Theo van Uem
We obtain expected number of arrivals, absorption probabilities and expected time before absorption for a discrete random walk on the integers with an infinite set of equidistant multiple function barriers
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We continue the line of research of random walks with barrier initiated by Iksanov and M{ö}hle (2008). Assuming that the tail of the step of the underlying random walk has a power-like behavior at infinity with exponent $-α$, $α\in(0,1)$, we prove that the number $V_n$ of zero increments in the random walk with barrier, properly centered and normalized, converges weakly to the standard normal law. This refines previously known weak law of large numbers for $V_n$ proved in Iksanov and Negadailov (2008).
We consider a branching random walk with an absorbing barrier, where the step of the associated one-dimensional random walk is in the domain of attraction of an $α$-stable law with $1<α<2$. We shall prove that there is a barrier $an^{\frac{1}{1+α}}$ and a critical value $a_α$ such that if $a<a_α$, then the process dies; if $a>a_α$, then the process survives. The results generalize previous results in literature for the case $α=2$.
The extinction rate of a branching random walk with a barrier in a time-inhomogeneous random environment
本文研究时间非齐次随机环境中带随机吸收屏障的超临界分支随机游走，给出屏障导致系统灭绝时在 $L^p$ 意义下的灭绝速率，并证明随机环境下随机游走小偏差概率的 $L^p$ 收敛性。
We study a branching random walk on $\r$ with an absorbing barrier. The position of the barrier depends on the generation. In each generation, only the individuals born below the barrier survive and reproduce. Given a reproduction law, Biggins et al. \cite{BLSW91} determined whether a linear barrier allows the process to survive. In this paper, we refine their result: in the boundary case in which the speed of the barrier matches the speed of the minimal position of a particle in a given generation, we add a second order term $a n^{1/3}$ to the position of the barrier for the $n^\mathrm{th}$ g…
We define the reflection of a random walk at a general barrier and derive, in case the increments are light tailed and have negative mean, a necessary and sufficient criterion for the global maximum of the reflected process to be finite a.s. If it is finite a.s., we show that the tail of the distribution of the global maximum decays exponentially fast and derive the precise rate of decay. Finally, we discuss an example from structural biology that motivated the interest in the reflection at a general barrier.
In this paper, we study the scaling limit of a class of random walks which behave like simple random walks outside of a bounded region around the origin and which are subject to a partial reflection near the origin. If the probability of crossing the barrier scales as $ 1/\sqrt{n} $ as we rescale space by $ \sqrt{n} $ and time by $ n $, we obtain a non trivial scaling limit which behaves like reflected Brownian motion until its local time at the origin reaches an independent exponential variable. It then follows reflected Brownian motion on the other side of the origin until its local time at …
We give sharp, uniform estimates for the probability that a random walk of n steps on the reals avoids a half-line [y,infinity) given that it ends at the point x. The estimates hold for general continuous or lattice distributions provided the 4th moment is finite.
Asymptotic behavior for the quenched survival probability of a supercritical branching random walk in random environment with a barrier
该研究针对带随机屏障的独立同分布随机环境中超临界分支随机游走，在可积条件下证明淬灭生存概率的对数依概率、几乎处处或依$L^p$收敛于依赖环境的显式负常数，将已有结果拓展至随机环境情形。
On the barrier problem of branching random walk in time-inhomogeneous random environment
本文研究带随机吸收障碍的时间非齐次随机环境分支随机游走，给出以 α 和临界常数 a_c 划分的灭绝与存活判据及灭绝速率，将既有确定性结果推广至随机环境。
We study the $L^1$-approximation of the log-Heston SDE at the terminal time point by arbitrary methods that use an equidistant discretization of the driving Brownian motion. We show that such methods can achieve at most order $ \min \{ ν, \tfrac{1}{2} \}$, where $ν$ is the Feller index of the underlying CIR process. As a consequence Euler-type schemes are optimal for $ν\geq 1$, since they have convergence order $\tfrac{1}{2}-ε$ for $ε>0$ arbitrarily small in this regime.
