Enrico Valdinoci
This note illustrates how a simple random walk with possibly long jumps is related to fractional powers of the Laplace operator. The exposition is elementary and self-contained.
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We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates.
We study random walk with unbounded jumps in random environment. The environment is stationary and ergodic, uniformly elliptic and decays polynomially with speed $Dj^{-(3+\varepsilon_0)}$ for some small $\varepsilon_0>0$ and proper $D>0.$ We prove a law of large number with positive velocity under the condition that the annealed mean of the hitting time of the positive half lattice is finite. Secondly, we consider birth and death process with bounded jumps in stationary and ergodic environment. Under the uniformly elliptic condition, we prove a law of large number and give the explicit formula…
In this paper we prove that under certain assumptions the transient random walk in random environment with bounded jumps (in $\mathbb{Z}$) grows much slower than the speed $n$. Precisely, there is $0<s<1$, such that although $X_n\rto$ we have $\frac{X_n}{n^{s'}}\rightarrow 0$ for $0<s<s'$ almost surely.
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched conditional invariance principle for the random walk, under the condition that it remains positive until time $n$. As a corollary of this result, we study the effect of conditioning the random walk to exceed level $n$ before returning to 0 as $n\to \infty$.
A zero-one law for random walks in random environments on $\mathbb{Z}^2$ with bounded jumps
本文将两类最近邻随机环境中随机游走的方向暂留性结果推广到有界跳跃情形，包括Zerner-Merkl的平面0-1律在Z^2非平面独立同分布环境中的推广，以及Sabot-Tournier的Dirichlet环境方向暂留性刻画在有界跳跃下的推广。
We consider a two dimensional reflecting random walk on the nonnegative integer quadrant. This random walk is assumed to be skip free in the direction to the boundary of the quadrant, but may have unbounded jumps in the opposite direction, which are referred to as upward jumps. We are interested in the tail asymptotic behavior of its stationary distribution, provided it exists. Assuming the upward jump size distributions have light tails, we completely find the rough tail asymptotics of the marginal stationary distributions in all directions. This generalizes the corresponding results for the …
In this paper, we first form a method to calculate the probability generating function of the total progeny of multitype branching process. As examples, we calculate probability generating function of the total progeny of the multitype branching processes within random walk which could stay at its position and (2-1) random walk. Consequently, we could give the probability generating functions and the distributions of the hitting time of corresponding random walks.
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched \textit{uniform} invariance principle for the random walk. This means that the rescaled trajectory of length $n$ is (in a certain sense) close enough to the Brownian motion, uniformly with respect to the choice of the starting location in an interval of length $O(\sqrt{n})$ around the origin.
We characterize ballistic behavior for general i.i.d. random walks in random environments on $\mathbb{Z}$ with bounded jumps. The two characterizations we provide do not use uniform ellipticity conditions. They are natural in the sense that they both relate to formulas for the limiting speed in the nearest-neighbor case. Note: This paper duplicates results from some versions of the preprint "Random walks in Dirichlet random environments on $\mathbb{Z}$ with bounded jumps." (arxiv: 2104.14950). The present paper is being split off for reasons of length, and the plan is to remove these results f…
In \cite{SzT}, D. Szász and A. Telcs have shown that for the diffusively scaled, simple symmetric random walk, weak convergence to the Brownian motion holds even in the case of local impurities if $d \ge 2$. The extension of their result to finite range random walks is straightforward. Here, however, we are interested in the situation when the random walk has unbounded range. Concretely we generalize the statement of \cite{SzT} to unbounded random walks whose jump distribution belongs to the domain of attraction of the normal law. We do this first: for diffusively scaled random walks on $\math…
We consider random walks indexed by arbitrary finite random or deterministic trees. We derive a simple sufficient criterion which ensures that the maximal displacement of the tree-indexed random walk is determined by a single large jump. This criterion is given in terms of four quantities : the tail and the expectation of the random walk steps, the height of the tree and the number of its vertices. The results are applied to critical Galton--Watson trees with offspring distributions in the domain of attraction of a stable law.
