Firas Rassoul-Agha

A minicourse on stochastic partial differential equations

A Primer on Stochastic Partial Differential Equations.- The Stochastic Wave Equation.- Application of Malliavin Calculus to Stochastic Partial Differential Equations.- Some Tools and Results for Parabolic Stochastic Partial Differential Equations.- Sample Path Properties of Anisotropic Gaussian Random Fields.

An almost sure invariance principle for random walks in a space-time random environment

Abstract.We consider a discrete time random walk in a space-time i.i.d. random environment. We use a martingale approach to show that the walk is diffusive in almost every fixed environment. We improve on existing results by proving an invariance principle and considering environments with an L2 averaged drift. We also state an a.s. invariance principle for random walks in general random environments whose hypothesis requires a subdiffusive bound on the variance of the quenched mean, under an ergodic invariant measure for the environment chain.

The point of view of the particle on the law of large numbers for random walks in a mixing random environment

The point of view of the particle is an approach that has proven very powerful in the study of many models of random motions in random media. We provide a new use of this approach to prove the law of large numbers in the case of one or higher-dimensional, finite range, transient random walks in mixing random environments. One of the advantages of this method over what has been used so far is that it is not restricted to i.i.d. environments.

A course on large deviations with an introduction to Gibbs measures

Large deviations: General theory and i.i.d. processes Introductory discussion The large deviation principle Large deviations and asymptotics of integrals Convex analysis in large deviation theory Relative entropy and large deviations for empirical measures Process level large deviations for i.i.d. fields Statistical mechanics Formalism for classical lattice systems Large deviations and equilibrium statistical mechanics Phase transition in the Ising model Percolation approach to phase transition Additional large deviation topics Further asymptotics for i.i.d. random variables Large deviations through the limiting generating function Large deviations for Markov chains Convexity criterion for large deviations Nonstationary independent variables Random walk in a dynamical random environment Appendixes: Analysis Probability Inequalities from statistical mechanics Nonnegative matrices Bibliography Notation index Author index General index

Process-level quenched large deviations for random walk in random environment

We consider a bounded step size random walk in an ergodic random environment with some ellipticity, on an integer lattice of arbitrary dimension. We prove a level 3 large deviation principle, under almost every environment, with rate function related to a relative entropy.

The Random Average Process and Random Walk in a Space-Time Random Environment in One Dimension

We study space-time fluctuations around a characteristic line for a one-dimensional interacting system known as the random average process. The state of this system is a real-valued function on the integers. New values of the function are created by averaging previous values with random weights. The fluctuations analyzed occur on the scale n1/4, where n is the ratio of macroscopic and microscopic scales in the system. The limits of the fluctuations are described by a family of Gaussian processes. In cases of known product-form invariant distributions, this limit is a two-parameter process whose time marginals are fractional Brownian motions with Hurst parameter 1/4. Along the way we study the limits of quenched mean processes for a random walk in a space-time random environment. These limits also happen at scale n1/4 and are described by certain Gaussian processes that we identify. In particular, when we look at a backward quenched mean process, the limit process is the solution of a stochastic heat equation.

Ratios of partition functions for the log-gamma polymer

We introduce a random walk in random environment associated to an underlying directed polymer model in 1 + 1 dimensions. This walk is the positive temperature counterpart of the competition in- terface of percolation and arises as the limit of quenched polymer measures. We prove this limit for the exactly solvable log-gamma polymer, as a consequence of almost sure limits of ratios of parti- tion functions. These limits of ratios give the Busemann functions of the log-gamma polymer, and furnish centered cocycles that solve a variational formula for the limiting free energy. Limits of ratios of point-to-point and point-to-line partition functions manifest a duality between tilt and velocity that comes from quenched large deviations under polymer measures. In the log-gamma case, we identify a fam- ily of ergodic invariant distributions for the random walk in random environment.

Quenched invariance principle for multidimensional ballistic random walk in a random environment with a forbidden direction

We consider a ballistic random walk in an i.i.d. random environment that does not allow retreating in a certain fixed direction. Homogenization and regeneration techniques combine to prove a law of large numbers and an averaged invariance principle. The assumptions are non-nestling and

Variational formulas and disorder regimes of random walks in random potentials

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A New Method for Generating Stochastic Simulations of Daily Air Temperature for Use in Weather Generators

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