (LIL) for Brownian motion. Our result can be viewed as a new improvement to the LIL.
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Let $X:=\{X(t)\}_{t\ge0}$ be a generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019): $$ \big\{X(t)\big\}_{t\ge0}\overset{d}{=}\left\{ \int_{\mathbb R} \left((t-u)_+^α-(-u)_+^α \right) |u|^{-γ} B(du) \right\}_{t\ge0}, $$ with parameters $γ\in (0, 1/2)$ and $α\in \left(-\frac12+ γ, \, \frac12+ γ\right)$. Continuing the studies of sample path properties of GFBM $X$ in Ichiba, Pang and Taqqu (2021) and Wang and Xiao (2021), we establish integral criteria for the lower functions of $X$ at $t=0$ and at infinity by modifying the arguments of Talagrand (1996). As a consequ…
The generalized fractional Brownian motion (GFBM) $X:=\{X(t)\}_{t\ge0}$ with parameters $γ\in [0, 1)$ and $α\in \left(-\frac12+\fracγ{2}, \, \frac12+\fracγ{2} \right)$ is a centered Gaussian $H$-self-similar process introduced by Pang and Taqqu (2019) as the scaling limit of power-law shot noise processes, where $H = α-\fracγ{2}+\frac12 \in(0,1)$. When $γ= 0$, $X$ is the ordinary fractional Brownian motion. For $γ\in (0, 1)$, GFBM $X$ does not have stationary increments, and its sample path properties such as Hölder continuity, path differentiability/non-differentiability, and the functional l…
The classical law of the iterated logarithm (LIL for short)as fundamental limit theorems in probability theory play an important role in the development of probability theory and its applications. Strassen (1964) extended LIL to large classes of functional random variables, it is well known as the invariance principle for LIL which provide an extremely powerful tool in probability and statistical inference. But recently many phenomena show that the linearity of probability is a limit for applications, for example in finance, statistics. As while a nonlinear expectation--- G-expectation has att…
Lower classes and Chung's LILs of the fractional integrated generalized fractional Brownian motion
本研究针对Pang和Taqqu提出的广义分数布朗运动的分数阶积分过程，借助Talagrand的论证方法与小球概率估计，得到其在零点和无穷远下类的积分准则及Chung型重对数律，丰富了分数布朗运动相关极限理论。
After establishing the moderate deviation principle by the Classical Azencott method, we prove the Strassen's compact law of the iterated logarithm (LIL) for a class of stochastic partial differential equations (SPDEs). As an application, we obtain this type of LIL for two population models known as super-Brownian motion and Fleming-Viot process. In addition, the classical LIL is shown for the class of SPDEs and the two population models.
By using a change of scale and space, we study a class of stochastic differential equations (SDEs) whose solutions are drift--perturbed and exhibit behaviour analogous to standard Brownian motion including to the Law of the Iterated Logarithm (LIL). Sufficient conditions ensuring that these processes obey the LIL are given.
We introduce a class of iterated processes called $α$-time Brownian motion for $0<α\leq 2$. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric $α$-stable process. We prove a Chung-type law of the iterated logarithm (LIL) for these processes which is a generalization of LIL proved in \cite{hu} for iterated Brownian motion. When $α=1$ it takes the following form $$ \liminf_{T\to\infty}T^{-1/2}(\log \log T) \sup_{0\leq t\leq T}|Z_{t}|=π^{2}\sqrt{λ_{1}} a.s. $$ where $λ_{1}$ is the first eigenvalue for the Cauchy process in the interval $[-1,1].$ We also…
Capacity of the range of random walk: The law of the iterated logarithm
本文研究 $d\ge 3$ 简单随机游走范围容量的极限行为，建立其上、下极限重对数律，并给出三维 Brown 路径结果与四维精确中偏差界。
We study the large-time behavior of the charged-polymer Hamiltonian $H_n$ of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths, and Higgs [Gaussian case], using strong approximations to Brownian motion. Our results imply, among other things, that in one dimension the process $\{H_{[nt]}\}_{0\le t\le 1}$ behaves like a Brownian motion, time-changed by the intersection local-time process of an independent Brownian motion. Chung-type LILs are also discussed.
