We prove an estimate for the probability that a simple random walk in a simply connected subset A of Z^2 starting on the boundary exits A at another specified boundary point. The estimates are uniform over all domains of a given inradius. We apply these estimates to prove a conjecture of S. Fomin in 2001 concerning a relationship between crossing probabilities of loop-erased random walk and Brownian motion.
The infinite two-sided loop-erased random walk (LERW) is a measure on infinite self-avoiding walks that can be viewed as giving the law of the `middle part' of an infinite LERW loop going through 0 and infinity. In this note we derive expressions for transition probabilities for this model in dimensions two and up. In the plane, the formula can be further expressed in terms of a Laplacian with signed weights acting on certain discrete harmonic functions at the tips of the walk, and taking a determinant. The discrete harmonic functions are closely related to a discrete version of the complex sq…
Let $M_{n}$ be the length (number of steps) of the loop-erasure of a simple random walk up to the first exit from a ball of radius $n$ centered at its starting point. It is shown in [18] that there exists $β\in (1, \frac{5}{3}]$ such that $E (M_{n} )$ is of order $n^β$ in 3 dimensions. In the present article, we show that the Hausdorff dimension of the scaling limit of the loop-erased random walk in 3 dimensions is equal to $β$ almost surely.
In this paper we consider the natural random walk on a planar graph and scale it by a small positive number $δ$. Given a simply connected domain $D$ and its two boundary points $a$ and $b$, we start the scaled walk at a vertex of the graph nearby $a$ and condition it on its exiting $D$ through a vertex nearby $b$, and prove that the loop erasure of the conditioned walk converges, as $δ\downarrow 0$, to the chordal SLE$_{2}$ that connects $a$ and $b$ in $D$, provided that an invariance principle is valid for both the random walk and the dual walk of it.
Let X and Y be independent transient Markov chains on the same state space that have the same transition probabilities. Let L denote the ``loop-erased path'' obtained from the path of X by erasing cycles when they are created. We prove that if the paths of X and Y have infinitely many intersections a.s., then L and Y also have infinitely many intersections a.s.
We show how to compute the probabilities of various connection topologies for uniformly random spanning trees on graphs embedded in surfaces. As an application, we show how to compute the "intensity" of the loop-erased random walk in ${\mathbb Z}^2$, that is, the probability that the walk from (0,0) to infinity passes through a given vertex or edge. For example, the probability that it passes through (1,0) is 5/16; this confirms a conjecture from 1994 about the stationary sandpile density on ${\mathbb Z}^2$. We do the analogous computation for the triangular lattice, honeycomb lattice and ${\m…
We use the coupling technique to prove that there exists a loop-erasure of a plane Brownian motion stopped on exiting a simply connected domain, and the loop-erased curve is the reversal of a radial SLE$_2$ curve.
The fundamental solution of a pseudo-differential equation for functions defined on the $d$-fold product of the $p$-adic numbers, $\mathbb{Q}_p$, induces an analogue of the Wiener process in $\mathbb{Q}_p^d$. As in the real setting, the components are $1$-dimensional $p$-adic Brownian motions with the same diffusion constant and exponent as the original process. Asymptotic analysis of the conditional probabilities shows that the vector components are dependent for all time. Exit time probabilities for the higher dimensional processes reveal a concrete effect of the component dependency.
For each prime $p$, a diffusion constant together with a positive exponent specify a Vladimirov operator and an associated $p$-adic diffusion equation. The fundamental solution of this pseudo-differential equation gives rise to a measure on the Skorokhod space of $p$-adic valued paths that is concentrated on the paths originating at the origin. We calculate the first exit probabilities of paths from balls and estimate these probabilities for the brownian bridges.
Tight estimates of exit/containment probabilities are of particular importance in many control problems. Yet, estimating the exit/containment probabilities is non-trivial: even for linear systems (Ornstein-Uhlenbeck processes), the containment probability can be computed exactly for only some particular values of the system parameters. In this paper, we derive tight bounds on the containment probability for a class of nonlinear stochastic systems. The core idea is to compare the "pull strength" (how hard the deterministic part of the system dynamics pulls towards the origin) experienced by the…
We provide Large Deviation estimates for the bridge of a $d$-dimensional general diffusion process as the conditioning time tends to $0$ and apply these results to the evaluation of the asymptotics of its exit time probabilities. We are motivated by applications to numerical simulation, especially in connection with stochastic volatility models.