We study how to sample paths of a random walk up to the first time it crosses a fixed barrier, in the setting where the step sizes are iid with negative mean and have a regularly varying right tail. We introduce a desirable property for a change of measure to be suitable for exact simulation. We study whether the change of measure of Blanchet and Glynn (2008) satisfies this property and show that it does so if and only if the tail index $α$ of the right tail lies in the interval $(1, \, 3/2)$.
Random walk with barriers on a graph
该研究针对带多个函数势垒的图上非对称离散随机游走，推导了期望到达数、吸收概率与吸收前期望时间，且为图的边和顶点势垒定义了特定概率。
The continuous-time random walk is defined as a Poissonization of discrete-time random walk. We study the noncolliding system of continuous-time simple and symmetric random walks on ${\mathbb{Z}}$. We show that the system is determinantal for any finite initial configuration without multiple point. The spatio-temporal correlation kernel is expressed by using the modified Bessel functions. We extend the system to the noncolliding process with an infinite number of particles, when the initial configuration has equidistant spacing of particles, and show a relaxation phenomenon to the equilibrium …
We consider the branching random walk in random environment with a random absorption wall. When we add this barrier, we discuss some topics related to the survival probability. We assume that the random environment is i.i.d., $S_i$ is a particular i.i.d. random walk depend on the random environment $\mathcal{L}$. Let the random barrier function (the random absorption wall) is $g_i(\mathcal{L}):=ai^α-S_i,$ where $i$ present the generation. We show that there exists a critical value $a_c>0$ such that if $a>a_c,α=\frac{1}{3}$, the survival probability is positive almost surly and if $a<a_c,α=\fra…
In this article we consider importance sampling (IS) and sequential Monte Carlo (SMC) methods in the context of 1-dimensional random walks with absorbing barriers. In particular, we develop a very precise variance analysis for several IS and SMC procedures. We take advantage of some explicit spectral formulae available for these models to derive sharp and explicit estimates; this provides stability properties of the associated normalized Feynman-Kac semigroups. Our analysis allows one to compare the variance of SMC and IS techniques for these models. The work in this article, is one of the few…
For the critical Galton--Watson process with geometric offspring distributions we provide sharp barrier estimates for barriers which are (small) perturbations of linear barriers. These are useful in analyzing the cover time of finite graphs in the critical regime by random walk, and the Brownian cover times of compact two dimensional manifolds. As an application of the barrier estimates, we prove that if $C_L$ denotes the cover time of the binary tree of depth $L$ by simple walk, then $\sqrt{C_L/2^{L+1}} -\sqrt{2\log 2} L+\log L/\sqrt{2\log 2}$ is tight. The latter improves results of Aldous (…
Least Squares as Random Walks
本文发现线性最小二乘法与基于数据样本的随机游走之间存在新联系，证明使残差数据游走净面积为零的直线斜率等同于最小二乘解，且对等距样本精确成立。
A Sharp Phase Transition for Killed Branching Random Walks with Heavy-Tailed Displacements
本文研究重尾位移分支随机游走在超线性杀死屏障下的存活相变，证明临界值为4而非一阶矩启发式的2，并分析临界值处的非确定性。
We consider a discrete random walk on a diagonal lattice in two and three dimensions and obtain explicit solutions of absorption probabilities and probabilities of return in several domains. In three dimensions we consider both the cube and the dodecahedron variant. In two dimensions we obtain explicit formula in case of rotated barriers.
Extremal regime for one-dimensional Mott variable-range hopping
该研究针对呈现严重阻塞的一维Mott随机游走的渐近行为展开，证明经适当重标度的游走位于两个环境可测势垒间且势垒位置有极值标度极限，同时给出了势垒间游走分布的渐近描述。
We consider a branching random walk on $\mathbb{R}$ with a killing barrier at zero. At criticality, the process becomes eventually extinct, and the total progeny $Z$ is therefore finite. We show that the tail distribution of $Z$ displays a typical behaviour in $(n\ln^2(n))^{-1}$, which confirms the prediction of Addario-Berry and Broutin.