The purpose of these notes is to clarify the duality between a natural class of jump processes on compact ultrametric spaces - studied in current work of Bendikov, Girgor'yan and Pittet - and nearest neighbour walks on trees. Processes of this type have appeared in recent work of Kigami. Every compact ultrametric space arises as the boundary of a locally finite tree. The duality arises via the Dirichlet forms: one on the tree associated with a random walk and the other on the boundary of the tree, which is given in terms of the Naïm kernel. Here, it is explained that up to a linear time change…
We study the relaxation time in the random walk with jumps. The random walk with jumps combines random walk based sampling with uniform node sampling and improves the performance of network analysis and learning tasks. We derive various conditions under which the relaxation time decreases with the introduction of jumps.
We examine a class of random walks in random environments on $\mathbb{Z}$ with bounded jumps, a generalization of the classic one-dimensional model. The environments we study have i.i.d. transition probability vectors drawn from Dirichlet distributions. For this model, we characterize recurrence and transience, and in the transient case we characterize ballisticity. For ballisticity, we give two parameters, $κ_0$ and $κ_1$. The parameter $κ_0$ governs finite trapping effects, and $κ_1$ governs repeated traversals of arbitrarily large regions of the graph. We show that the walk is right-transie…
A {\it scenery} is a coloring $ξ$ of the integers. Let $\{S_t\}_{t\geq 0}$ be a recurrent random walk on the integers. Observing the scenery $ξ$ along the path of this random walk, one sees the color $χ_t:=ξ(S_t)$ at time $t$. The {\it scenery reconstruction problem} is concerned with recovering the scenery $ξ$, given only the sequence of observations $χ:=(χ_t)_{t\geq 0}$. The scenery reconstruction methods presented to date require the random walk to have bounded increments. Here, we present a new approach for random walks with unbounded increments which works when the tail of the increment d…
Foss and Zachary (2003) and Foss, Palmowski and Zachary (2005) studied the probability of achieving a receding boundary on a time interval of random length by a random walk with a heavy-tailed jump distribution. They have proposed and developed a new approach that allows to generalise results of Asmussen (1998) onto the case of arbitrary stopping times and a wide class of nonlinear boundaries, and to obtain uniform results over all stopping times. In this paper, we consider a class of branching random walks with fading branching and obtain results on the tail asymptotics for the maximum of a b…
Level numbers preserving transformations on excursionsdefined by random walks with state dependent jump laws
本文研究了状态相关跳跃律随机游走游程定义的层号，计算了固定层号的游程数量并证明平移类可生成所有同层号游程，同时分析了层号在 Vervaat 和 Doob 变换下的行为。
We consider a variant of random walks on finite groups. At each step, we choose an element from a set of generators ("directions") uniformly, and an integer from a power law ("speed") distribution associated with the chosen direction. We show that if the finite group is nilpotent, the mixing time of this walk is of the same order of magnitude as the diameter of a suitable pseudo-metric, D(S, a), on the group, which depends only on the generators and speeds. Additionally, we give sharp bounds on the $\ell^2$-distance between the distribution of the position of the walker and the stationary dist…
In this paper we investigate the asymptotic properties of the wait-first and jump-first Lévy walk with rest, which is a generalization of standard jump-first and jump-first Lévy walk that assumes each waiting time in the model is a sum of two positive random variables. We investigate the asymptotic properties of the theses new-type waiting times. Next we use the previous results of this paper together with continuous mapping approach to establish the main result, which is a functional convergence in Skorokhod $\mathbb{J}_1$ topology for the Lévy walks with rests.
Continuous time random walks have random waiting times between particle jumps. We define the correlated continuous time random walks (CTRWs) that converge to fractional Pearson diffusions (fPDs). The jumps in these CTRWs are obtained from Markov chains through the Bernoulli urn-scheme model and Wright-Fisher model. The jumps are correlated so that the limiting processes are not Lévy but diffusion processes with non-independent increments. The waiting times are selected from the domain of attraction of a stable law
We analyse a random walk on the ring of integers mod $n$, which at each time point can make an additive `step' or a multiplicative `jump'. When the probability of making a jump tends to zero as an appropriate power of $n$ we prove the existence of a total variation pre-cutoff for this walk. In addition, we show that the process obtained by subsampling our walk at jump times exhibits a true cutoff, with mixing time dependent on whether the step distribution has zero mean.