In this paper, we consider a diffusion process pertaining to a chain of distributed control systems with small random perturbation. The distributed control system is formed by n subsystems that satisfy an appropriate Hormander condition, i.e., the second subsystem assumes the random perturbation entered into the first subsystem, the third subsystem assumes the random perturbation entered into the first subsystem then was transmitted to the second subsystem and so on, such that the random perturbation propagates through the entire distributed control system. Note that the random perturbation en…
Let $X$ be the constrained random walk on ${\mathbb Z}_+^d$ representing the queue lengths of a stable Jackson network and $x$ its initial position. Let $τ_n$ be the first time the sum of the components of $X$ equals $n$. $p_n \doteq P_x(τ_n < τ_0)$ is a key performance measure for the queueing system represented by $X$, stability implies $p_n\rightarrow 0$ exponentially. Currently the only analytic method available to approximate $p_n$ is large deviations analysis, which gives the exponential decay rate of $p_n$. Finer results are available via rare event simulation. The present article devel…
We derive asymptotics for the quenched probability that a critical branching Brownian motion killed at a small rate in Poissonian obstacles exits a large domain. Results are formulated in terms of the solution to a semilinear partial differential equation with singular boundary conditions. The proofs depend on a quenched homogenization theorem for branching Brownian motion among mild obstacles.
The paper presents two results. The first one provides separate conditions for the upper and lower estimate of the distribution of the exit time from balls of a random walk on a weighted graph. The main result of the paper is that the lower estimate follows from the elliptic Harnack inequality. The second result is an off-diagonal lower bound for the transition probability of the random walk.
We consider the tail probabilities for Brownian exit time from a class of perturbed multi-strips in Euclidean plane. Under some assumptions we prove that the long stays in a perturbed multi-strip are more likely than those in a strip of the same width. This effect is related to the existence of the trapped modes in waveguides.
Let $X$ be the constrained random walk on ${\mathbb Z}_+^2$ having increments $(1,0)$, $(-1,1)$, $(0,-1)$ with jump probabilities $λ(M_k)$, $μ_1(M_k)$, and $μ_2(M_k)$ where $M$ is an irreducible aperiodic finite state Markov chain. The process $X$ represents the lengths of two tandem queues with arrival rate $λ(M_k)$, and service rates $μ_1(M_k)$, and $μ_2(M_k)$. We assume that the average arrival rate with respect to the stationary measure of $M$ is less than the average service rates, i.e., $X$ is assumed stable. Let $τ_n$ be the first time when the sum of the components of $X$ equals $n$ fo…
We study exit laws from large balls in $\mathbb{Z}^d$, $d\geq3$, of random walks in an i.i.d. random environment that is a small perturbation of the environment corresponding to simple random walk. Under a centering condition on the measure governing the environment, we prove that the exit laws are close to those of a symmetric random walk, which we identify as a perturbed simple random walk. We obtain bounds on total variation distances as well as local results comparing exit probabilities on boundary segments. As an application, we prove transience of the random walks in random environment. …
We find explicit formulas for the probabilities of general boundary visit events for planar loop-erased random walks, as well as connectivity events for branches in the uniform spanning tree. We show that both probabilities, when suitably renormalized, converge in the scaling limit to conformally covariant functions which satisfy partial differential equations of second and third order, as predicted by conformal field theory. The scaling limit connectivity probabilities also provide formulas for the pure partition functions of multiple $\mathrm{SLE}_κ$ at $κ=2$.
