线上扩散过程的小时间行为和时变扩散过程

时变Pearson扩散的关联结构

Correlation structure of time-changed Pearson diffusions

多项式系数扩散方程的随机解称为皮尔逊扩散。如果时间导数被一个分布的分数阶导数所代替，则随机解被称为分数阶皮尔逊扩散。本文利用广义Mittag-Leffler函数，导出了稳态分数Pearson扩散的协方差函数的计算公式。该公式表明，分数皮尔逊扩散是长程相关的，相关性像幂律一样下降，其指数等于分布分数导数的最小阶。

时间变换反射扩散：亚扩散、超扩散与爆炸现象

Time-changed reflected diffusions: subdiffusion, superdiffusion and explosion

本文研究广义抛物域内受独立逆次ordinator时间变换驱动的斜反射扩散，强化了相关增长律，得到二阶矩渐近，分析了不同参数下的扩散 regime 与爆炸条件，确立了强存在性等性质。

老年随机游动的扩散常数和鞅

Diffusion constants and martingales for senile random walks

利用时变的方法导出了老年随机游动的扩散常数和鞅。我们提供了直接计算的扩散常数随时间变化的步行。或者，这些常数的值可以从与时变游动相关联的鞅中导出。利用逆时变，通过这些鞅得到了老年随机游动的扩散常数。当游动是扩散的，弱收敛到布朗运动可以证明使用鞅泛函极限定理。

一类扩散时间变换随机SIS传染病模型：适定性、长时间行为与数值逼近

A diffusion time-changed stochastic SIS epidemic model: well-posedness, long-time behavior, and numerical approximation

提出并分析由时变布朗运动驱动的扩散时变随机SIS传染病模型，证明其全局正解唯一性，探究疾病灭绝与持久性，构造保正对数欧拉-丸山数值方法并证明强收敛阶，通过数值实验验证收敛率与保正性。

流形上的坐标变换随机场

Coordinate changed random fields on manifolds

在紧致黎曼流形上引入了一类不定常随机场。这些都是由随时间变化的布朗运动。这些过程是时变扩散，或者是涉及Laplace-Beltrami算子和$β\in (0,1)$阶时间分数导数的方程的随机解。因此，我们在这项工作中提出的不定常的随机场，可以实现通过组合，并可以被视为随机变化的流形上的随机场。

Reflections at infinity of time changed RBMs on a domain with Liouville branches

Let $Z$ be the transient reflecting Brownian motion on the closure of an unbounded domain $D\subset {\mathbb R}^d$ with $N$ number of Liouville branches. We consider a diffusion $X$ on $\overline D$ having finite lifetime obtained from $Z$ by a time change. We show that $X$ admits only a finite number of possible symmetric conservative diffusion extensions $Y$ beyond its lifetime characterized by possible partitions of the collection of $N$ ends and we identify the family of the extended Dirichlet spaces of all $Y$ (which are independent of time change used) as subspaces of the space ${\rm BL}…

具有双边均值回复跳跃的时变CIR缺省强度

Time-changed CIR default intensities with two-sided mean-reverting jumps

本文利用Bochner意义下的从属关系，将经典的扩散违约强度模型推广到跳扩散情形。我们从二元过程$(X,D)$开始，扩散状态变量$X$驱动默认强度和默认指示器过程$D$，并且时间用Lévy从属子${\mathcal{T}}$改变它。我们将时变过程$(X^ϕ_t,D^ϕ_t)=(X({\mathcal{T}}_t),D({\mathcal{T}}_t))$刻画为Markovian-Itô半鞅，并通过对$D^ϕ$的Doob-Meyer分解证明了时变模型中的缺省时间具有跳扩散或纯跳强度.当$X$是具有均值回复漂移的CIR扩散时，从属模型（SubCIR）的默认强度是跳跃扩散或纯跳跃过程，在两个方向上具有均值回复跳跃，并且保持非负。SubCIR违约强度模型通过显式计算相关半群的本征函数展开，得到信用敏感证券的封闭形式定价。

Exponential and strong ergodicity for one-dimensional symmetric stable jump diffusions

We obtain explicit criteria for both exponential ergodicity and strong ergodicity for one-dimensional time-changed symmetric stable processes with $α\in(1,2)$. Explicit lower bounds for ergodic convergence rates are given.

