Open Access
2019 Dense blowup for parabolic SPDEs
Le Chen, Jingyu Huang, Davar Khoshnevisan, Kunwoo Kim
Electron. J. Probab. 24: 1-33 (2019). DOI: 10.1214/19-EJP372


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.


Download Citation

Le Chen. Jingyu Huang. Davar Khoshnevisan. Kunwoo Kim. "Dense blowup for parabolic SPDEs." Electron. J. Probab. 24 1 - 33, 2019.


Received: 22 March 2019; Accepted: 9 October 2019; Published: 2019
First available in Project Euclid: 29 October 2019

zbMATH: 07142912
MathSciNet: MR4029421
Digital Object Identifier: 10.1214/19-EJP372

Primary: 35R60 , 60H15
Secondary: 60G15

Keywords: blowup , Intermittency , Stochastic partial differential equations

Vol.24 • 2019
Back to Top