Mathew Joseph

Initial measures for the stochastic heat equation

We consider a family of nonlinear stochastic heat equations of the form @tu = Lu + (u) _ W , where _ W denotes space-time white noise, L the generator of a symmetric L evy process on R, and is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-eld solution for every nite initial measure u0. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that Lf = cf 00 for some c > 0, we prove that if u0 is a nite measure of compact support, then the solution is with probability one a bounded function for all times t > 0.

On the chaotic character of the stochastic heat equation, before the onset of intermitttency

We consider a nonlinear stochastic heat equation @tu = 1 @xxu + (u)@xtW , where @xtW denotes space-time white noise and : R ! R is Lipschitz continuous. We establish that, at every xed time t > 0, the global behavior of the solution depends in a critical manner on the structure of the initial function u0: Under suitable technical conditions on u0 and , supx2Rut(x) is a.s. nite when u0 has compact support, whereas with probability one, lim supjxj!1ut(x)=(logjxj) 1=6 > 0 when u0 is bounded uniformly away from zero. The mentioned sensitivity to the initial data of the stochastic heat equation is a way to state that the solution to the stochastic heat equation is chaotic at xed times, well before the onset of intermittency.

Strong invariance and noise-comparison principles for some parabolic stochastic PDEs

We consider a system of interacting diffusions on the integer lattice. By letting the mesh size go to zero and by using a suitable scaling, we show that the system converges (in a strong sense) to a solution of the stochastic heat equation on the real line. As a consequence, we obtain comparison inequalities for product moments of the stochastic heat equation with different nonlinearities.

Intermittency and Chaos for a Nonlinear Stochastic Wave Equation in Dimension 1

Consider a nonlinear stochastic wave equation driven by space-time white noise in dimension one. We discuss the intermittency of the solution, and then use those intermittency results in order to demonstrate that in many cases the solution is chaotic. For the most part, the novel portion of our work is about the two cases where (1) the initial conditions have compact support, where the global maximum of the solution remains bounded, and (2) the initial conditions are positive constants, where the global maximum is almost surely infinite. Bounds are also provided on the behavior of the global maximum of the solution in Case (2).

Semi-discrete semi-linear parabolic SPDEs.

Consider the semi-discrete semi-linear Ito stochastic heat equation, ∂tut(x) = (Lut)(x) + σ(ut(x))∂tBt(x), started at a non-random bounded initial profile u0 : Z d → R+. Here: {B(x)}x2Zd is an field of i.i.d. Brownian motions; L denotes the generator of a continuous-time random walk on Z d ; and σ : R → R is Lipschitz continuous and non-random with σ(0) = 0. The main findings of this paper are: (i) The kth moment Lyapunov exponent of u grows exactly as k 2 ; (ii) The following random Radon-Nikodym theorem holds: lim#0 u t+�(x) − ut(x) Bt+�(x) − Bt(x) = σ(u t(x)) in probability;