Multidimensional Sticky Brownian Motions: Tail Behaviour of the Joint Stationary Distribution

Sticky Brownian motions, as time-changed semimartingale reflecting Brownian motions, have various applications in many fields, including queuing theory and mathematical finance. In this paper, we are concerned about the stationary distributions of a multidimensional sticky Brownian motion, provided it is stable. We will study the large deviations principle for stationary distribution and the tail behaviour of the joint stationary distribution.

时间变换多维Wiener过程的泛函极限定理

Functional limit theorems for a time-changed multidimensional Wiener process

本文研究强度函数在各卦限无穷远有不同极限的时间变换多维Wiener过程，利用随机过程叠加的泛函极限定理证明其归一化后的泛函极限定理，极限包含斜Brown运动的多维类似物。

时变Bessel过程与信用风险

Time-Changed Bessel Processes and Credit Risk

常数方差弹性（CEV）模型的数学提出，然后在一个信用股权混合框架。接下来，我们提出了扩展的CEV模型与默认：首先通过添加一个随机波动扩散不相关的股票价格过程，然后通过更一般的时间改变贝塞尔过程，最后通过相关的随机波动移动的股票。本文讨论了严格局部鞅和真鞅的性质。分析公式提供和傅立叶和拉普拉斯变换技术，然后可以用来计算期权价格和违约概率。

时间变换Lévy过程在无穷远处的进入与退出结构

The structure of entrance and exit at infinity for time-changed Lévy processes

本文研究时间变换Lévy过程在无穷远处的进入与退出结构，提出适用于所有这类过程的广义尺度性质，给出其在无穷远具备进入性与正则边界点的精确条件，可推广自相似马尔可夫过程的尺度论证。

带纵向漂移的时间变换球面布朗运动

Time changed spherical Brownian motions with longitudinal drifts

本文研究带纵向漂移的二维球面扩散过程的时间变换，通过谱分解法求解时间非局部Kolmogorov方程强解，构造复合过程得到解的随机表示与谱分解，给出低正则初始数据下的极弱解，并证明该过程为反常扩散，改进了已有结果。

可逆化学反应扩散方程的渐近性态

Asymptotic behaviour of reversible chemical reaction-diffusion equations

研究了一类具有相同扩散项的可逆化学反应扩散方程的渐近性态。特别是，我们证明了在两种情况下的最佳速率：当没有扩散和在经典的“两两”的情况下。

1/2 Hölder扩散布居过程的长时间行为

Long time behaviour of 1/2 Hölder diffusion population processes

本文研究了紧集上一类具有Hölder连续扩散项的扩散过程的长时间行为，这些过程自然地出现在常微分方程的随机逼近中.我们将证明这些过程在有限时间内到达边界，并证明存在准平稳分布

分数阶时空扩散方程的连续时间随机游动模型

Continuous time random walk models for fractional space-time diffusion equations

本文研究了分数维时空扩散过程的连续时间随机游动模型。与所考虑的方程相关联的随机过程表示时变过程，其中时变过程是Lévy稳定从属子，其稳定性指标为$β\in (0,1).$。本文用一种新的解析方法证明了所构造的CTRW收敛于与相应的分数阶扩散方程相关联的时变过程。

Self-interacting diffusions: long-time behaviour and exit-problem in the convex case

We study a class of time-inhomogeneous diffusion: the self-interacting one. We show a convergence result with a rate of convergence that does not depend on the diffusion coefficient. Finally, we establish a so-called Kramers' type law for the first exit-time of the process from domain of attractions when the landscapes are uniformly convex.

