Dmitry Dolgopyat, Carlangelo Liverani
We consider a random walk with transition probabilities weakly dependent on an environment with a deterministic, but strongly chaotic, evolution. We prove that for almost all initial conditions of the environment the walk satisfies the CLT.
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Elephant Random Walk Conditioned to Avoid Zero
该研究探讨避免零值条件下的大象随机游走的长时间极限，将其嵌入三维贝塞尔过程并建立相关极限定理，丰富了随机游走领域的理论成果。
Random walk in slowly changing environments
针对变化环境中随机游走（RWCE）的常返暂留判定难题，提出适用于任意图的慢变化边权有界RWCE的判定条件，证明其常返/暂留性与初始加权图一致。
In this paper we introduce the notion of Random Walk in Changing Environment - a random walk in which each step is performed in a different graph on the same set of vertices, or more generally, a weighted random walk on the same vertex and edge sets but with different (possibly 0) weights in each step. This is a very wide class of RW, which includes some well known types of RW as special cases (e.g. reinforced RW, true SAW). We define and explore various possible properties of such walks, and provide criteria for recurrence and transience when the underlying graph is $\mathbb{N}$ or a tree. We…
As a strategy to complete games quickly, we investigate one-dimensional random walks where the step length increases deterministically upon each return to the origin. When the step length after the kth return equals k, the displacement of the walk x grows linearly in time. Asymptotically, the probability distribution of displacements is a purely exponentially decaying function of |x|/t. The probability E(t,L) for the walk to escape a bounded domain of size L at time t decays algebraically in the long time limit, E(t,L) ~ L/t^2. Consequently, the mean escape time <t> ~ L ln L, while <t^n> ~ L^{…
We analyze the long-term stability of a stochastic model designed to illustrate the adaptation of a population to variation in its environment. A piecewise-deterministic process modeling adaptation is coupled to a Feller logistic diffusion modeling population size. As the individual features in the population become further away from the optimal ones, the growth rate declines, making population extinction more likely. Assuming that the environment changes deterministically and steadily in a constant direction, we obtain the existence and uniqueness of the quasi-stationary distribution, the ass…
Recurrence and transience of random walks in monotonically changing environments
本文研究单调变化电导环境上的随机游走，证明电导单调递增时游走常返性/暂态性与极限环境一致，并给出单调递减时的反例，解决多个猜想。
Random walk on changing graphs is considered. For sequences of finite graphs increasing monotonically towards a limiting infinite graph, we establish transition probability upper bounds. It yields sufficient transience criteria for simple random walk on slowly growing graphs, upon knowing the volume and Cheeger constant of each graph. For much more specialized cases, we establish matching lower bounds, and deduce sufficient (weak) recurrence criteria. We also address recurrence directly in relation to a universality conjecture of [DHS]. We answer a related question of [SZ, Problem 1.8] about "…
On Elephant Random Walk with Random Memory
本文提出了一种具有随机记忆的大象随机游走模型，通过随机选取历史步骤构成记忆，并推导了其条件分布性质及平均增量与位移的递归和显式表达式。
Mixing cutoff for simple random walks on the Chung-Lu digraph
本文研究Chung-Lu有向图上简单随机游走的混合行为，证明在平均度随图大小对数增长的弱稠密状态下，总变差距离在熵时间处呈现截断行为且收敛于高斯尾函数，将已有硬约束模型的结果扩展至软约束模型。
Transience of continuous-time conservative random walks
本文研究连续时间保守随机游走的两种推广，证明在 d≥2 且方向改变速率满足特定幂律或对数衰减条件时，正交模型与 random flights 均为瞬态。
Vertex-Reinforced Random Walk (VRRW), defined by Pemantle (1988a), is a random process in a continuously changing environment which is more likely to visit states it has visited before. We consider VRRW on arbitrary graphs and show that on almost all of them, VRRW visits only finitely many vertices with a positive probability. We conjecture that on all graphs of bounded degree, this happens a.s., and provide a proof only for trees of this type. We distinguish between several different patterns of localization and explicitly describe the long-run structure of VRRW, which depends on whether a gr…
We propose a simple model of columnar growth through {\it diffusion limited aggregation} (DLA). Consider a graph $G_N\times\N$, where the basis has $N$ vertices $G_N:=\{1,\dots,N\}$, and two vertices $(x,h)$ and $(x',h')$ are adjacent if $|h-h'|\le 1$. Consider there a simple random walk {\it coming from infinity} which {\it deposits} on a growing cluster as follows: the cluster is a collection of columns, and the height of the column first hit by the walk immediately grows by one unit. Thus, columns do not grow laterally. We prove that there is a critical time scale $N/\log(N)$ for the maxima…
Renewal structure of the Tree Builder Random Walk
本文为树构建随机游走建立更新框架，并在独立同分布假设下证明距离的SLLN、LIL和不变性原理，且极限速度连续依赖于加叶分布。
We establish and generalise several bounds for various random walk quantities including the mixing time and the maximum hitting time. Unlike previous analyses, our derivations are based on rather intuitive notions of local expansion properties which allows us to capture the progress the random walk makes through $t$-step probabilities. We apply our framework to dynamically changing graphs, where the set of vertices is fixed while the set of edges changes in each round. For random walks on dynamic connected graphs for which the stationary distribution does not change over time, we show that the…
We will investigate a random mass splitting model and the closely related random walk in a random environment (RWRE). The heat kernel for the RWRE at time t is the mass splitting distribution at t. We prove a quenched invariance principle for the RWRE which gives us a quenched central limit theorem for the mass splitting model. Our RWRE has an environment which is changing with time. We follow the outline for proving a quenched invariant process for a random walk in a space-time random environment laid out by Rassoul-Agha and Seppäläinen which in turn was based on the work of Kipnis and Varadh…
The Fiedler vector of a graph, namely the eigenvector corresponding to the second smallest eigenvalue of a graph Laplacian matrix, plays an important role in spectral graph theory with applications in problems such as graph bi-partitioning and envelope reduction. Algorithms designed to estimate this quantity usually rely on a priori knowledge of the entire graph, and employ techniques such as graph sparsification and power iterations, which have obvious shortcomings in cases where the graph is unknown, or changing dynamically. In this paper, we develop a framework in which we construct a stoch…
Hierarchy of KPZ limits arising from directed random walk models in random media
本文研究了随机介质中定向随机游走模型的KPZ极限层次结构，通过削弱假设条件，证明了SHE在多种模型中的出现，并揭示了中等偏差指数的层次结构。
$K$-Branching Random Walk with Noisy Selection: Large Population Limits and Phase Transitions
本文研究带噪声选择的 $K$-分支随机游走，利用随机 Hopf-Cole 变换刻画大种群极限，并证明指数尾波动下唯一且局部稳定的行波解及其随选择压力变化的相变。
In this paper we consider a telegraph equation with time-dependent coefficients, governing the persistent random walk of a particle moving on the line with a time-varying velocity $c(t)$ and changing direction at instants distributed according to a non-stationary Poisson distribution with rate $λ(t)$. We show that, under suitable assumptions, we are able to find the exact form of the probability distribution. We also consider the space-fractional counterpart of this model, finding the characteristic function of the related process. A conclusive discussion is devoted to the potential applicatio…
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