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Dissipation and high disorder. (English) Zbl 1382.60103
Consider the system of stochastic differential equations $du_t(x)=(\mathcal{G}u_t)(x)dt + \lambda \sigma(u_t(x)) dB_t(x), \quad t>0,\quad x\in\mathbb{Z}^d,$ with initial condition $$u_0=c_0\delta_0$$ for some $$c_0>0$$, where $$\{B(x)\}_{x\in\mathbb{Z}^d}$$ are independent standard Brownian motions, $(\mathcal{G}h)(x)=\sum_{y\in\mathbb{Z}^d} \tau(y)\big(h(x+y)-h(x)\big), \quad x\in\mathbb{Z}^d$ is a generator of a suitable Markov process on $$\mathbb{Z}^d$$, $$\sigma:[0,+\infty)\to[0,+\infty)$$ is a sufficiently nice function, and $$\lambda>0$$ is a real parameter. The special case $$\sigma(u)=u$$ is called the parabolic Anderson model and has been amply studied in the literature.
Denote by $m_t(\lambda) = \|u_t\|_{\ell^1(\mathbb{Z}^d)} = \sum_{x\in\mathbb{Z}^d} |u_t(x)|$ the total mass process.
The authors call the system introduced above globally dissipative if $$\lim_{t\to\infty} m_t(\lambda) =0$$ a.s.
The principle result of the paper states that the system is globally dissipative if and only if there is strong disorder. More precisely, it is shown that the system is always globally dissipative if $$d\in\{1,2\}$$, and that for $$d\geq 3$$ there exists a critical value $$\lambda\in(0,+\infty)$$ such that the system is globally dissipative for $$\lambda>\lambda_c$$, and not globally dissipative for $$\lambda\in (0,\lambda_c)$$. The case $$\lambda=\lambda_c$$ remains open.
The proof of the main theorem provides quantitative estimates for the decay of the local mass process.
##### MSC:
 60J60 Diffusion processes 47B80 Random linear operators 60H25 Random operators and equations (aspects of stochastic analysis) 60K37 Processes in random environments
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