Fine properties of fractional Brownian motions on Wiener space
本文在经典 Wiener 空间的 $(p,r)$-容量下，利用分数 Brown 运动的积分表示与 Malliavin 分析，证明其样本轨道的不可微、连续模、LIL 和自回避等精细性质。
On asymptotic behavior of almost surely extreme values of independent random variables
本文研究离散同分布随机变量极值的几乎必然渐近行为，建立了极值等于给定递增整数序列加固定整数的无限频繁发生概率为1的充分必要条件，并针对不同尾部衰减速度给出了比对数迭代律和三重对数律更具信息量的结果。
Law of the Iterated Logarithm for $p$-Walks on $\mathbb{Z}$
研究 $p$-转子游走的几乎必然渐近边界，通过分解为鞅与运行极值扰动，建立函数型迭代对数定律并求解变分问题得到经典常数。
In this paper, we consider a two-dimensional sticky Brownian motion. Sticky Brownian motions can be viewed as time-changed semimartingale reflecting Brownian motions, which find applications in many areas including queueing theory and mathematical finance. For example, a sticky Brownian motion can be used to model a storage system.with exceptional services. In this paper, we focus on stationary distributions for sticky Brownian motions. The main results obtained here include tail asymptotic properties in boundary stationary distributions, marginal distributions, and joint distributions. The ke…
Let $B_{H}(t), t\geq [0,T], T\in(0,\infty)$ be the standard Multifractional Brownian Motion(mBm), in this contribution we are concerned with the exact asymptotics of \begin{eqnarray*} \mathbb{P}\left\{\sup_{t\in[0,T]}B_{H}(t)>u\right\} \end{eqnarray*} as $u\rightarrow\infty$. Mainly depended on the structures of $H(t)$, the results under several important cases are investigated.
Let $\{B_β(x), x \in \mathbb{S}^N\}$ be a fractional Brownian motion on the $N$-dimensional unit sphere $\mathbb{S}^N$ with Hurst index $β$. We study the excursion probability $\mathbb{P}\{\sup_{x\in T} B_β(x) > u \}$ and obtain the asymptotics as $u\to \infty$, where $T$ can be the entire sphere $\mathbb{S}^N$ or a geodesic disc on $\mathbb{S}^N$.
This work is a continuation of the manuscript "the structure of extreme level sets in branching Brownian motion", in which the same authors studied the fine structure of the extreme level sets of branching Brownian motion, namely the sets of particles whose height is within a finite distance from the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. Our main finding here is that most of the particles in an extreme level set come from only a small fraction of th…
Brownian motion conditioned to spend limited time outside a bounded interval -- an extreme example of entropic repulsion
研究半直线上布朗运动在有界区间外停留时间受限的现象，证明其完全不离开区间，并给出该概率随时间趋于无穷时的精确渐近式。
Extremes of Brownian Decision Trees
研究布朗决策树在有限时间内的极值行为，分析高超越概率的渐进行为及多棵树同时超越阈值的概率。
Let $X$ be a super-Brownian motion (SBM) defined on a domain $E\subset R^n$ and $(X_D)$ be its exit measures indexed by sub-domains of $E$. The relationship between the equation $1/2 Δu=2 u^2$ and Super-Brownian motion (SBM) is analogous to the relationship between Brownian motion and the Laplace's equation, and substantial progress has been made on the study of the solutions of this semi-linear p.d.e. exploring this analogy. An area that remains to be explored is Martin boundary theory. Martin boundary in the semi-linear case is defined as the convex set of extreme $X$-harmonic functions whic…
Extreme value theory for geometric Brownian motion and pricing of short maturity options
研究几何布朗运动在运行最大值取大值条件下的极限分布，证明样本路径强收敛并给出收敛速率，应用于短期限障碍期权定价，得到含定量误差的渐近公式。
This paper is concerned with the limit laws of the extreme order statistics derived from a symmetric Laplace walk. We provide two different descriptions of the point process of the limiting extreme order statistics: a branching representation and a squared Bessel representation. These complementary descriptions expose various hidden symmetries in branching processes and Brownian motion which lie behind some striking formulas found by Schehr and Majumdar (Phys. Rev. Lett., 108:040601). In particular, the Bessel process of dimension $4 = 2+2$ appears in the descriptions as a path decomposition o…
This contribution derives the exact asymptotic behaviour of the supremum of alpha(t)-locally stationary Gaussian random fields over a finite hypercube. We present two applications of our result; the first one deals with extremes of ggregate multifractional Brownian motions, whereas the second application establishes the exact asymptotics of the supremum of chi-processes generated by multifractional Brownian motions.