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…
Application of simplest random walk algorithms for pricing barrier options
本文展示了一阶算法在障碍期权定价中的有效性，该算法基于弱欧拉近似和线性插值，易于实现且通用，无需触发概率知识。
Block and Residue Exclusion in Kernel-Root Expansions for Two-Barrier Walks
针对整数上步长为 +y 和 -b 的双 barrier 游走，提出两阶段修剪方法构造核根展开，在示例中有效排除大量非必要项，提升计算效率。
We consider a family X^{(n)}, n \in \bbN_+, of continuous-time nearest-neighbor random walks on the one dimensional lattice Z. We reduce the spectral analysis of the Markov generator of X^{(n)} with Dirichlet conditions outside (0,n) to the analogous problem for a suitable generalized second order differential operator -D_{m_n} D_x, with Dirichlet conditions outside a given interval. If the measures dm_n weakly converge to some measure dm_*, we prove a limit theorem for the eigenvalues and eigenfunctions of -D_{m_n}D_x to the corresponding spectral quantities of -D_{m_*} D_x. As second result,…
We consider a discrete random walk (RW) in n dimensions . The RW is adapted with a geometric absorption process: at any discrete time there is a constant probability that absorption occurs in the current state. To model the RW with geometric absorption we use the concept of a multiple function barrier (MFB). In a MFB there is a modification of the original RW: each transition probability in the original RW is multiplied by β and there is an additional probability (1-β) of absorption, where 0<β<1. We study three cases: one-dimensional simple asymmetric RW, n-dimensional simple symmetric RW (n>1…
We identify a fundamental phenomenon of heterogeneous one dimensional random walks: the escape (traversal) time is maximized when the heterogeneity in transition probabilities forms a pyramid-like potential barrier. This barrier corresponds to a distinct arrangement of transition probabilities, sometimes referred to as the pendulum arrangement. We reduce this problem to a sum over products, combinatorial optimization problem, proving that this unique structure always maximizes the escape time. This general property may influence studies in epidemiology, biology, and computer science to better …
True self-repelling motion above a general barrier
本文研究了在一般屏障上构造的真实自避运动，并探讨了其在不规则屏障下的证明扩展难题。
We develop the theory of the $W$ and $Z$ scale functions for right-continuous (upwards skip-free) discrete-time discrete-space random walks, along the lines of the analogue theory for spectrally negative Lévy processes. Notably, we introduce for the first time in this context the one and two-parameter scale functions $Z$, which appear for example in the joint problem of deficit at ruin and time of ruin, and in problems concerning the walk reflected at an upper barrier. Comparisons are made between the various theories of scale functions as one makes time and/or space continuous. The theory is …
We study the asymptotic behaviour of the probability that a weighted sum of centered i.i.d. random variables X_k does not exceed a constant barrier. For regular random walks, the results follow easily from classical fluctuation theory, while this theory does not carry over to weighted random walks, where essentially nothing seems to be known. First we discuss the case of a polynomial weight function and determine the rate of decay of the above probability for Gaussian X_k. This rate is shown to be universal over a larger class of distributions that obey suitable moment conditions. Finally we d…
Consider a branching random walk on the real line with a killing barrier at zero: starting from a nonnegative point, particles reproduce and move independently, but are killed when they touch the negative half-line. The population of the killed branching random walk dies out almost surely in both critical and subcritical cases, where by subcritical case we mean that the rightmost particle of the branching random walk without killing has a negative speed, and by critical case, when this speed is zero. We investigate the total progeny of the killed branching random walk and give their precise ta…
We obtain expected number of arrivals, probability of arrival, absorption probabilities and expected time before absorption for a modified discrete random walk on the (sub)set of integers. In a [pqrs] random walk the particle can move one step forward or backward, stay for a moment in the same state or it can be absorbed immediately in the current state. M[pqrs] is a modified version, where probabilities on both sides of a multiple function barrier M are of different [pqrs] type.