In this paper we present a weak approximation scheme for BSDEs driven by a Wiener process and an (in)finite activity Poisson random measure with drivers that are general Lipschitz functionals of the solution of the BSDE. The approximating backward stochastic difference equations (BSΔEs) are driven by random walks that weakly approximate the given Wiener process and Poisson random measure. We establish the weak convergence to the solution of the BSDE and the numerical stability of the sequence of solutions of the BSΔEs. By way of illustration we analyse explicitly a scheme with discrete step-si…
Large deviations for the branching random walk with heavy-tailed associated random walk - a principle of one big jump
本文研究分支随机游走的大偏差问题，证明了在重尾关联随机游走条件下，通过大跳跃原理得到随机测度的渐进行为。
We consider a random walk in a stationary ergodic environment in $\mathbb Z$, with unbounded jumps. In addition to uniform ellipticity and a bound on the tails of the possible jumps, we assume a condition of strong transience to the right which implies that there are no "traps". We prove the law of large numbers with positive speed, as well as the ergodicity of the environment seen from the particle. Then, we consider Knudsen stochastic billiard with a drift in a random tube in ${\mathbb R}^d$, $d\geq 3$, which serves as environment. The tube is infinite in the first direction, and is a statio…
We propose a class of strongly efficient rare event simulation estimators for random walks and compound Poisson processes with a regularly varying increment/jump-size distribution in a general large deviations regime. Our estimator is based on an importance sampling strategy that hinges on the heavy-tailed sample path large deviations result recently established in Rhee, Blanchet, and Zwart (2016). The new estimators are straightforward to implement and can be used to systematically evaluate the probability of a wide range of rare events with bounded relative error. They are "universal" in the…
For Laplacian models in dimension $(1+1)$ we derive sample path large deviations for the profile height function, that is, we study scaling limits of Gaussian integrated random walks and Gaussian integrated random walk bridges perturbed by an attractive force towards the zero-level, called pinning. We study in particular the regime when the rate functions of the corresponding large deviation principles admit more than one minimiser, in our models either two, three, or five minimiser depending on the pinning strength and the boundary conditions. This study complements corresponding large deviat…
Laplacians and Random Walks on CW Complexes
本文在有限型自由G-CW复形的k胞腔上构造随机游走，建立其链算子与胞腔拉普拉斯算子的关系，并据此由返回概率恢复Novikov-Shubin不变量。
Let $N$ and $M$ be positive integers satisfying $1\le M\le N$, and let $0<p_0<p_1<1$. Define a process $\{X_n\}_{n=0}^\infty$ on $\mathbb{Z}$ as follows. At each step, the process jumps either one step to the right or one step to the left, according to the following mechanism. For the first $N$ steps, the process behaves like a random walk that jumps to the right with probability $p_0$ and to the left with probability $1-p_0$. At subsequent steps the jump mechanism is defined as follows: if at least $M$ out of the $N$ most recent jumps were to the right, then the probability of jumping to the …
We study the limiting occupation density process for a large number of critical and driftless branching random walks. We show that the rescaled occupation densities of $\lfloor sN\rfloor$ branching random walks, viewed as a function-valued, increasing process $\{g_{s}^{N}\}_{s\ge 0}$, converges weakly to a pure jump process in the Skorohod space $\mathbb D([0, +\infty), \mathcal C_{0}(\mathbb R))$, as $N\to\infty$. Moreover, the jumps of the limiting process consist of i.i.d. copies of an Integrated super-Brownian Excursion (ISE) density, rescaled and weighted by the jump sizes in a real-value…
We consider a modulated process S which, conditional on a background process X, has independent increments. Assuming that S drifts to -infinity and that its increments (jumps) are heavy-tailed (in a sense made precise in the paper), we exhibit natural conditions under which the asymptotics of the tail distribution of the overall maximum of S can be computed. We present results in discrete and in continuous time. In particular, in the absence of modulation, the process S in continuous time reduces to a Levy process with heavy-tailed Levy measure. A central point of the paper is that we make ful…
Asymmetric discrete random walk and drift-diffusion with unequal jump times, lengths, and probabilities
针对跳跃概率、时间和长度均不等的非对称离散随机游走，本文推导其一般理论表达式并通过模拟验证，结果可在等跳跃时间时退化为已有理论。
For $λ>0$, we define a $λ$-damped random walk to be a random walk that is started from a random vertex of a graph and stopped at each step with probability $\fracλ{1+λ}$, otherwise continued with probability $\frac{1}{1+λ}$. We use the Aldous-Broder algorithm (\cite{aldous, broder}) of generating a random spanning tree and the Matrix-tree theorem to relate the values of the characteristic polynomial of the Laplacian at $\pm λ$ and the stationary measures of the sets of nodes visited by $i$ independent $λ$-damped random walks for $i \in \N$. As a corollary, we obtain a new characterization of t…
We study a discrete-time random walk on the non-negative integers, such that when 0 is reached a jump occurs to an arbitrary location, with given probabilities. We obtain an asymptotic formula for the expected position at large times, in dependence on the jump probabilities and on the starting position. Our proof of this result displays the relevance of the spectral analysis of the transition operator associated to the stochastic process, both of its eigenvalues and of its resonances.