We consider open quantum walks on a graph, and consider the random variables defined as the passage time and number of visits to a given point of the graph. We study in particular the probability that the passage time is finite, the expectation of that passage time, and the expectation of the number of visits, and discuss the notion of recurrence for open quantum walks. We also study exit times and exit probabilities from a finite domain, and use them to solve Dirichlet problems and to determine harmonic measures. We consider in particular the case of irreducible open quantum walks. The result…
The conditions $(T)_γ,$ $γ\in (0,1),$ which have been introduced by Sznitman in 2002, have had a significant impact on research in random walk in random environment. Among others, these conditions entail a ballistic behaviour as well as an invariance principle. They require the stretched exponential decay of certain slab exit probabilities for the random walk under the averaged measure and are asymptotic in nature. The main goal of this paper is to show that in all relevant dimensions (i.e., $d \ge 2$), in order to establish the conditions $(T)_γ$, it is actually enough to check a correspondin…
Conditional splitting probabilities for hidden-state inference in drift-diffusive processes
Imperial College London（帝国理工学院）；Laboratoire de Physique de l’Ecole Normale Supérieure (LPENS)（巴黎高等师范学院物理实验室）；Max Planck Insitute for Dynamics and Self-Organization（马克斯·普朗克动力学与自组织研究所）
Let {M_n}_{n\ge 0}$ be a nonnegative Markov process with stationary transition probabilities. The quasistationary distributions referred to in this note are of the form Q_A(x) = lim_{n\to\infty} P(M_n \le x | M_0 \le A, M_1 \le A, ..., M_n \le A) . Suppose that $M_0$ has distribution $\Qb_A$ and define T_A^{Q_A} = \min\{n | M_n > A, n\ge 1\}, the first time when M_n exceeds A. We provide sufficient conditions for E T_A^{Q_A}$ to be an increasing function of A.
Consider a random walk in an i.i.d. uniformly elliptic environment in dimensions larger than one. In 2002, Sznitman introduced for each $γ\in(0,1)$ the ballisticity condition $(T)_γ$ and the condition $(T')$ defined as the fulfillment of $(T)_γ$ for each $γ\in(0,1)$. Sznitman proved that $(T')$ implies a ballistic law of large numbers. Furthermore, he showed that for all $γ\in (0.5,1)$, $(T)_γ$ is equivalent to $(T')$. Recently, Berger has proved that in dimensions larger than three, for each $γ\in (0,1)$, condition $(T)_γ$ implies a ballistic law of large numbers. On the other hand, Drewitz a…
We study simple random walk on the uniform spanning tree on Z^2 . We obtain estimates for the transition probabilities of the random walk, the distance of the walk from its starting point after n steps, and exit times of both Euclidean balls and balls in the intrinsic graph metric. In particular, we prove that the spectral dimension of the uniform spanning tree on Z^2 is 16/13 almost surely.
The Aldous--Broder algorithm provides a way of sampling a uniformly random spanning tree for finite connected graphs using simple random walk. Namely, start a simple random walk on a connected graph and stop at the cover time. The tree formed by all the first-entrance edges has the law of a uniform spanning tree. Here we show that the tree formed by all the last-exit edges also has the law of a uniform spanning tree. This answers a question of Tom Hayes and Cris Moore from 2010. The proof relies on a bijection that is related to the BEST theorem in graph theory. We also give other applications…
This paper concerns a random walk on a planar graph and presents certain estimates concerning the harmonic measures for the walk in a grid domain which estimates are useful for showing the convergence of a LERW (loop-erased random walk) to an SLE (stochastic Loewner evolution). We assume that the walk started at a fixed vertex of the graph satisfies the invariance principle as in Yadin and Yehudayoff [16] in which the convergence of LERW to a radial SLE is established in this setting. Our main concern is chordal case, where a random walk is started at a boundary vertex of a simply connected gr…
We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On ${\mathbb Z}^d$, RWDE are parameterized by a 2d-uplet of positive reals called weights. In this paper, we characterize for $d\ge 3$ the weights for which there exists an absolutely continuous invariant probability for the process viewed from the particle. We can deduce from this result and from [27] a complete description of the ballistic regime for $d\ge 3$.
We review some of the results that have been derived in the last years on conformal invariance, scaling limits and properties of some two-dimensional random curves. In particular, we describe the intuitive ideas that lead to the definition of the Schramm-Loewner evolutions SLE, we define these objects, study its various properties, show how to compute (probabilities, critical exponents) using SLE, relate SLE to planar Brownian motions (i.e. the determination of the critical exponents), planar self-avoiding walks, critical percolation, loop-erased random walks and uniform spanning trees.