具有对称重叠的无穷多等位扩散的渐近性

Asymptotic Behaviour of an Infinitely-Many-Alleles Diffusion with Symmetric Overdonminance

本文研究了具有对称超显性\cite{MR1626158}的无限多等位基因扩散的平稳分布$π_{λ,θ}$的极限分布。在\cite{MR2519357}中，$π_{λ,θ}$的大偏差原理表明，对于$π_{λ,θ}$的极限分布，存在可数个相变，临界点是$λ=k(k+1), k\geq1$ .然而，在这些临界点的渐近行为是不清楚的。本文对危重病例作了明确的描述。

Asymptotic behaviour for a time-inhomogeneous Kolmogorov type diffusion

We study a kinetic stochastic model with a non-linear time-inhomogeneous drag force and a Brownian-type random force. More precisely, the Kolmogorov type diffusion $(V,X)$ is considered: here $X$ is the position of the particle and $V$ is its velocity and is solution of a stochastic differential equation driven by a one-dimensional Brownian motion, with the drift of the form $t^{-β}F(v)$. The function $F$ satisfies some homogeneity condition and $β$ is positive. The behaviour of the process $(V,X)$ in large time is proved by using stochastic analysis tools.

时变反射布朗运动的唯一延拓

On unique extension of time changed reflecting Brownian motions

设$D$是$\RR^d$中的无界域，其中有$d\geq 3$ .证明了若$D$包含无界一致区域，则$\overline D$上的对称反射布朗运动（RBM）是瞬态的.接下来，假设$\overline D$上的RBM $X$是瞬态的，并且对于$\overline D$上的严格正连续可积函数$m$，令$Y$是其通过Revuz测度${\bf 1}_D(x) m(x)dx$的时间变化。进一步证明了如果存在$r>0$使得$D\setminus \overline {B(0, r)}$是一个无界一致域，则$Y$允许一个且只有一个对称扩散真正地扩展它并且不允许杀戮.换句话说，在这种情况下，$X$（或等效地，$Y$）在无穷远处具有唯一的马丁边界点。

一类非时齐扩散方程的存在性和渐近性态

Existence and asymptotic behaviour of some time-inhomogeneous diffusions

考虑由漂移系数为$b(t,x)=ρ\,{\rm sgn}(x)|x|^α/t^β$的布朗运动驱动的非时齐随机微分方程的解。这个过程可以被看作是一个扭曲的布朗运动的潜力，可能是奇异的，取决于时间。在得到解的存在唯一性的结果后，我们研究了它的渐近性态，并用参数$ρ,α$和$β$对常返性、瞬时性和收敛性作了精确的描述.更确切地说，渐近分布，重对数型法律和速度的瞬态和爆炸证明了这样的过程。

小噪声Levy驱动扩散的亚稳行为

Metastable Behaviour of Small Noise Levy-Driven Diffusions

考虑R中一个由向量场-U '驱动的动力系统，其中U是满足某些正则性条件的多阱势。我们扰动这个动力系统的Levy噪声强度小，使其Levy措施的最重的尾巴是经常变化的。我们表明，扰动的动力系统表现出亚稳态行为，即在适当的时间尺度，它提醒的马尔可夫跳跃过程采取的潜在的U的局部极小值的值。由于随机扰动的重尾性质，结果与研究充分的纯高斯情况有很大不同。

生灭过程是经过时间变换的 Feller 布朗运动

Birth-death processes are time-changed Feller's Brownian motions

本文通过特定时间变换建立 Feller 布朗运动与生灭过程的双向等价关系，并给出参数精确对应与构造性路径表示框架。

Non-Local Pearson diffusions

In this paper we focus on strong solutions of some heat-like problems with a non-local derivative in time induced by a Bernstein function and an elliptic operator given by the generator or the Fokker-Planck operator of a Pearson diffusion. Such kind of non-local equations naturally arise in the treatment of particle motion in heterogeneous media. In particular, we use spectral decomposition results for the usual Pearson diffusion to exploit explicit solutions of the aforementioned equations. Moreover, we provide stochastic representation of such solutions in terms of time-changed Pearson diffu…

Non-Local Solvable Birth-Death Processes

In this paper we study strong solutions of some non-local difference-differential equations linked to a class of birth-death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth-death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth-death processes.