Sub-Gaussian estimates for the natural random walk is typical of many regular fractal graphs. Subordination shows that there exist heavy tailed jump processes whose jump indices are greater than or equal to two. However, the existing machinery used to prove heat kernel bounds for such heavy tailed random walks fail in this case. In this work we extend Davies' perturbation method to obtain transition probability bounds for these anomalous heavy tailed random walks. We prove global upper and lower bounds on the transition probability density that are sharp up to constants. An important feature o…
Consider a simple random walk on the integers with the following transition mechanism. At each site $x$, the probability of jumping to the right is $ω(x)\in[\frac12,1)$, until the first time the process jumps to the left from site $x$, from which time onward the probability of jumping to the right is $\frac12$. We investigate the transience/recurrence properties of this process in both deterministic and stationary, ergodic environments $\{ω(x)\}_{x\in Z}$. In deterministic environments, we also study the speed of the process.
A continuous time random walk (CTRW) is a random walk in which both spatial changes represented by jumps and waiting times between the jumps are random. The CTRW is coupled if a jump and its preceding or following waiting time are dependent random variables, respectively. The aim of this paper is to explain the occurrence of different limit processes for CTRWs with forward- or backward-coupling in Straka and Henry (2011) using marked point processes. We also establish a series representation for the different limits. The methods used also allow us to solve an open problem concerning residual o…
Quantitative stochastic homogenization for long-range random walks with critical jump index
本文研究了一类对称随机游走的随机同质化，其一步转移概率与|x−y|^{-d−2}成正比。在临界跳跃指数下，标量过程收敛于布朗运动，并给出了标量算子收敛速率。
Given a probability measure on a finitely generated group, its Martin boundary is a natural way to compactify the group using the Green function of the corresponding random walk. For finitely supported measures in hyperbolic groups, it is known since the work of Ancona and Gou{ë}zel-Lalley that the Martin boundary coincides with the geometric boundary. The goal of this paper is to weaken the finite support assumption. We first show that, in any non-amenable group, there exist probability measures with exponential tails giving rise to pathological Martin boundaries. Then, for probability measur…
Let $σ$ be a permutation of $\{0,\ldots,n\}$. We consider the Markov chain $X$ which jumps from $k\neq 0,n$ to $σ(k+1)$ or $σ(k-1)$, equally likely. When $X$ is at 0 it jumps to either $σ(0)$ or $σ(1)$ equally likely, and when $X$ is at $n$ it jumps to either $σ(n)$ or $σ(n-1)$, equally likely. We show that the identity permutation maximizes the expected hitting time of n, when the walk starts at 0. More generally, we prove that the hitting time of a random walk on a strongly connected $d$-directed graph is maximized when the graph is the line $[0,n]\cap\Z$ with $d-2$ self-loops at every verte…
In this note, we make explicit the law of the renormalized supercritical branching random walk, giving credit to a conjecture formulated in a previous works of the authors for a continuous analog of the branching random walk. Also, in the case of a branching random walk on a homogeneous tree, we express the law of the corresponding limiting renormalized Gibbs measures, confirming, in this discrete model, conjectures formulated by physicists about the Poisson-Dirichlet nature of the jumps in the limit, and precising the conjecture by giving the spatial distribution of these jumps.