In this article we establish the magnitude of fluctuations of the extreme particle in the model of binary branching Brownian motion with a single catalytic point at the origin.
We study the structure of extreme level sets of a standard one dimensional branching Brownian motion, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the dist…
In this paper, by using the exact tail asymptotics derived by Debicki, Hashorva and Ji (Ann. Probab. 2014), we proved the Gumbel limit theorem for the maximum of a class of non-homogeneous Gaussian random fields. By using the obtained results, we also derived the Gumbel laws for Shepp statistics of fractional Brownian motion and Gaussian integrated process as well as the Gumbel law for Storage process with fractional Brownian motion as input.
This paper provides a multivariate extension of Bertoin's pathwise construction of a Lévy process conditioned to stay positive/negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Br…
In this paper we study fluctuations of extreme particles of nonintersecting Brownian bridges starting from $a_1\leq a_2\leq \cdots \leq a_n$ at time $t=0$ and ending at $b_1\leq b_2\leq \cdots\leq b_n$ at time $t=1$, where $μ_{A_n}=(1/n)\sum_{i}δ_{a_i}, μ_{B_n}=(1/n)\sum_i δ_{b_i}$ are discretization of probability measures $μ_A, μ_B$. Under regularity assumptions of $μ_A, μ_B$, we show as the number of particles $n$ goes to infinity, fluctuations of extreme particles at any time $0<t<1$, after proper rescaling, are asymptotically universal, converging to the Airy point process.
In this paper, we consider the drawdown and drawup of the fractional Brownian motion with trend, which corresponds to the logarithm of geometric fractional Brownian motion representing the stock price in financial market. We derive the asymptotics of tail probabilities of the maximum drawdown and maximum drawup as the threshold goes to infinity, respectively. It turns out that the extremes of drawdown leads to new scenarios of asymptotics depending on Hurst index of fractional Brownian motion.
We use the Real Space Renormalization Group (RSRG) method to study extreme value statistics for a variety of Brownian motions, free or constrained such as the Brownian bridge, excursion, meander and reflected bridge, recovering some standard results, and extending others. We apply the same method to compute the distribution of extrema of Bessel processes. We briefly show how the continuous time random walk (CTRW) corresponds to a non standard fixed point of the RSRG transformation.
