Dense blowup for parabolic SPDEs

Abstract

The main result of this paper is that there are examples of stochastic partial differential equations [hereforth, SPDEs] of the type ∂tu=12Δu+σ(u)ηon (0,∞)×R3 ∂tu=12Δu+σ(u)ηon (0,∞)×R3 such that the solution exists and is unique as a random field in the sense of Dalang [6] and Walsh [31], yet the solution has unbounded oscillations in every open neighborhood of every space-time point. We are not aware of the existence of such a construction in spatial dimensions below 33.

En route, it will be proved that when σ(u)=uσ(u)=u there exist a large family of parabolic SPDEs whose moment Lyapunov exponents grow at least sub exponentially in its order parameter in the sense that there exist A1,β∈(0,1)A1,β∈(0,1) such that γ––(k):=lim inft→∞t−1infx∈R3logE(|u(t,x)|k)⩾A1exp(A1kβ)for all k⩾2. γ_(k):=lim inft→∞t−1infx∈R3log⁡E(|u(t,x)|k)⩾A1exp⁡(A1kβ)for all k⩾2. This sort of “super intermittency” is combined with a local linearization of the solution, and with techniques from Gaussian analysis in order to establish the unbounded oscillations of the sample functions of the solution to our SPDE.