Study of random networks generally requires the nodes to be independently and uniformly distributed such as a Poisson point process. In this work, we venture beyond this standard paradigm and investigate a stochastic forest obtained from a drainage network model constructed on a randomly perturbed subset of $\mathbb{Z}^2$, where both horizontal and vertical perturbations are given by exponentially decaying unbounded discrete random variables and vertical perturbations are allowed in the upward direction only. We show that the resultant stochastic network is a single tree a.s. We further establish that as a collection of paths, under diffusive scaling the resultant network converges to the Brownian web.
We consider the tributary structure of Howard's drainage model studied by Gangopadhyay et. al. Conditional on the event that the tributary survives up to time $n$, we show that, as a sequence of random metric spaces, scaled tributary converges in distribution to a continuum random tree with respect to Gromov Hausdorff topology. This verifies a prediction made by Aldous for a simpler model (where paths are independent till they coalesce) but for a different conditional set up. The limiting continuum tree is slightly different from what was surmised earlier. Our proof uses the fact that there ex…
We introduce a system of one-dimensional coalescing nonsimple random walks with long range jumps allowing crossing paths and exibiting dependence before coalescence. We show that under diffusive scaling this system converges in distribution to the Brownian Web.
We consider a linearized dynamical system modelling the flow rate of water along the rivers and hillslopes of an arbitrary watershed. The system is perturbed by a random rainfall in the form of a compound Poisson process. The model describes the evolution, at daily time scales, of an interconnected network of linear reservoirs and takes into account the differences in flow celerity between hillslopes and streams as well as their spatial variation. The resulting stochastic process is a piece-wise deterministic Markov process of the Orstein-Uhlembeck type. We provide an explicit formula for the …
Hack [Studies of longitudinal stream profiles in Virginia and Maryland (1957). Report], while studying the drainage system in the Shenandoah valley and the adjacent mountains of Virginia, observed a power law relation $l\sim a^{0.6}$ between the length $l$ of a stream from its source to a divide and the area $a$ of the basin that collects the precipitation contributing to the stream as tributaries. We study the tributary structure of Howard's drainage network model of headward growth and branching studied by Gangopadhyay, Roy and Sarkar [Ann. Appl. Probab. 14 (2004) 1242-1266]. We show that th…
We study a system of coalescing random walks on the integer lattice $\mathbb{Z}^{d}$ in which the walk is oriented in the $d$-th direction and follows certain specified rules. We first study the geometry of the paths and show that, almost surely, the paths from a graph consisting of just one tree for dimentions $d=2,3$ and infinitely many disjoint trees for dimensions $d\geq 4$. Also, there is no bi-infinite path in the graph almost surely for $d\geq 2$. Subsequently, we prove that for $d=2$ the diffusive scaling of this system converges in distribution to the Brownian web.
Walsh's Brownian Motion and Donsker Scaling Limits of Perturbed Random Walks
Institute of Mathematics, Friedrich Schiller University Jena（耶拿弗里德里希·席勒大学数学研究所）；Institute of Mathematics, National Academy of Sciences of Ukraine（乌克兰国家科学院数学研究所）；National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”（乌克兰国立技术大学“伊戈尔·西科尔斯基基辅理工学院”）
We study a limit behavior of a sequence of Markov processes (or Markov chains) such that their distributions outside of any neighborhood of a "singular" point attract to some probability law. In any neighborhood of this point the behavior may be irregular. As an example of the general result we consider a symmetric random walk with the unit jump that is perturbed in a neighborhood of 0. The invariance principle is obtained under standard scaling of time and space. The limit process turns out to be a skew Brownian motion.
We consider the asymmetric zero range process in dimensions $d \geq 2$. Assume the initial density profile is a perturbation of the constant density, which has order $N^{-α}$, $α\in (0,1)$, and is constant along the drift direction. Here, $N$ is the scaling parameter. We show that under some constraints on the jump rate of the zero range process, the perturbed quantity macroscopically obeys the heat equation under diffusive scaling.
We consider the convex hull of the perturbed point process comprised of $n$ i.i.d. points, each distributed as the sum of a uniform point on the unit sphere $§^{d-1}$ and a uniform point in the $d$-dimensional ball centered at the origin and of radius $n^α, α\in (-\infty, \infty)$. This model, inspired by the smoothed complexity analysis introduced in computational geometry \cite{DGGT,ST}, is a perturbation of the classical random polytope. We show that the perturbed point process, after rescaling, converges in the scaling limit to one of five Poisson point processes according to whether $α$ b…
We study the stochastic diffusive limit of a kinetic radiative transfer equation, which is non-linear, involving a small parameter and perturbed by a smooth random term. Under an appropriate scaling for the small parameter, using a generalization of the perturbed test-functions method, we show the convergence in law to a stochastic non-linear fluid limit.
