论文标题
三个随机扩散率模型的确切的先进时间分布
Exact first-passage time distributions for three random diffusivity models
论文作者
论文摘要
我们研究由langevin方程$ \ dot {x} _t = \ sqrt {2 d_0 v（b_t）} \，ξ_t$定义的随机过程$ x_t $的极端特性$ x_t $ “扩散性”（噪声强度）本身是独立的布朗运动$ b_t $的功能。我们从固定位置$ x_0 $从固定位置$ x_0 $的概率密度函数（fpt）$ t $的概率密度函数（PDF）得出精确，紧凑的表达方式，以实现随机扩散率的三种不同实现：截止情况$ v（b_t）=θ（b_t）=θ（b_t）$（b_t）$（b_t）$（model i）几何布朗运动$ v（b_t）= \ exp（b_t）$（型号II）;以及$ v（b_t）= b_t^2 $（型号III）的情况。我们意识到，令人惊讶的是，FPT PDF完全具有Model II的Lévy-Smirnov形式（特定于标准的Brownian运动），该形式同时表现出强烈反常的扩散。对于模型I和III，与Lévy-Smirnov密度相比，左尾或右尾部（或两者都）具有不同的功能依赖性。在所有情况下，PDF都广泛，因此已经不存在第一刻了。 FPT PDF的三个维度与吸收球靶相似。
We study the extremal properties of a stochastic process $x_t$ defined by a Langevin equation $\dot{x}_t=\sqrt{2 D_0 V(B_t)}\,ξ_t$, where $ξ_t$ is a Gaussian white noise with zero mean, $D_0$ is a constant scale factor, and $V(B_t)$ is a stochastic "diffusivity" (noise strength), which itself is a functional of independent Brownian motion $B_t$. We derive exact, compact expressions for the probability density functions (PDFs) of the first passage time (FPT) $t$ from a fixed location $x_0$ to the origin for three different realisations of the stochastic diffusivity: a cut-off case $V(B_t) =Θ(B_t)$ (Model I), where $Θ(x)$ is the Heaviside theta function; a Geometric Brownian Motion $V(B_t)=\exp(B_t)$ (Model II); and a case with $V(B_t)=B_t^2$ (Model III). We realise that, rather surprisingly, the FPT PDF has exactly the Lévy-Smirnov form (specific for standard Brownian motion) for Model II, which concurrently exhibits a strongly anomalous diffusion. For Models I and III either the left or right tails (or both) have a different functional dependence on time as compared to the Lévy-Smirnov density. In all cases, the PDFs are broad such that already the first moment does not exist. Similar results are obtained in three dimensions for the FPT PDF to an absorbing spherical target.
