Kunwoo Kim

Non-linear noise excitation and intermittency under high disorder

Consider the semilinear heat equation@tu = @ 2 xu+ (u) on the interval [0;L] with Dirichlet zero-boundary condition and a nice non-random initial function, where the forcing is space-time white noise and > 0 denotes the level of the noise. We show that, when the solution is intermittent [that is, when infz j (z)=zj > 0], the expected L 2 -energy of the solution grows at least as expfc 2 g and at most as expfc 4 g as ! 1. In the case that the Dirichlet boundary condition is replaced by a Neumann boundary condition, we prove that the L 2 -energy of the solution is in fact of sharp exponential order expfc 4 g. We show also that, for a large family of one-dimensional randomly-forced wave equations on R, the energy of the solution grows as expfc g as ! 1. Thus, we observe the surprising result that the stochastic wave equation is, quite typically, significantly less noise-excitable than its parabolic counterparts.

Intermittency and multifractality: A case study via parabolic stochastic PDES

Let denote space-time white noise, and consider the following stochastic partial dierential equations: (i) _ u = 1 u 00 +u , started identically at one; and (ii) _ Z = 1 Z 00 + , started identically at zero. It is well known that the solution to (i) is intermittent, whereas the solution to (ii) is not. And the two equations are known to be in dierent universality classes. We prove that the tall peaks of both systems are multifractals in a natural large-scale sense. Some of this work is extended to also establish the multifractal behavior of the peaks of stochastic PDEs on R+ R d with d > 2. G. Lawler has asked us if intermittency is the same as multifractality. The present work gives a negative answer to this question. As a byproduct of our methods, we prove also that the peaks of the Brownian motion form a large-scale monofractal, whereas the peaks of the Ornstein{Uhlenbeck process on R are multifractal. Throughout, we make extensive use of the macroscopic fractal theory of M.T. Barlow and S.J. Taylor [3, 4]. We expand on aspects of the Barlow{Taylor theory, as well.

Numerical Analysis of the Stochastic Moving Boundary Problem

We consider a numerical solution of the stochastic moving boundary value problem, whose existence and uniqueness of solution are proved in [16]. Numerical approximations are based on the transformation, which transforms the stochastic moving boundary problem whose spatial domain is a priori unknown to a nonlinear stochastic partial differential equation which has a fixed spatial domain. We construct a numerical solution of the nonlinear stochastic partial differential equation and investigate the convergence theory.

Dissipation and high disorder

Given a fieldfB(x)gx2Zd of independent standard Brownian motions, indexed by Z d , the generator of a suitable Markov process on Z d ; G; and suciently nice function : [0;1)! [0;1); we consider the influence of the parameter on the behavior of the system, dut(x) = (Gut)(x) dt + (ut(x))dBt(x) [t > 0; x2 Z d ]; u0(x) = c0 0(x); We show that for any