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Register a new userScaling limit for a family of random paths with radial behavior
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We introduce a system of coalescing random paths with radialbehavior in a subsetof the plane. We call it theDiscrete Radial Poissonian Web. We show that underdiffusive scaling this family converges in distribution toa mapping of a restrictionof the Brownian Web.
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We consider the limit behavior of a one-dimensional random walk with unit jumps whose transition probabilities are modified every time the walk hits zero. The invariance principle is proved in the scheme of series where the size of modifications depends on the number of series. For the natural scaling of time and space arguments the limit process is (i) a Brownian motion if modifications are small, (ii) a linear motion with a random slope if modifications are large, and (iii) the limit process satisfies an SDE with a local time of unknown process in a drift if modifications are moderate.
We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We show that the system of particles, rescaled in an appropriate way, converges in distribution to a scaling limit that is interesting in its own right. We describe the limit object as a growing collection of lilypads built on a Poisson point process in $mathbb{R}^d$. As an application of our main theorem, we show that the maximizer of the system displays the ageing property.
We consider gradient fields $(phi_x:xin mathbb{Z}^d)$ whose law takes the Gibbs--Boltzmann form $Z^{-1}exp{-sum_{< x,y>}V(phi_y-phi_x)}$, where the sum runs over nearest neighbors. We assume that the potential $V$ admits the representation [V(eta):=-logintvarrho({d}kappa)expbiggl[-{1/2}kappaet a^2biggr],] where $varrho$ is a positive measure with compact support in $(0,infty)$. Hence, the potential $V$ is symmetric, but nonconvex in general. While for strictly convex $V$s, the translation-invariant, ergodic gradient Gibbs measures are completely characterized by their tilt, a nonconvex potential as above may lead to several ergodic gradient Gibbs measures with zero tilt. Still, every ergodic, zero-tilt gradient Gibbs measure for the potential $V$ above scales to a Gaussian free field.
We consider the two dimensional version of a drainage network model introduced by Gangopadhyay, Roy and Sarkar, and show that the appropriately rescaled family of its paths converges in distribution to the Brownian web. We do so by verifying the convergence criteria proposed by Fontes, Isopi, Newman and Ravishankar.
We consider on the torus the scaling limit of stochastic 2D (inviscid) fluid dynamical equations with transport noise to deterministic viscous equations. Quantitative estimates on the convergence rates are provided by combining analytic and probabilistic arguments, especially heat kernel properties and maximal estimates for stochastic convolutions. Similar ideas are applied to the stochastic 2D Keller-Segel model, yielding explicit choice of noise to ensure that the blow-up probability is less than any given threshold. Our approach also gives rise to some mixing property for stochastic linear transport equations and dissipation enhancement in the viscous case.
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