We study the statistics of near-extreme events of Brownian motion (BM) on the time interval [0,t]. We focus on the density of states (DOS) near the maximum ρ(r,t) which is the amount of time spent by the process at a distance r from the maximum. We develop a path integral approach to study functionals of the maximum of BM, which allows us to study the full probability density function (PDF) of ρ(r,t) and obtain an explicit expression for the moments, \langle [ρ(r,t)]^k \rangle, for arbitrary integer k. We also study near-extremes of constrained BM, like the Brownian bridge. Finally we also pre…
The extreme statistics of some noncolliding Brownian processes
研究非碰撞粒子系统极限理论，揭示布朗运动最大特征值的极限行为、Airy过程极限及非碰撞布朗桥最大路径的Fredholm行列式公式。
Edge Rigidity of Dyson Brownian Motion with General Initial Data
本文在仅要求平均初始密度具有平方根剖面下界的条件下，证明一般 β 的 Dyson 布朗运动极端粒子涨落界为 (log n)^{O(1)}n^{-2/3}，并由此得到短时间内 Tracy-Widom β 分布的收敛性。
The search for hidden targets is a fundamental problem in many areas of science, engineering, and other fields. Studies of search processes often adopt a probabilistic framework, in which a searcher randomly explores a spatial domain for a randomly located target. There has been significant interest and controversy regarding optimal search strategies, especially for superdiffusive processes. The optimal search strategy is typically defined as the strategy that minimizes the time it takes a given single searcher to find a target, which is called a first hitting time (FHT). However, many systems…
In this paper we derive weak limits for the discretization errors of sampling barrier-hitting and extreme events of Brownian motion by using the Euler discretization simulation method. Specifically, we consider the Euler discretization approximation of Brownian motion to sample barrier-hitting events, i.e. hitting for the first time a deterministic "barrier" function; and to sample extreme events, i.e. attaining a minimum on a given compact time interval or unbounded closed time interval. For each case we study the discretization error between the actual time the event occurs versus the time t…
In this paper, we study the weak convergence of the extremes of supercritical branching Lévy processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are Lévy processes with regularly varying tails. The result is drastically different from the case of branching Brownian motions. We prove that, when properly renormalized, $\mathbb{X}_t$ converges weakly. As a consequence, we obtain a limit theorem for the order statistics of $\mathbb{X}_t$.
First passage time (FPT) theory is often used to estimate timescales in cellular and molecular biology. While the overwhelming majority of studies have focused on the time it takes a given single Brownian searcher to reach a target, cellular processes are instead often triggered by the arrival of the first molecule out of many molecules. In these scenarios, the more relevant timescale is the FPT of the first Brownian searcher to reach a target from a large group of independent and identical Brownian searchers. Though the searchers are identically distributed, one searcher will reach the target…
Entrance laws for coalescing and annihilating Brownian motions
本文研究了凝聚与湮灭布朗运动的入口定律，证明其极值点为Pfaffian点过程并确定了其核结构。
In this paper, we compare the solutions of Dyson Brownian motion with general $β$ and potential $V$ and the associated McKean-Vlasov equation near the edge. Under suitable conditions on the initial data and potential $V$, we obtain the optimal rigidity estimates of particle locations near the edge for short time $t=\text{o}(1)$. Our argument uses the method of characteristics along with a careful estimate involving an equation of the edge. With the rigidity estimates as an input, we prove a central limit theorem for mesoscopic statistics near the edge which, as far as we know, have been done f…
We consider finite collections of $N$ non-intersecting Brownian paths on the line and on the half-line with both absorbing and reflecting boundary conditions (corresponding to Brownian excursions and reflected Brownian motions) and compute in each case the joint distribution of the maximal height of the top path and the location at which this maximum is attained. The resulting formulas are analogous to the ones obtained in [MFQR13] for the joint distribution of $\mathcal{M}={\rm max}_{x\in\mathbb{R}}\{\mathcal{A}_2(x)-x^2\}$ and $\mathcal{T}={\rm argmax}_{x\in\mathbb{R}}\{\mathcal{A}_2(x)-x^2\…
Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent copies of a random process $\{X(t), t\ge0\}$. For a given positive constant $u$, define the set of $r$th conjunctions $C_r(u):=\{t\in[0,1]: X_{r:n}(t)>u\}$ with $ X_{r:n}$ the $r$th largest order statistics of $X_i, 1\le i\le n$. In numerical applications such as brain mapping and digital communication systems, of interest is the approximation of $p_r(u)=\mathbb P\{C_r(u)\neqϕ\}$. Instead of stationary processes dealt with by Dȩbicki et al. (2014), we consider in this paper $X$ a self-similar $\mathbb R$-valued process with $P$-continuous samp…
We consider a Gaussian random matrix with correlated entries that have a power law decay of order $d>2$ and prove universality for the extreme eigenvalues. A local law is proved using the self-consistent equation combined with a decomposition of the matrix. This local law along with concentration of eigenvalues around the edge allows us to get an bound for extreme eigenvalues. Using a recent result of the Dyson-Brownian motion, we prove universality of extreme eigenvalues.