Let $ξ_1$, $ξ_2,\ldots$ be i.i.d. random variables of zero mean and finite variance and $η_1$, $η_2,\ldots$ positive i.i.d. random variables whose distribution belongs to the domain of attraction of an $α$-stable distribution, $α\in (0,1)$. The two collections are assumed independent. We consider a Markov chain with jumps of two types. If the present position of the Markov chain is positive, then the jump $ξ_k$ occurs; if the present position of the Markov chain is nonpositive, then its next position is $η_j$. We prove a functional limit theorem for this Markov chain under Donsker's scaling. T…
Functional limit theorems for perturbed random walks
本文研究扰动随机游走的标度极限，在膜外跳跃为有限方差零均值时给出反射布朗运动等极限，并探讨稳定律吸引域情形下的可能极限。
Quenched local limit theorem for random conductance models with long-range jumps
该研究针对带任意长度跳跃的平稳遍历随机电导模型，利用弱抛物Harnack不等式与对角热核估计，建立了d≥2维整数格点上可逆随机游走的淬火局部极限定理，部分解决了相关开放问题，还得到带额外尾项的极大不等式与校正子的处处次线性性质。
Evolution problems with perturbed $1$-Laplacian type operators on random walk spaces
本文在随机游走空间框架下研究含受扰动 $1$-Laplacian 型算子的演化问题，分析泛函在两种随机游走结构或其划分上呈现不同增长性的情形。
For a given one-dimensional random walk $\{S_n\}$ with a subexponential step-size distribution, we present a unifying theory to study the sequences $\{x_n\}$ for which $\mathsf{P}\{S_n>x\}\sim n\mathsf{P}\{S_1>x\}$ as $n\to\infty$ uniformly for $x\ge x_n$. We also investigate the stronger "local" analogue, $\mathsf{P}\{S_n\in(x,x+T]\}\sim n\mathsf{P}\{S_1\in(x,x+T]\}$. Our theory is self-contained and fits well within classical results on domains of (partial) attraction and local limit theory. When specialized to the most important subclasses of subexponential distributions that have been stud…
The $N$-particle branching random walk is a discrete time branching particle system with selection. We have $N$ particles located on the real line at all times. At every time step each particle is replaced by two offspring, and each offspring particle makes a jump of non-negative size from its parent's location, independently from the other jumps, according to a given jump distribution. Then only the $N$ rightmost particles survive; the other particles are removed from the system to keep the population size constant. Inspired by work of J. Bérard and P. Maillard, we examine the long term behav…
For a symmetric random walk in $Z^2$ which does not necessarily have bounded jumps we study those points which are visited an unusually large number of times. We prove the analogue of the Erdős-Taylor conjecture and obtain the asymptotics for the number of visits to the most visited site. We also obtain the asymptotics for the number of points which are visited very frequently by time $n$. Among the tools we use are Harnack inequalities and Green's function estimates for random walks with unbounded jumps; some of these are of independent interest.
The paper consists of two parts. In the first part we review recent work on limit theorems for random walks in random environment (RWRE) on a strip with jumps to the nearest layers. In the second part, we prove the quenched Local Limit Theorem (LLT) for the position of the walk in the transient diffusive regime. This fills an important gap in the literature. We then obtain two corollaries of the quenched LLT. The first one is the annealed version of the LLT on a strip. The second one is the proof of the fact that the distribution of the environment viewed from the particle (EVFP) has a limit f…
We study one-dimensional excited random walks with non-nearest neighbor jumps. When the process is at a vertex that has not been visited before, its next transition has a positive drift to the right, possibly with long jumps. Whenever the process visits a vertex that has already been visited in the past, its next transition is the one of a simple symmetric random walk. We give a sufficient condition for the process to have positive speed.
We study Markovian continuous-time random walk models for Lévy flights and we show an example in which the convergence to stable densities is not guaranteed when jumps follow a bi-modal power-law distribution that is equal to zero in zero. The significance of this result is two-fold: i) with regard to the probabilistic derivation of the fractional diffusion equation and also ii) with regard to the concept of site fidelity in the framework of Lévy-like motion for wild animals.
