## Electronic Journal of Probability

### 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 $\partial _{t} u=\tfrac{1} {2}\Delta u +\sigma (u)\eta \qquad \text{on (0\,,\infty )\times \mathbb {R}^{3}}$ 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.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 118, 33 pp.

Dates
Accepted: 9 October 2019
First available in Project Euclid: 29 October 2019

https://projecteuclid.org/euclid.ejp/1572314778

Digital Object Identifier
doi:10.1214/19-EJP372

Mathematical Reviews number (MathSciNet)
MR4029421

Zentralblatt MATH identifier
07142912

#### Citation

Chen, Le; Huang, Jingyu; Khoshnevisan, Davar; Kim, Kunwoo. Dense blowup for parabolic SPDEs. Electron. J. Probab. 24 (2019), paper no. 118, 33 pp. doi:10.1214/19-EJP372. https://projecteuclid.org/euclid.ejp/1572314778

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