Inverse local time of one-dimensional diffusions and its comparison theorem

具有两个边界的一维反射扩散问题和一个逆首击问题

One-dimensional reflected diffusions with two boundaries and an inverse first-hitting problem

本文研究了一维时间均匀扩散$X(t)$的逆首击问题，该问题从一个随机位置$η.$开始，在两个边界$a$和$b,$之间反射。设$a \le S \le b$为给定阈值，使得$P( η\in [a,S])=1,$和$F$为指定的分布函数。该问题包括寻找分布的$η$，使第一次击中时间的$X$ $S$有分布$F.$这是一个推广的类似问题，为普通的扩散，即没有反映，以前考虑的作者。

Some examples of solutions to an inverse problem for the first-passage place of a jump-diffusion process

We report some additional examples of explicit solutions to an inverse first-passage place problem for one-dimensional diffusions with jumps, introduced in a previous paper. If $X(t)$ is a one-dimensional diffusion with jumps, starting from a random position $η\in [a,b],$ let be $τ_{a,b}$ the time at which $X(t)$ first exits the interval $(a,b),$ and $π_a = P(X(τ_{a,b}) \le a)$ the probability of exit from the left of $(a,b).$ Given a probability $q \in (0,1),$ the problem consists in finding the density $g$ of $η$ (if it exists) such that $π_a = q;$ it can be seen as a problem of optimization…

关于扩散逆局部时的Lévy测度的次指数性及其在惩罚中的应用

On subexponentiality of the Lévy measure of the diffusion inverse local time; with applications to penalizations

对于$\R_+$上的常返线性扩散，我们研究了当时间参数趋于无穷大时，其局部时间在0处的分布的渐近性。在逆当地时间的Lévy测度是次指数的假设下，这个分布表现为Lévy测度的倍数。利用谱表示，我们可以找到倍数的精确值。为此，我们还需要一个结果的卷积的次指数分布和任意分布的渐近行为的$\R_+.$的确切知识的渐近行为的分布的本地时间允许我们分析的过程中，通过一个惩罚程序与本地时间。这一结果推广了Roynette，Vallois和Yor \cite{rvyV}对Bessel过程的惩罚.

Inverse source problem for time fractional diffusion with discrete random noise

In this paper, we deal with the inverse source problem of determining a source in a time fractional diffusion equation where data are given at a fixed time. This problem is ill-posed, i.e., the solution does not depend continuously on the data. To regularize the instable solution, we use the trigonometric method in nonparametric regression associated with the truncated expansion method. We also investigate the convergence rate.

An inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion

We study the inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion with Hurst index $H\in(0,1)$. With the aid of a novel estimate, by using the operator approach we propose regularity analyses for the direct problem. Then we provide a reconstruction scheme for the source terms $f$ and $g$ up to the sign. Next, combining the properties of Mittag-Leffler function, the complete uniqueness and instability analyses are provided. It's worth mentioning that all the analyses are unified for $H\in(0,1)$.

α-分集过程和归一化逆高斯扩散

Alpha-diversity processes and normalized inverse-Gaussian diffusions

无限多中性等位基因模型最近被推广到一类与两参数Poisson-Dirichlet型Gibbs划分相关的扩散过程。本文引入了一类由归一化逆高斯随机概率测度引起的与Gibbs划分的不同子类相关联的无穷维扩散。这种扩散描述了无限多类型的频率的演变以及由α多样性扩散驱动的随时间变化的突变率的动态。作为一个动态的版本，相对于这个框架，吉布斯分区的相应概念，后者是明确地从一个基本的人口模型，并显示出一致的，在一个特殊的情况下，与扩散近似的一个关键的高尔顿-沃森分支过程。一类无穷维过程的特点是在适当的域上的无穷小生成元，并证明是有限多类型的Feller扩散的某一序列的分布极限。此外，通过适当变换的Moran型粒子过程提供离散表示，其中粒子是来自归一化的逆高斯随机概率测度的样本。讨论了极限扩散与双参数模型的关系。