Dense blowup for parabolic SPDEs

摘要

The main result of this paper is that there are examples of stochastic partial differential equations [hereforth, SPDEs] of the type [ partial _{t} u=frac{1} {2}Delta u +sigma (u)eta qquad ext{on $(0,,infty )imes mathbb {R}^{3}$} ] 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 $3$. En route, it will be proved that when $sigma (u)=u$ 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 $A_{1},eta in (0,,1)$ such that [ underline{gamma } (k) := liminf _{to infty }t^{-1}inf _{xin mathbb{R} ^{3}} log mathrm{E} left ( u(t,,x) ^{k}ight ) geqslant A_{1}exp (A_{1} k^{eta }) qquad ext{for all $kgeqslant 2$} . ] 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.