arXiv 0711.09822007-11math.PR

具有指数光跳跃的Lévy驱动扩散的首次出射时间

First exit times for Lévy-driven diffusions with exponentially light jumps

Peter Imkeller, Ilya Pavlyukevich, Torsten Wetzel

备注：: Published in at http://dx.doi.org/10.1214/08-AOP412 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
发表信息：: Annals of Probability 2009, Vol. 37, No. 2, 530-564
DOI：: 10.1214/08-AOP412

摘要：

我们考虑一个由微分方程$\dot{Y}_t=-U'(Y_t)$描述的动力系统，在原点有唯一的稳定点。我们用强度为$\varepsilon$的Lévy噪声扰动系统，得到随机微分方程$dX^{\varepsilon}_t=-U'(X^{\varepsilon}_{t-}) dt+\varepsilon dL_t.$过程$L$是一个对称Lévy过程，其跳测度$ν$具有指数轻尾，$ν([u,\infty))\sim\exp(-u^α)$，$α>0$，$u\to \infty$ .研究了一类随机微分方程解的轨道在区间$(-1,1)$上的第一出口问题。在小噪声极限$\varepsilon\to0$下，第一次出射时间$σ_x$，$x\in(-1,1)$的规律具有指数尾，在临界指数$α=1$处的平均值表现出有趣的相变，即$0<α<1$为$\ln\mathbf{E}σ\sim\varepsilon^{-α}$，而$α>1$为$\ln\mathbf{E}σ\sim\varepsilon^{- 1}|\ln\varepsilon|^{1-{1}/α}$.

英文摘要：

We consider a dynamical system described by the differential equation $\dot{Y}_t=-U'(Y_t)$ with a unique stable point at the origin. We perturb the system by the Lévy noise of intensity $\varepsilon$ to obtain the stochastic differential equation $dX^{\varepsilon}_t=-U'(X^{\varepsilon}_{t-}) dt+\varepsilon dL_t.$ The process $L$ is a symmetric Lévy process whose jump measure $ν$ has exponentially light tails, $ν([u,\infty))\sim\exp(-u^α)$, $α>0$, $u\to \infty$. We study the first exit problem for the trajectories of the solutions of the stochastic differential equation from the interval $(-1,1)$. In the small noise limit $\varepsilon\to0$, the law of the first exit time $σ_x$, $x\in(-1,1)$, has exponential tail and the mean value exhibiting an intriguing phase transition at the critical index $α=1$, namely, $\ln\mathbf{E}σ\sim\varepsilon^{-α}$ for $0<α<1$, whereas $\ln\mathbf{E}σ\sim\varepsilon^{- 1}|\ln\varepsilon|^{1-{1}/α}$ for $α>1$.

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摘要：

英文摘要：

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