arXiv math/06094432006-09math.PR

随机环境中标量扩散方程的大偏差

Large deviations for a scalar diffusion in random environment

P. Chigansky, R. Liptser

备注：: 15 pages
发表信息：: Theory of Prob. Appl., 54 (2010) No. 1, pp. 29-50

摘要：

设$σ(u)$，$u\in \mathbb{R}$是遍历平稳马尔可夫链，$a_1,...,a_m$，和$b(u)=g(σ(u))$取有限个值，其中$g$是有界可测函数.本文考虑扩散型过程$$ dX^ε_t = b(X^ε_t/ε)dt + ε^κσ\big(X^ε_t/ε\big)dB_t, t\le T $$服从$X^ε_0=x_0$，其中$ε$是一个小的正参数，$B_t$是一个与$σ$无关的布朗运动，$κ> 0$是一个固定常数.我们表明，对于$κ<1/6$，家庭$\{X^ε_t\}_{ε\to 0}$满足Freidlin-Wentzell型的大偏差原理（LDP）的恒定漂移$\mathbf{b}$和扩散$\mathbf{a}$，由$$ \mathbf{b}=\sum\limits_{i=1}^m\dfrac{g(a_i)}{a^2_i}π_i\Big/ \sum\limits_{i=1}^m\dfrac{1}{a^2_i}π_i, \quad \mathbf{a}=1\Big/\sum\limits_{i=1}^m\dfrac{1}{a^2_i}π_i, $$，其中$\{π_1,...,π_m\}$是链$σ(u)$的不变分布。

英文摘要：

Let $σ(u)$, $u\in \mathbb{R}$ be an ergodic stationary Markov chain, taking a finite number of values $a_1,...,a_m$, and $b(u)=g(σ(u))$, where $g$ is a bounded and measurable function. We consider the diffusion type process $$ dX^ε_t = b(X^ε_t/ε)dt + ε^κσ\big(X^ε_t/ε\big)dB_t, t\le T $$ subject to $X^ε_0=x_0$, where $ε$ is a small positive parameter, $B_t$ is a Brownian motion, independent of $σ$, and $κ> 0$ is a fixed constant. We show that for $κ<1/6$, the family $\{X^ε_t\}_{ε\to 0}$ satisfies the Large Deviations Principle (LDP) of the Freidlin-Wentzell type with the constant drift $\mathbf{b}$ and the diffusion $\mathbf{a}$, given by $$ \mathbf{b}=\sum\limits_{i=1}^m\dfrac{g(a_i)}{a^2_i}π_i\Big/ \sum\limits_{i=1}^m\dfrac{1}{a^2_i}π_i, \quad \mathbf{a}=1\Big/\sum\limits_{i=1}^m\dfrac{1}{a^2_i}π_i, $$ where $\{π_1,...,π_m\}$ is the invariant distribution of the chain $σ(u)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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