9. chen.cranston.ea:17:dissipation

9. chen.cranston.ea:17:dissipation#

Dissipation and high disorder

Le Chen, Michael Cranston, Davar Khoshnevisan, and Kunwoo Kim

Abstract: Given a field \(\{B(x)\}_{x\in\mathbb{Z}^d}\) of independent standard Brownian motions, indexed by \(\mathbb{Z}^d\), the generator of a suitable Markov process on \(\mathbb{Z}^d,\,\,\mathcal{G},\) and sufficiently nice function \(\sigma:[0,\infty)\mapsto[0,\infty),\) we consider the influence of the parameter \(\lambda\) on the behavior of the system,

\[\begin{split}d u_t(x) &= (\mathcal{G}\,u_t)(x)\, d t + \lambda\sigma(u_t(x)) d B_t(x) \qquad[t>0,\ x\in\mathbb{Z}^d],\\ u_0(x) &= c_0\delta_0(x),\end{split}\]

We show that for any \(\lambda>0\) in dimensions one and two the total mass \(\sum_{x\in\mathbb{Z}^d}u_t(x)\) converges to zero as \(t\to\infty\) while for dimensions greater than two there is a phase transition point \(\lambda_c\in(0,\infty)\) such that for

\[\lambda>\lambda_c,\, \sum_{x\in\mathbb{Z}^d}u_t(x)\to 0, \quad\text{as } t\to\infty,\]

while for

\[\lambda<\lambda_c,\,\sum_{x\in \mathbb{Z}^d}u_t(x)\not\to 0,\quad\text{as } t\to\infty.\]

MSC 2010 subject classifications: Primary: 60J60, 60K35, 60K37, Secondary: 47B80, 60H25.

Keywords: Parabolic Anderson model, strong disorder, stochastic partial differential equations.

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[CCKK17] Chen, Le, Cranston, Michael, Khoshnevisan, Davar & Kim, Kunwoo (2017) ‘Dissipation and high disorder’, Ann. Probab. 45, 82–99. <https://doi.org/10.1214/15-AOP1040>

@article{chen.cranston.ea:17:dissipation,
  author        = {Chen, Le and Cranston, Michael and Khoshnevisan, Davar and Kim, Kunwoo},
  title         = {Dissipation and high disorder},
  journal       = {Ann. Probab.},
  fjournal      = {The Annals of Probability},
  volume        = {45},
  year          = {2017},
  number        = {1},
  pages         = {82--99},
  issn          = {0091-1798},
  mrclass       = {60J60 (47B80 60H25 60K35 60K37)},
  mrnumber      = {3601646},
  mrreviewer    = {Khoa L\^{e}},
  doi           = {10.1214/15-AOP1040},
  url           = {https://doi.org/10.1214/15-AOP1040}
}

References: Borodin and Corwin [BC14c]; Carmona and Hu [CH06]; Carmona et al. [CKM01]; Carmona and Molchanov [CM94]; Chung and Fuchs [CF51]; Cox et al. [CFG96]; Cranston et al. [CMS02]; Dalang and Mueller [DM03]; Dawson and Perkins [DP12]; Foondun and Khoshnevisan [FK09]; Georgiou et al. [GJKS15]; Hoeffding [Hoe63]; Liggett [Lig05]; Mueller [Mue91b]; Mueller [Mue09]; Mueller and Nualart [MN08]; Mueller and Tribe [MT04]; Shiga [Shi92]; Shiga and Shimizu [SS80]; Spitzer [Spi81]; Walsh [Wal86];

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