The epsilon-cover time of the two dimensional torus by Brownian motion is the time it takes for the process to come within distance epsilon>0 from any point. Its leading order in the small epsilon-regime has been established by Dembo, Peres, Rosen and Zeitouni [Ann. of Math., 160 (2004)]. In this work, the second order correction is identified. The approach relies on a multi-scale refinement of the second moment method, and draws on ideas from the study of the extremes of branching Brownian motion.
Estimates for exponential functionals of continuous Gaussian processes with emphasis on fractional Brownian motion
本文为带漂移分数布朗运动及其他连续高斯过程的指数泛函的分布函数和p阶矩提供显式上下界估计，并证明其关于Hurst参数的分布连续性。
On the Growth of the Extremal and Cluster Level Sets in Branching Brownian Motion
本研究针对分支布朗运动，证明了极值点过程上水平集质量几乎必然以 $C_\star Z v e^{\sqrt{2} v}$ 增长，并揭示簇点过程对数质量增长为 $\sqrt{2}v$ 减去 $v^{2/3}$ 阶随机波动，改进先前概率结果并严格化物理推导。
Let $\{X_i,i=1,2,...\}$ be i.i.d. standard gaussian variables. Let $S_n=X_1+...+X_n$ be the sequence of partial sums and $$ L_n=\max_{0\leq i<j\leq n}\frac{S_j-S_i}{\sqrt{j-i}}. $$ We show that the distribution of $L_n$, appropriately normalized, converges as $n\to\infty$ to the Gumbel distribution. In some sense, the the random variable $L_n$, being the maximum of $n(n+1)/2$ dependent standard gaussian variables, behaves like the maximum of $Hn \log n$ independent standard gaussian variables. Here, $H\in (0,\infty)$ is some constant. We also prove a version of the above result for the Brownia…
A class of interacting particle systems on $\mathbb{Z}$, involving instantaneously annihilating or coalescing nearest neighbour random walks, are shown to be Pfaffan point processes for all deterministic initial conditions. As diffusion limits, explicit Pfaffan kernels are derived for a variety of coalescing and annihilating Brownian systems. For Brownian motions on $\mathbb{R}$, depending on the initial conditions, the corresponding kernels are closely related to the bulk and edge scaling limits of the Pfaffan point process for real eigenvalues for the real Ginibre ensemble of random matrices…
Noncolliding diffusion processes reported in the present paper are $N$-particle systems of diffusion processes in one-dimension, which are conditioned so that all particles start from the origin and never collide with each other in a finite time interval $(0, T)$, $0 < T < \infty$. We consider four temporally inhomogeneous processes with duration $T$, the noncolliding Brownian bridge, the noncolliding Brownian motion, the noncolliding three-dimensional Bessel bridge, and the noncolliding Brownian meander. Their particle distributions at each time $t \in [0, T]$ are related to the eigenvalue di…
The generalized Rosenblatt process is obtained by replacing the single critical exponent characterizing the Rosenblatt process by two different exponents living in the interior of a triangular region. What happens to that generalized Rosenblatt process as these critical exponents approach the boundaries of the triangle? We show by two different methods that on each of the two symmetric boundaries, the limit is non-Gaussian. On the third boundary, the limit is Brownian motion. The rates of convergence to these boundaries are also given. The situation is particularly delicate as one approaches t…
We consider the discrete Gaussian Free Field in a square box in $\mathbb Z^2$ of side length $N$ with zero boundary conditions and study the joint law of its properly-centered extreme values ($h$) and their scaled spatial positions ($x$) in the limit as $N\to\infty$. Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an $r_N$-neighborhood thereof, we prove that the associated process tends, whenever $r_N\to\infty$ and $r_N/N\to0$, to a Poisson point process with intensity measure $Z(dx)e^{-αh}dh$, where $α:= 2/\sqrt{g}$ with $g:=2/π$ and where $Z(dx)$ i…
The first of $N$ identical independently distributed (i.i.d.) Brownian trajectories that arrives to a small target, sets the time scale of activation, which in general is much faster than the arrival to the target of only a single trajectory. Analytical asymptotic expressions for the minimal time is notoriously difficult to compute in general geometries. We derive here asymptotic laws for the probability density function of the first and second arrival times of a large number of i.i.d. Brownian trajectories to a small target in 1,2, and 3 dimensions and study their range of validity by stochas…
Universality and ambiguity in extremes of anomalous diffusion
研究异常和正常扩散模型中最快首次通过时间的普遍性与模型特异性，揭示亚扩散可能比正常扩散更快，但参数范围依赖具体模型存在歧义。
From 1 to infinity: The log-correction for the maximum of variable-speed branching Brownian motion
研究变速分支布朗运动极值，针对两类速度函数，确定最大值对数修正范围，并证明极限分布与标准BBM一致，推广了双速情形结果。
We compute exactly the mean perimeter and area of the convex hull of N independent planar Brownian paths each of duration T, both for open and closed paths. We show that the mean perimeter < L_N > = α_N, \sqrt{T} and the mean area <A_N> = β_N T for all T. The prefactors α_N and β_N, computed exactly for all N, increase very slowly (logarithmically) with increasing N. This slow growth is a consequence of extreme value statistics and has interesting implication in ecological context in estimating the home range of a herd of animals with population size N.
