17. chen.huang.ea:19:dense

Dense blowup for parabolic SPDEs

Le Chen, Jingyu Huang, Davar Khoshnevisan, and Kunwoo Kim

Abstract: The main result of this paper is that there are examples of stochastic partial differential equations [hereforth, SPDEs] of the type

\[\partial_t u=\tfrac12\Delta u +\sigma(u)\eta \quad\text{on } (0\,,\infty)\times\mathbb{R}^3\]

such that the solution exists and is unique as a random field in the sense of Dalang [Dal99] and Walsh [Wal86], 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

\[\underline{\gamma}(k) := \liminf_{t\to\infty}t^{-1}\inf_{x\in\mathbb{R}^3} \log\mathbb{E}\left( |u(t\,,x)|^k \right) \ge A_1\exp(A_1 k^\beta) \qquad\text{for all } k\ge 2.\]

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.

MSC 2010 subject classifications: Primary 35R60, 60H15; Secondary 60G15.

Keywords: stochastic partial differential equations, blowup, intermittency.

Download

[CHKK19] Chen, Le, Huang, Jingyu, Khoshnevisan, Davar & Kim, Kunwoo (2019) ‘Dense blowup for parabolic SPDEs’, Electron. J. Probab. 24, Paper No. 118, 33. <https://doi.org/10.1214/19-ejp372>

@article{chen.huang.ea:19:dense,
  author        = {Chen, Le and Huang, Jingyu and Khoshnevisan, Davar and Kim, Kunwoo},
  title         = {Dense blowup for parabolic {SPDE}s},
  journal       = {Electron. J. Probab.},
  fjournal      = {Electronic Journal of Probability},
  volume        = {24},
  year          = {2019},
  pages         = {Paper No. 118, 33},
  mrclass       = {60H15 (35B44 35K15 35K91 35R60 60G15)},
  mrnumber      = {4029421},
  mrreviewer    = {Guangqu Zheng},
  doi           = {10.1214/19-ejp372},
  url           = {https://doi.org/10.1214/19-ejp372}
}

References: Borell [Bor75]; Chen and Huang [CH19a]; Ciesielski and Taylor [CT62]; Conus [Con13]; Conus et al. [CJKS13b]; Dalang [Dal99]; Dalang and Frangos [DF98]; Dalang and Lévêque [DL04]; Dalang and Lévêque [DL06]; Dudley [Dud67]; Fernique [Fer71]; Foondun and Khoshnevisan [FK09]; Foondun et al. [FKM15]; Foondun et al. [FKN11]; Hairer [Hai14]; Hairer [Hai13]; Hairer and Pardoux [HP15]; Hu and Nualart [HN09]; Khoshnevisan [Kho02]; Khoshnevisan [Kho14]; Khoshnevisan and Xiao [KX03]; Khoshnevisan et al. [KSXZ13]; Mueller [Mue91b]; Mueller [Mue09]; Mytnik and Perkins [MP03]; Sato [Sat13]; Schilling et al. [SSV10]; Sudakov and Cirelson [SC74]; Walsh [Wal86];

This page