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 the modified Surface Quasi-Geostrophic (mSQG) equation on the 2D torus $\mathbb{T}^2$, perturbed by multiplicative transport noise. The equation admits the white noise measure on $\mathbb{T}^2$ as the invariant measure. We first prove the existence of white noise solutions to the stochastic equation via the method of point vortex approximation, then, under a suitable scaling limit of the noise, we show that the solutions converge weakly to the unique stationary solution of the dissipative mSQG equation driven by space-time white noise. The weak uniqueness of the latter equation is …
We describe scaling limits of recurrent excited random walks (ERWs) on integers in i.i.d. cookie environments with a bounded number of cookies per site. We allow both positive and negative excitations. It is known that ERW is recurrent if and only if the expected total drift per site, delta, belongs to the interval [-1,1]. We show that if |delta|<1 then the diffusively scaled ERW under the averaged measure converges to a (delta,-delta)-perturbed Brownian motion. In the boundary case, |delta|=1, the space scaling has to be adjusted by an extra logarithmic term, and the weak limit of ERW happens…
We explore the relation between fast waves, damping and imposed noise for different scalings by considering the singularly perturbed stochastic nonlinear wave equations νu_{tt}+u_t=\D u+f(u)+ν^α\dot{W} on a bounded spatial domain. An asymptotic approximation to the stochastic wave equation is constructed by a special transformation and splitting of $νu_{t}$. This splitting gives a clear description of the structure of $u$. The approximating model, for small $ν>0$\,, is a stochastic nonlinear heat equation for exponent $0\leqα<1$\,, and is a deterministic nonlinear wave equation for exponent $α…
We consider one-dimensional excited random walks (ERWs) with i.i.d. markovian cookie stacks in the non-boundary recurrent regime. We prove that under diffusive scaling such an ERW converges in the standard Skorokhod topology to a multiple of Brownian motion perturbed at its extrema (BMPE). All parameters of the limiting process are given explicitly in terms of those of the cookie markov chain at a single site. While our results extend the results of Dolgopyat and Kosygina (2012, ERWs with boundedly many cookies per site) and Kosygina and Peterson (2016, ERWs with periodic cookie stacks), the a…
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…
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 review, an outline of the so called Freidlin-Wentzell theory and its recent extensions is given. Broadly, this theory studies the exponential rate at which the probabilities of rare events related to random perturbation of ODE decays. The typical situation is when an ODE has several stable equilibria, in which case, the theory predicts the most likely paths in which the randomly perturbed system goes from one equilibria to another. In recent developments I will outline how recent approaches allows to distinguish between paths that are otherwise exponentially equivalent and provide an o…
An averaging method is applied to derive effective approximation to the following singularly perturbed nonlinear stochastic damped wave equation νu_{tt}+u_t=\D u+f(u)+ν^α\dot{W} on an open bounded domain $D\subset\R^n$\,, $1\leq n\leq 3$\,. Here $ν>0$ is a small parameter characterising the singular perturbation, and $ν^α$\,, $0\leq α\leq 1/2$\,, parametrises the strength of the noise. Some scaling transformations and the martingale representation theorem yield the following effective approximation for small $ν$, u_t=\D u+f(u)+ν^α\dot{W} to an error of $\ord{ν^α}$\,.
We study the stochastic fractional diffusive limit of a kinetic equation involving a small parameter and perturbed by a smooth random term. Generalizing the method of perturbed test functions, under an appropriate scaling for the small parameter, and with the moment method used in the deterministic case, we show the convergence in law to a stochastic fluid limit involving a fractional Laplacian.
The main purpose of the present paper is to establish a link between quadrature surfaces (potential theoretic concept) and sandpile dynamics (Laplacian growth models). For this aim, we introduce a new model of Laplacian growth on the lattice $\mathbb{Z}^d$ $(d\geq 2)$ which continuously deforms occupied regions of the \emph{divisible sandpile} model of Levine and Peres, by redistributing the total mass of the system onto $\frac 1m$-sub-level sets of the odometer which is a function counting total emissions of mass from lattice vertices. In free boundary terminology this goes in parallel with s…
We study the evolution in equilibrium of the fluctuations for the conserved quantities of a chain of anharmonic oscillators in the hyperbolic space-time scaling. Boundary conditions are determined by applying a constant tension at one side, while the position of the other side is kept fixed. The Hamiltonian dynamics is perturbed by random terms conservative of such quantities. We prove that these fluctuations evolve macroscopically following the linearized Euler equations with the corresponding boundary conditions, even in some time scales larger than the hyperbolic one.
We consider point vortex systems on the two dimensional torus perturbed by environmental noise. It is shown that, under a suitable scaling of the noises, weak limit points of the empirical measures are solutions to the vorticity formulation of deterministic 2D Navier-Stokes equations.
In this article, we prove that a small random perturbation of dynamical system with multiple stable equilibria converges to a Markov chain whose states are neighborhoods of the deepest stable equilibria, under a suitable time-rescaling, provided that the perturbed dynamics is reversible in time. Such a result has been anticipated from 1970s, when the foundation of mathematical treatment for this problem has been established by Freidlin and Wentzell. We solve this long-standing problem by reducing the entire analysis to an investigation of the solution of an associated Poisson equation, and fur…
We consider a chain of $n$ coupled oscillators placed on a one-dimensional lattice with periodic boundary conditions. The interaction between particles is determined by a weakly anharmonic potential $V_n = r^2/2 + σ_nU(r)$, where $U$ has bounded second derivative and $σ_n$ vanishes as $n \to \infty$. The dynamics is perturbed by noises acting only on the positions, such that the total momentum and length are the only conserved quantities. With relative entropy technique, we prove for dynamics out of equilibrium that, if $σ_n$ decays sufficiently fast, the fluctuation field of the conserved qua…