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…
If L is a partition of n, the rank of L is the size of the largest part minus the number of parts. Under the uniform distribution on partitions, Bringmann, Mahlburg, and Rhoades showed that the rank statistic has a limiting distribution. We identify the limit as the difference between two independent extreme value distributions and as the distribution of B(T) where B(t) is standard Brownian motion and T is the first time that an independent three-dimensional Brownian motion hits the unit sphere. The same limit holds for the crank.
In the framework of a stochastic picture for the one-dimensional branching Brownian motion, we compute the probability density of the number of particles near the rightmost one at a time $T$, that we take very large, when this extreme particle is conditioned to arrive at a predefined position $x_T$ chosen far ahead of its expected position $m_T$. We recover the previously-conjectured fact that the typical number density of particles a distance $Δ$ to the left of the lead particle, when both $Δ$ and $x_T-Δ-m_T$ are large, is smaller than the mean number density by a factor proportional to $e^{-…
Extreme value statistics and some applications in statistical physics
本文探讨了极值统计在强相关系统中的应用，包括随机游走、布朗运动及随机矩阵理论中的特征值统计，用于解决统计物理和无序系统中的基础问题。
We study the one dimensional branching Brownian motion starting at the origin and investigate the correlation between the rightmost ($X_{\max}\geq 0$) and leftmost ($X_{\min} \leq 0$) visited sites up to time $t$. At each time step the existing particles in the system either diffuse (with diffusion constant $D$), die (with rate $a$) or split into two particles (with rate $b$). We focus on the regime $b \leq a$ where these two extreme values $X_{\max}$ and $X_{\min}$ are strongly correlated. We show that at large time $t$, the joint probability distribution function (PDF) of the two extreme poi…
We compute exactly the mean perimeter and the mean area of the convex hull of a $2$-d Brownian motion of duration $t$ and diffusion constant $D$, in the presence of resetting to the origin at a constant rate $r$. We show that for any $t$, the mean perimeter is given by $\langle L(t)\rangle= 2 π\sqrt{\frac{D}{r}}\, f_1(rt)$ and the mean area is given by $\langle A(t) \rangle= 2π\frac{D}{r}\, f_2(rt)$ where the scaling functions $f_1(z)$ and $f_2(z)$ are computed explicitly. For large $t\gg 1/r$, the mean perimeter grows extremely slowly as $\langle L(t)\rangle \propto \ln (rt)$ with time. Likew…
Consider non-intersecting Brownian motions on the real line, starting from the origin at t=0, with a number of particles forced to reach p distinct target points at time t=1. This work shows that the transition probability, that is the probability for the particles to pass through windows E_k at times t_k, satisfies, in a new set of variables, a non-linear PDE which can be expressed as a near-Wronskian; that is a determinant of a matrix of size p+1, with each row being a derivative of the previous, except for the last column. It is an interesting open question to understand those equations fro…
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