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arXiv 1405.0901math.PR
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移动随机介质中重整化势布朗运动的退火渐近性

Annealed asymptotics for Brownian motion of renormalized potential in mobile random medium

Xia Chen, Jie Xiong

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43 pages
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摘要：

受具有移动Poissonian陷阱或催化剂的定向聚合物模型和具有含时势的随机抛物Anderson模型研究的启发，我们研究了\[\mathbb{E}\otimes\mathbb{E}_0\exp\left\{\pmθ\int^t_0\bar{V}(s,B_s)ds\right\}\qquad (t\to\infty)\]的渐近行为，其中\[\mathbb{E}\otimes\mathbb{E}_0\exp\left\{\pmθ\int^t_0\bar{V}(s,B_s)ds\right\}\qquad (t\to\infty)\]为常数，$\overline{V}$是形式为\[\overline{V}(s,x)=\int_{\mathbb{R}^d}\frac{1}{|y-x|^p}\left(ω_s(dy)-dy\right),\]的重整化泊松势，$ω_s$是测量值。由独立的布朗粒子组成的取值过程，其初始位置形成$\mathbb{R}^d$上的Poisson随机测度，Lebesgue测度作为其强度。根据参数$p$和尺寸$d$，得到不同的标度极限。对于负指数矩的对数，$\frac{d}{2}<p<d$的范围被划分为5个区域，分别具有$t^{d/p}$、$t^{3/2}$、$t^{(4-d-2p)/2}$、$t\log t$和$t$阶的不同缩放率。对于正指数矩，根据参数$p$和$d$，在三个区域研究了极限行为。在亚临界区（$p<2$），指数矩的双对数有一个$t$的速率.在临界区域（$p=2$），根据$p=2$与Hardy不等式中最佳常数的比较，决定了它在两个部分上有不同的行为。在超临界区$(p>2)$，所有$t>0$的指数矩都变为无穷大.

英文摘要：

Motivated by the study of the directed polymer model with mobile Poissonian traps or catalysts and the stochastic parabolic Anderson model with time dependent potential, we investigate the asymptotic behavior of \[\mathbb{E}\otimes\mathbb{E}_0\exp\left\{\pmθ\int^t_0\bar{V}(s,B_s)ds\right\}\qquad (t\to\infty)\] where $þ>0$ is a constant, $\overline{V}$ is the renormalized Poisson potential of the form \[\overline{V}(s,x)=\int_{\mathbb{R}^d}\frac{1}{|y-x|^p}\left(ω_s(dy)-dy\right),\] and $ω_s$ is the measure-valued process consisting of independent Brownian particles whose initial positions form a Poisson random measure on $\mathbb{R}^d$ with Lebesgue measure as its intensity. Different scaling limits are obtained according to the parameter $p$ and dimension $d$. For the logarithm of the negative exponential moment, the range of $\frac{d}{2}<p<d$ is divided into 5 regions with various scaling rates of the orders $t^{d/p}$, $t^{3/2}$, $t^{(4-d-2p)/2}$, $t\log t$ and $t$, respectively. For the positive exponential moment, the limiting behavior is studied according to the parameters $p$ and $d$ in three regions. In the sub-critical region ($p<2$), the double logarithm of the exponential moment has a rate of $t$. In the critical region ($p=2$), it has different behavior over two parts decided according to the comparison of $θ$ with the best constant in the Hardy inequality. In the super-critical region $(p>2)$, the exponential moments become infinite for all $t>0$.

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