Asymptotic behaviour of randomised fractional volatility models

We study the asymptotic behaviour of a class of small-noise diffusions driven by fractional Brownian motion, with random starting points. Different scalings allow for different asymptotic properties of the process (small-time and tail behaviours in particular). In order to do so, we extend some results on sample path large deviations for such diffusions. As an application, we show how these results characterise the small-time and tail estimates of the implied volatility for rough volatility models, recently proposed in mathematical finance.

Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes

We study the long time behaviour of a large class of diffusion processes on $R^N$, generated by second order differential operators of (possibly) degenerate type. The operators that we consider {\em need not} satisfy the Hörmander condition. Instead, they satisfy the so-called UFG condition, introduced by Herman, Lobry and Sussman in the context of geometric control theory and later by Kusuoka and Stroock, this time with probabilistic motivations. In this paper we study UFG diffusions and demonstrate the importance of such a class of processes in several respects: roughly speaking i) we show t…

星图上的粘性扩散：刻画与 Itô 公式

Sticky diffusions on star graphs : characterization and It{ô} formula

本文研究星图顶点处具有粘性行为的连续扩散，利用时间变换及 Itô-McKean 技术加以刻画，并证明 Freidlin-Sheu 型 Itô 公式与径向分量 SDE。

时变系数分数阶扩散

Fractional diffusions with time-varying coefficients

研究了分数布朗运动$B_H(t)$的分数阶扩散方程.我们得到的这些方程的解决方案，这是概率定律的推广$B_H(t)$。我们的分析是基于McBride分数算子推广的超贝塞尔算子$L$和转换其分数幂$L^α$到Erdélyi-Kober分数次积分。我们还研究了r. v.的概率性质。的分布满足涉及Caputo和Riesz分数阶导数的时空分数阶方程。在分析时变系数分数阶方程时得到的一些结果具有时变r. v.分布的形式。's.

Properties at infinity of diffusion semigroups and stochastic flows via weak uniform covers

A unified treatment is given of some results of H. Donnelly-P. Li and L. Schwartz concerning the behaviour of heat semigroups on open manifolds with given compactifications, on one hand, and the relationship with the behaviour at infinity of solutions of related stochastic differential equations on the other. A principal tool is the use of certain covers of the manifold: which also gives a non-explosion test. As a corollary we obtain known results about the behaviour of Brownian motions on a complete Riemannian manifold with Ricci curvature decaying at most quadratically in the distance functi…

Semi-implicit Euler-Maruyama method for non-linear time-changed stochastic differential equations

The semi-implicit Euler-Maruyama (EM) method is investigated to approximate a class of time-changed stochastic differential equations, whose drift coefficient can grow super-linearly and diffusion coefficient obeys the global Lipschitz condition. The strong convergence of the semi-implicit EM is proved and the convergence rate is discussed. When the Bernstein function of the inverse subordinator (time-change) is regularly varying at zero, we establish the mean square polynomial stability of the underlying equations. In addition, the numerical method is proved to be able to preserve such an asy…

具有超线性漂移的测度依赖非线性扩散：首次退出时间的渐近行为

Measure-dependent non-linear diffusions with superlinear drifts: asymptotic behaviour of the first exit-times

本文在无凸性假设的可逆 McKean-Vlasov SDE 中，利用大偏差原理和改进耦合方法证明了首次退出时间的 Kramers 型指数渐近律，显著推广了已有结果。

大波动稳定市场

Large volatility-stabilized markets

我们研究了相互作用的扩散过程系统的行为，在数学金融文献中称为波动稳定的市场模型，当扩散的数量趋于无穷大。我们表明，经过适当的时间参数的重新缩放，系统的经验措施收敛到退化抛物型偏微分方程的解决方案。一个随机表示后者的一维分布的时变平方贝塞尔过程允许我们给出一个明确的描述的限制。

一些基于 Mittag-Leffler 等待时间的再生过程的概率性质及其时间改变版本

Some probabilistic properties and time-changed versions of a renewal process based on Mittag-Leffler waiting times

本文研究了基于 Mittag-Leffler 等待时间的再生过程的概率性质及其时间改变版本，探讨了其分布特性、极限行为及在破产理论中的应用。

优化可变速率扩散以在设定时间击中无穷小目标

Optimizing a variable-rate diffusion to hit an infinitesimal target at a set time

