Dissipation and high disorder
| Autor: | Davar Khoshnevisan, Le Chen, Kunwoo Kim, Michael Cranston |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Statistics and Probability
stochastic partial differential equations Mathematics::Analysis of PDEs Markov process 47B80 01 natural sciences 010104 statistics & probability symbols.namesake Primary: 60J60 60K35 60K37 Secondary: 47B80 60H25 60H25 FOS: Mathematics Parabolic Anderson model 0101 mathematics Brownian motion Mathematics 60J60 Generator (computer programming) 010102 general mathematics Mathematical analysis Probability (math.PR) Function (mathematics) Dissipation Stochastic partial differential equation strong disorder 60K37 60K35 symbols Statistics Probability and Uncertainty Mathematics - Probability |
| Zdroj: | Ann. Probab. 45, no. 1 (2017), 82-99 |
| Popis: | Given a field $\{B(x)\}_{x\in\mathbf{Z}^d}$ of independent standard Brownian motions, indexed by $\mathbf{Z}^d$, the generator of a suitable Markov process on $\mathbf{Z}^d,\,\,\mathcal{G},$ and sufficiently nice function $\sigma:[0,\infty)\to[0,\infty),$ we consider the influence of the parameter $\lambda$ on the behavior of the system, \begin{align*} \rm{d} u_t(x) = & (\mathcal{G}u_t)(x)\,\rm{d} t + \lambda\sigma(u_t(x))\rm{d} B_t(x) \qquad[t>0,\ x\in\mathbf{Z}^d], &u_0(x)=c_0\delta_0(x). \end{align*} We show that for any $\lambda>0$ in dimensions one and two the total mass $\sum_{x\in\mathbf{Z}^d}u_t(x)\to 0$ as $t\to\infty$ while for dimensions greater than two there is a phase transition point $\lambda_c\in(0,\infty)$ such that for $\lambda>\lambda_c,\, \sum_{\mathbf{Z}^d}u_t(x)\to 0$ as $t\to\infty$ while for $\lambda Comment: 20 pages |
| Databáze: | OpenAIRE |
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