We consider weighted graphs satisfying sub-Gaussian estimate for the natural random walk. On such graphs, we study symmetric Markov chains with heavy tailed jumps. We establish a threshold behavior of such Markov chains when the index governing the tail heaviness (or jump index) equals the escape time exponent (or walk dimension) of the sub-Gaussian estimate. In a certain sense, this generalizes the classical threshold corresponding to the second moment condition.
In this paper, we reveal the branching structure for a non-homogeneous random walk with bounded jumps. The ladder time $T_1,$ the first hitting time of $[1,\infty)$ by the walk starting from $0,$ could be expressed in terms of a non-homogeneous multitype branching process. As an application of the branching structure, we prove a law of large numbers of random walk in random environment with bounded jumps and specify the explicit invariant density for the Markov chain of ``the environment viewed from the particle" .The invariant density and the limit velocity could be expressed explicitly in te…
A particle moves randomly over the integer points of the real line. Jumps of the particle outside the membrane (a fixed "locally perturbating set") are i.i.d., have zero mean and finite variance, whereas jumps of the particle from the membrane have other distributions with finite means which may be different for different points of the membrane; furthermore, these jumps are mutually independent and independent of the jumps outside the membrane. Assuming that the particle cannot jump over the membrane we prove that the weak scaling limit of the particle position is a skew Brownian motion with p…
In this note, we consider random walks in the quarter plane with arbitrary big jumps. We announce the extension to that class of models of the analytic approach of [G. Fayolle, R. Iasnogorodski, and V. Malyshev, Random walks in the quarter plane, Springer-Verlag, Berlin (1999)], initially valid for walks with small steps in the quarter plane. New technical challenges arise, most of them being tackled in the framework of generalized boundary value problems on compact Riemann surfaces.
Mixing of fast random walks on dynamic random permutations
研究动态随机排列上快速随机游走的混合行为，证明全变差距离在缩放后以单次跳跃下降，跳跃时间分布和跳跃后衰减函数与Erdős-Rényi图最大分量相关。
We consider a random walk on top of the contact process on $\mathbb{Z}^d$ with $d\geq 1$. In particular, we focus on the "contact process as seen from the random walk". Under the assumption that the infection rate of the contact process is large or the jump rate of the random walk is small, we show that this process has at most two extremal measures. Moreover, the convergence to these extremal measures is characterised by whether the contact process survives or dies out, similar to the complete convergence theorem known for the ordinary contact process. Using this, we furthermore provide a law…
We define the probability structure of a continuous-time time-homogeneous Markov jump process, on a finite graph, that represents the continuous-time counterpart of the so-called Ruelle-Bowen discrete-time random walk. It constitutes the unique jump process having maximal entropy rate. Moreover, it has the property that, given the number of jumps between any two specified end-points on the graph, the probability of traversing any one of the alternative paths that are consistent with the specified number of jumps and end-points, is the same for all, and thereby depends only on the number of jum…
We consider operators of the form $L u(x) = \sum_{y \in \mathbb{Z}} k(x-y) \big( u(y) - u(x)\big)$ on the one-dimensional lattice with symmetric, integrable kernel $k$. We prove several results stating that under certain conditions on the kernel the operator $L$ satisfies the curvature-dimension condition $CD_Υ(0,F)$ (recently introduced by two of the authors) with some $CD$-function $F$, where attention is also paid to the asymptotic properties of $F$ (exponential growth at infinity and power-type behaviour near zero). We show that $CD_Υ(0,F)$ implies a Li-Yau inequality for positive solution…
We prove that the only nearest neighbor jump process with local dependence on the occupation times satisfying the partial exchangeability property is the vertex reinforced jump process, under some technical conditions. This result gives a counterpart to the characterization of edge reinforced random walk given by Rolles.
We consider a discrete time simple symmetric random walk on Z^d, d>=1, where the path of the walk is perturbed by inserting deterministic jumps. We show that for any time n and any deterministic jumps that we insert, the expected number of sites visited by the perturbed random walk up to time n is always larger than or equal to that for the unperturbed walk. This intriguing problem arises from the study of a particle among a Poisson system of moving traps with sub-diffusive trap motion. In particular, our result implies a variant of the Pascal principle, which asserts that among all determinis…
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