考虑了扩散率限制在两个正值$r_{1}<r_{2}$之间的时变Bessel过程的随机最优化问题。问题是找到一个最佳的自适应策略选择的扩散率，以最大限度地提高机会击中一个无穷小区域周围的原点在一个设定的时间在未来。更确切地说，与“击中原点的机会”相关的参数是在最终时间概率密度的原点处诱导的奇点的指数。我表明，最佳的指数解决了超越方程依赖于比率$\frac{r_{2}}{r_{1}}$和贝塞尔过程的尺寸。

中间时间尺度快细胞流中的反常扩散

Anomalous diffusion in fast cellular flows at intermediate time scales

众所周知，在长时间尺度上，示踪粒子在细胞流中扩散的行为实际上是布朗运动的行为。本文研究了扩散开始前“中间”时间尺度上的行为。各种物理学表明，应该观察到一种反常的扩散行为。我们证明了中间时间尺度的方差增长像$O(\sqrt{t})$。因此，在这些时间尺度上，有效行为不可能是纯粹的扩散行为，而是与异常扩散行为相一致。

时间变换的马尔可夫过程与时空耦合非局部方程

Time-changed Markov processes and space-time coupled non-local equations

研究耦合完全非局部方程，通过基于异常过程的随机表示建立解的存在唯一性，利用最大值原理推导唯一性，还得到布莱克 - 斯科尔斯方程的非局部类似物，解决特定条件下衍生证券定价问题。

Asymptotic behaviour of a conservative reaction-diffusion system associated with a Markov process algebra model

This paper demonstrates a lower and upper solution method to investigate the asymptotic behaviour of the conservative reaction-diffusion systems associated with Markovian process algebra models. In particular, we have proved the uniform convergence of the solution to its constant equilibrium for a case study as time tends to infinity, together with experimental results illustrations.

平均原理在叶状扩散中的应用：叶的拓扑

Application of an averaging principle on foliated diffusions: topology of the leaves

本文考虑$\R^3$中嵌入紧致子流形的叶状管状邻域中定义的叶状布朗运动中的$εK$横向扰动向量场.我们研究了系统在这种$ε$扰动下的有效行为。如果扰动矢量场$K$与相应叶的高斯曲率成比例，则当$ε$趋于零时，通过$t/ε$重新标度时间后，横向分量接近与$M$的欧拉示性数成比例的线性增加行为。估计的收敛速率。

使用时间变更马尔可夫过程进行采样

Sampling with time-changed Markov processes

本文提出通过时间变更马尔可夫过程加速MCMC算法收敛，通过状态依赖函数调整基础过程时间速度，证明了其理论性质并在多峰分布和稀有事件模拟中实现性能提升。

基于秩的相互作用扩散的混沌传播和标量拟线性抛物型方程的长时间行为

Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation

研究了一类拟线性抛物型Cauchy问题，其初始条件为实直线上的累积分布函数。我们称“概率解”为始终保持累积分布函数的弱解。我们证明了这样的解决方案的唯一性，我们推导出的存在性从一个系统的标量扩散过程的混沌结果的传播，其中的相互作用只取决于他们的排名。然后，我们调查的解决方案的长期行为。使用概率参数和弱假设下，我们表明，两个解决方案之间的Wasserstein距离流是压缩的。在保证概率解的正则性的更严格条件下，我们导出了流的时间导数的显式公式，并推导出解收敛到平衡点的条件.

Self-similar behaviour of a non-local diffusion equation with time delay

We study the asymptotic behaviour of solutions of a class of linear non-local measure-valued differential equations with time delay. Our main result states that the solutions asymptotically exhibit a parabolic like behaviour in the large times, that is precisely expressed in term of heat kernel. Our proof relies on the study of a-self-similar-rescaled family of solutions. We first identify the asymptotic behaviour of the solutions by deriving a convergence result in the sense of the Young measures. Then we strengthen this convergence by deriving suitable fractional Sobolev compactness estimate…

