16. chen.hu.ea:19:nonlinear

Nonlinear stochastic time-fractional slow and fast diffusion equations on \(\mathbb{R}^d\)

Le Chen, Yaozhong Hu, David Nualart

Abstract: This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables:

\[\left(\partial^\beta+\frac{\nu}{2}(-\Delta)^{\alpha/2}\right)u(t,x) = I_t^\gamma\left[\rho(u(t,x))\dot{W}(t,x)\right],\quad t>0,\: x\in\mathbb{R}^d,\]

where \(\dot{W}\) is the space-time white noise, \(\alpha\in(0,2]\), \(\beta\in(0,2)\), \(\gamma\ge 0\) and \(\nu>0\). Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang’s condition:

\[d<2\alpha+\frac{\alpha}{\beta}\min(2\gamma-1,0).\]

In some cases, the initial data can be measures. When \(\beta\in (0,1]\), we prove the sample path regularity of the solution.

MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.

Keywords: nonlinear stochastic fractional diffusion equations, measure-valued initial data, Hölder continuity, intermittency, the Fox H-function.

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[CHN19] Chen, Le, Hu, Yaozhong & Nualart, David (2019) ‘Nonlinear stochastic time-fractional slow and fast diffusion equations on \(\mathbb{R}^d\)’, Stochastic Process. Appl. 129, 5073–5112. <https://doi.org/10.1016/j.spa.2019.01.003>

@article{chen.hu.ea:19:nonlinear,
  author        = {Chen, Le and Hu, Yaozhong and Nualart, David},
  title         = {Nonlinear stochastic time-fractional slow and fast diffusion equations on {$\Bbb R^d$}},
  journal       = {Stochastic Process. Appl.},
  fjournal      = {Stochastic Processes and their Applications},
  volume        = {129},
  year          = {2019},
  number        = {12},
  pages         = {5073--5112},
  issn          = {0304-4149},
  mrclass       = {60H15 (35R11 35R60 60G60)},
  mrnumber      = {4025700},
  mrreviewer    = {Latifa Debbi},
  doi           = {10.1016/j.spa.2019.01.003},
  url           = {https://doi.org/10.1016/j.spa.2019.01.003}
}

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