Abstract
The main result of this paper is that there are examples of stochastic partial differential equations [hereforth, SPDEs] of the type \[ \partial _{t} u=\tfrac{1} {2}\Delta u +\sigma (u)\eta \qquad \text{on $(0\,,\infty )\times \mathbb {R}^{3}$} \] 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 $A_{1},\beta \in (0\,,1)$ such that \[ \underline{\gamma } (k) := \liminf _{t\to \infty }t^{-1}\inf _{x\in \mathbb{R} ^{3}} \log \mathrm{E} \left (|u(t\,,x)|^{k}\right ) \geqslant A_{1}\exp (A_{1} k^{\beta }) \qquad \text{for all $k\geqslant 2$} . \] 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.
Citation
Le Chen. Jingyu Huang. Davar Khoshnevisan. Kunwoo Kim. "Dense blowup for parabolic SPDEs." Electron. J. Probab. 24 1 - 33, 2019. https://doi.org/10.1214/19-EJP372
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