The maximal displacement of radially symmetric branching random walk in $\mathbb{R}^d$
本文研究径向对称分支随机游走的最大位移，证明在生存条件下其增长与分支布朗运动一致，主要工具为带移动障碍的投票定理。
We are interested in predator-prey dynamics on infinite trees, which can informally be seen as particular two-type branching processes where individuals may die (or be infected) only after their parent dies (or is infected). We study two types of such dynamics: the chase-escape process, introduced by Kordzakhia with a variant by Bordenave, and the birth-and-assassination process, introduced by Aldous & Krebs. We exhibit a coupling between these processes and branching random walks starting from a random point and killed at the barrier 0. This sheds new light on the chase-escape and birth-and-a…
We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift $c$. At the point $x > 0$, we add an absorbing barrier, i.e.\ individuals touching the barrier are instantly killed without producing offspring. It is known that there is a critical drift $c_0$, such that this process becomes extinct almost surely if and only if $c \ge c_0$. In this case, if $Z_x$ denotes the number of individuals absorbed at the barrier, we give an asymptotic for $P(Z_x=n)$ as $n$ goes to infinity. If $c=c_0$ and the reproduction is deterministic, this improves upo…
We study the noncolliding random walk (RW), which is a particle system of one-dimensional, simple and symmetric RWs starting from distinct even sites and conditioned never to collide with each other. When the number of particles is finite, $N < \infty$, this discrete process is constructed as an $h$-transform of absorbing RW in the $N$-dimensional Weyl chamber. We consider Fujita's polynomial martingales of RW with time-dependent coefficients and express them by introducing a complex Markov process. It is a complexification of RW, in which independent increments of its imaginary part are in th…
We consider stochastic differential equations driven by a general Lévy processes (SDEs) with infinite activity and the related, via the Feynman-Kac formula, Dirichlet problem for parabolic integro-differential equation (PIDE). We approximate the solution of PIDE using a numerical method for the SDEs. The method is based on three ingredients: (i) we approximate small jumps by a diffusion; (ii) we use restricted jump-adaptive time-stepping; and (iii) between the jumps we exploit a weak Euler approximation. We prove weak convergence of the considered algorithm and present an in-depth analysis of …
Do biological cells sense time by the number of their divisions, a process that ends at senescence? We consider the question "can the cell's perception of time be expressed through the length of the shortest telomere?" The answer is that the absolute time before senescence cannot be expressed by the telomere's length and that a cell can survive many more divisions than intuitively expected. This apparent paradox is due to shortening and elongation of the telomere, which suggests a random walk model of the telomere's length. The model indicates two phases, first, a determinist drift of the leng…
To implement the Wigner branching random walk, the particle carrying a signed weight, either $-1$ or $+1$, is more friendly to data storage and arithmetic manipulations than that taking a real-valued weight continuously from $-1$ to $+1$. The former is called a signed particle and the latter a weighted particle. In this paper, we propose two efficient strategies to realize the signed-particle implementation. One is to interpret the multiplicative functional as the probability to generate pairs of particles instead of the incremental weight, and the other is to utilize a bootstrap filter to adj…
Strategic geometry of competing first-passage random walks
该研究探讨竞争型首通随机游走博弈的策略几何，通过数学分析推导有限路径、特定图结构等场景下的最优策略与均衡，证明其与连续布朗运动博弈的收敛性。
Given a Lipschitz or smooth convex function $\, f:K \to \mathbb{R}$ for a bounded polytope $K \subseteq \mathbb{R}^d$ defined by $m$ inequalities, we consider the problem of sampling from the log-concave distribution $π(θ) \propto e^{-f(θ)}$ constrained to $K$. Interest in this problem derives from its applications to Bayesian inference and differentially private learning. Our main result is a generalization of the Dikin walk Markov chain to this setting that requires at most $O((md + d L^2 R^2) \times md^{ω-1}) \log(\frac{w}δ))$ arithmetic operations to sample from $π$ within error $δ>0$ in t…
Ballistic Speed and Potential-Theoretic Recurrence for Homogeneous Open Quantum Random Walks
本研究证明齐次开放量子随机游走中均方根弹道速度等于漂移范数，非零漂移导致非常返性，中心化游走在低维常返而高维暂态，并给出势核渐近与可约分解。
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