随机抛物算子的均匀化。扩散近似

Homogenization of random parabolic operators. Diffusion approximation

本文研究了系数在空间上为周期函数，在时间上为随机平稳的散度型二阶抛物算子的均匀化问题。在适当的混合假设下，我们研究了原始和均匀化问题的解之间的归一化差的极限行为。这种差异的渐近行为关键取决于空间和时间的比例因子之间的比率。在这里，我们研究的情况下，自相似抛物扩散标度。

A note on generalized fractional diffusion equations on Poincarè half plane

In this paper we study generalized time-fractional diffusion equations on the Poincarè half plane $\mathbb{H}_2^+$. The time-fractional operators here considered are fractional derivatives of a function with respect to another function, that can be obtained by starting from the classical Caputo-derivatives essentially by means of a deterministic change of variable. We obtain an explicit representation of the fundamental solution of the generalized-diffusion equation on $\mathbb{H}_2^+$ and provide a probabilistic interpretation related to the time-changed hyperbolic Brownian motion. We finally…

A fractional kinetic process describing the intermediate time behaviour of cellular flows

This paper studies the intermediate time behaviour of a small random perturbation of a periodic cellular flow. Our main result shows that on time scales shorter than the diffusive time scale, the limiting behaviour of trajectories that start close enough to cell boundaries is a fractional kinetic process: A Brownian motion time changed by the local time of an independent Brownian motion. Our proof uses the Freidlin-Wentzell framework, and the key step is to establish an analogous averaging principle on shorter time scales. As a consequence of our main theorem, we obtain a homogenization result…

某些特殊的自相互作用扩散：遍历性态和几乎必然收敛

Some particular self-interacting diffusions: Ergodic behaviour and almost sure convergence

本文研究了生活在$\mathbb{R}^d$上的自相互作用扩散$(X_t,t\geq 0)$.这些扩散是随机微分方程的解：\[\mathrm{d}X_t=\mathrm{d}B_t-g(t)\nabla V(X_t-\barμ_t)\,\mathrm{d}t,\]其中$\barμ_t$是过程$X$的经验平均值，$V$是渐近严格凸势，$g$是给定函数。我们研究了$X$的遍历行为，证明了它与$g$强相关.实际上，我们证明了$X$是遍历的（在极限商意义下）当且仅当$\barμ_t$ a.s.我们还给出了$X$的几乎必然收敛的一些条件（关于$g$和$V$）.

连续时间扩散的行为伪度量

Behavioural pseudometrics for continuous-time diffusions

本文针对连续时间扩散过程提出了两种行为伪度量，它们分别是1有界伪度量空间上不同泛函的不动点，并通过实值模态逻辑进行刻画，从而定量量化过程的行为差异。

Space-time fractional Dirichlet problems

This paper establishes explicit solutions for fractional diffusion problems on bounded domains. It also gives stochastic solutions, in terms of Markov processes time-changed by an inverse stable subordinator whose index equals the order of the fractional time derivative. Some applications are given, to demonstrate how to specify a well-posed Dirichlet problem for space-time fractional diffusions in one or several variables. This solves an open problem in numerical analysis.

面向具有遍历行为的随机混合系统的近似贝叶斯计算

Approximate Bayesian computation for stochastic hybrid systems with empirically observed ergodic behaviour

针对分段扩散马尔可夫过程（PDifMPs）参数估计难题，提出基于近似贝叶斯计算的推断框架，配套仿真算法与适配混合特性的汇总统计量，在多类遍历PDifMP算例中实现了可靠的参数恢复。

时间分数阶扩散的一个极限定理

A limit theorem to a time-fractional diffusion

证明了一个关于马氏过程的积分泛函的极限定理。马尔可夫动力学的特点是由线性玻尔兹曼方程建模的质量$λ^{-1}\gg 1$在外部周期性的潜力和经历碰撞的粒子与质量为1的背景气体的一维测试粒子。我们的极限定理的对象是势作用在测试粒子上的力的时间积分，我们考虑这个量的极限，即在$λ^{-1}$尺度上，$λ$在时间间隔内趋于零。在适当的重新标度下，由势产生的动量的总漂移收敛于由Ornstein-Uhlenbeck过程的零点处的本地时间改变的布朗运动。