6. chen.hu.ea:17:space-time

Space-time fractional diffusions in Gaussian noisy environment

Le Chen, Guannan Hu, Yaozhong Hu, and Jingyu Huang

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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[CHHH17] Chen, Le, Hu, Guannan, Hu, Yaozhong & Huang, Jingyu (2017) ‘Space-time fractional diffusions in Gaussian noisy environment’, Stochastics 89, 171–206. <https://doi.org/10.1080/17442508.2016.1146282>

@article{chen.hu.ea:17:space-time,
  author        = {Chen, Le and Hu, Guannan and Hu, Yaozhong and Huang, Jingyu},
  title         = {Space-time fractional diffusions in {G}aussian noisy environment},
  journal       = {Stochastics},
  fjournal      = {Stochastics. An International Journal of Probability and Stochastic Processes},
  volume        = {89},
  year          = {2017},
  number        = {1},
  pages         = {171--206},
  issn          = {1744-2508},
  mrclass       = {60H15 (60G22)},
  mrnumber      = {3574699},
  mrreviewer    = {Xiliang Fan},
  doi           = {10.1080/17442508.2016.1146282},
  url           = {https://doi.org/10.1080/17442508.2016.1146282}
}

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