We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On $\Z^d$, RWDE are parameterized by a $2d$-uplet of positive reals. We prove that for all values of the parameters, RWDE are transient in dimension $d\ge 3$. We also prove that the Green function has some finite moments and we characterize the finite moments. Our result is more general and applies for example to finitely generated symmetric transient Cayley graphs. In terms of reinforced r…
We are concerned with random walks on $\mathbb{Z}^d$, $d\geq 3$, in an i.i.d. random environment with transition probabilities $ε$-close to those of simple random walk. We assume that the environment is balanced in one fixed coordinate direction, and invariant under reflection in the coordinate hyperplanes. The invariance condition was used in Baur and Bolthausen (2014) as a weaker replacement of isotropy to study exit distributions. We obtain precise results on mean sojourn times in large balls and prove a quenched invariance principle, showing that for almost all environments, the random wal…
We study the probabilities with which chordal Schramm-Loewner Evolutions (SLE) visit small neighborhoods of boundary points. We find formulas for general chordal SLE boundary visiting probability amplitudes, also known as SLE boundary zig-zags or order refined SLE multi-point Green's functions on the boundary. Remarkably, an exact answer can be found to this important SLE question for an arbitrarily large number of marked points. The main technique employed is a spin chain-Coulomb gas correspondence between tensor product representations of a quantum group and functions given by Dotsenko-Fatee…
We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non-existence of moments for first-passage and last-exit times. In our proofs we also make use of estimates for hitting probabilities and large deviations bounds. Our results are more general than existing results in the literature, which consider only the case of sums of independent (typically, identically distributed) random variables. We do not assume the Markov property. Existing results that we generalize include a circle of i…
We consider a random walk in a uniformly elliptic i.i.d. random environment in $\mathbb Z^d$ for $d\ge 2$. It is believed that whenever the random walk is transient in a given direction it is necessarily ballistic. In order to quantify the gap which would be needed to prove this equivalence, several ballisticity conditions have been introduced. In particular, in 2001 and 2002, Sznitman defined the so called conditions $(T)$ and $(T')$. The first one is the requirement that certain unlikely exit probabilities from a set of slabs decay exponentially fast with their width $L$. The second one is t…
Let $S_n =X_1+\cdots +X_n$ be an irreducible random walk (r.w.) on the one dimensional integer lattice with zero mean, infinite variance and i.i.d. increments $X_n$. We obtain an upper and lower bounds of the potential function, $a(x)$, of $S_n$ in the form $a(x)\asymp x/m(x)$ under a reasonable condition on the distribution of $X_n$; we especially show that as $x\to\infty$ $$a(x) \asymp \frac{x}{m_-(x)} \quad\mbox{and}\quad \frac{a(-x)}{a(x)} \to 0 \quad\;\;\mbox{if}\quad \lim_{x\to +\infty} \frac{m_+(x)}{m_-(x)} =0,$$ where $m_\pm(x) = \int_0^xdy\int_y^\infty P[\pm X_1>u]du$ and $m=m_++m_-$.…
We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on $Z^d \times Z_+$. In dimensions $d>6$, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that the spectral dimension of the incipient infinite cluster is 4/3, and thereby prove a version of the Alexander--Orbach conjecture in this setting. The proof divides into two parts. One part establishes general estimates for simple random walk on an arbitrary infinite random graph, given suitable bounds on volume and effective resi…
In a number of applications, particularly in financial and actuarial mathematics, it is of interest to characterize the tail distribution of a random variable $V$ satisfying the distributional equation $V\stackrel{\mathcal{D}}{=}f(V)$, where $f(v)=A\max\{v,D\}+B$ for $(A,B,D)\in(0,\infty)\times {\mathbb{R}}^2$. This paper is concerned with computational methods for evaluating these tail probabilities. We introduce a novel importance sampling algorithm, involving an exponential shift over a random time interval, for estimating these rare event probabilities. We prove that the proposed estimator…
