We study the extremal properties of a stochastic process x _t defined by a Langevin equation , 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 in one and three dimensions 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(z) is the Heaviside theta function; a geometric Brownian motion V(B _t ) = exp(B _t ) (model II); and a case with (model III). We realise that, rather surprisingly, the FPT PDF has exactly the Lvy–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 Lvy–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.

我们研究一个随机过程的极值特性X _吨由Langevin方程定义，其中ξ_吨是高斯白噪声具有零均值，d ₀为常数的比例因子，和V（乙_吨）是一个随机“扩散”（噪声强度），它本身是独立布朗运动B _t的函数。我们从固定位置x ₀得出第一遍时间（FPT）t的概率密度函数（PDF）的一维和三维精确精确表达式 随机扩散率的三种不同实现的起点：截止情况V（B _t）=Θ（B _t）（模型I），其中Θ（z）是Heaviside theta函数；几何布朗运动V（B _t）= exp（B _t）（模型II）; 和一个案例 （模型III）。我们意识到，令人惊讶的是，FPT PDF恰好具有模型II的Lvy–Smirnov形式（特定于标准布朗运动），并同时表现出强烈的异常扩散。对于模型I和III，与Lvy–Smirnov密度相比，左或右尾部（或两者）对时间的功能依赖性不同。在所有情况下，PDF都很宽泛，以至于第一时间已经不存在。FPT PDF在吸收球体目标的三个维度上获得了相似的结果。