Adam Gregosiewicz
We consider diffusion processes on metric graphs with semipermeable sticky membranes in each vertex. We prove that the process is governed by a Feller semigroup and find its asymptotic behavior as diffusion's speed increases to infinity with the same rate as permeability coefficients decreases to zero.
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Super and Weak Poincaré Inequalities for Sticky-Reflected Diffusion Processes
本文为粘性反射扩散过程建立超和弱Poincaré不等式，并刻画相关扩散半群的收敛速率和一致可积性。
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。
On the number of crossings and bouncings of a diffusion at a sticky threshold
本文研究了粘性扩散在阈值处交叉和反弹数量的渐近行为，并提出了粘性参数的估计方法。
Large deviations for sticky-reflecting Brownian motion with boundary diffusion
本文研究具有边界扩散的粘性反射布朗运动在短时间极限下的大偏差原理，揭示了速率函数随边界与内部扩散速度比较的尖锐转变，并由此提出一种新的最优输运模型。
Limit theorems for one-dimensional diffusions: reflected, skew, sticky, skew-sticky, and snapping-out limits
本文建立正则一维扩散弱收敛的一般条件，基于Itô–McKean表示将随机收敛化为确定性问题，并应用于半透膜介质均匀化，得到反射、偏斜、黏性及断裂等极限过程。
Functional convergence to the local time of a sticky diffusion
本文证明了基于高频观测的粘性扩散局部时间近似的一致性，并开发了粘性参数的一致估计量，通过数值实验评估了其统计性质。
Markov Chain Approximation of One-Dimensional Sticky Diffusions
本文提出了一种基于连续时间马尔可夫链的粘性扩散近似方法，用于高效计算粘性扩散的首次通过概率和债券定价。
We develop a new simulation method for multidimensional diffusions with sticky boundaries. The challenge comes from simulating the sticky boundary behavior, for which standard methods like the Euler scheme fail. We approximate the sticky diffusion process by a multidimensional continuous time Markov chain (CTMC), for which we can simulate easily. We develop two ways of constructing the CTMC: approximating the infinitesimal generator of the sticky diffusion by finite difference using standard coordinate directions, and matching the local moments using the drift and the eigenvectors of the covar…
Functional Inequalities for Brownian Motion on Riemannian Manifolds with Sticky-Reflecting Boundary Diffusion
本文在一般曲率假设下，利用边界与内部能量相互作用的插值方法，证明了具有粘性反射边界扩散的流形上布朗运动的Poincaré和对数Sobolev常数的几何上界，并附带得到Steklov特征值及边界迹算子的显式几何界。
Functional Inequalities for doubly weighted Brownian Motion with Sticky-Reflecting Boundary Diffusion
本文研究了具有粘性反射边界的流形上双重加权布朗运动的Poincaré和Logarithmic Sobolev常数上界，采用能量交互和加权Reilly公式进行插值分析，同时得到Steklov特征值和边界迹算子的上界。
A modified Euler-Maruyama method to simulate a one-dimensional sticky diffusion
针对一维粘性扩散过程，提出一种改进的Euler-Maruyama方法，通过概率选择反射更新或跳至粘性点，并证明该方案以1阶弱收敛。
We present a novel approach of coupling two multidimensional and non-degenerate Itô processes $(X_t)$ and $(Y_t)$ which follow dynamics with different drifts. Our coupling is sticky in the sense that there is a stochastic process $(r_t)$, which solves a one-dimensional stochastic differential equation with a sticky boundary behavior at zero, such that almost surely $|X_t-Y_t|\leq r_t$ for all $t\geq 0$. The coupling is constructed as a weak limit of Markovian couplings. We provide explicit, non-asymptotic and long-time stable bounds for the probability of the event $\{X_t=Y_t\}$.
Spectral gap estimates for Brownian motion on domains with sticky-reflecting boundary diffusion
本文通过插值方法推导出带有粘性反射边界扩散的布朗运动谱隙下界，并在流形情况下应用了 Reilly 公式。
Limit theorems for sticky SDEs with local times and applications to stochastic homogenization
本文研究了具有局部时间的粘性随机微分方程的极限定理，揭示了在存在半渗透界面时扩散过程的随机正则化机制。
Starting with a Brownian motion, we define and study a novel diffusion process by combining stickiness and oscillation properties. The associated stochastic differential equation, resolvent and semigroup are provided. Also the trivariate density of position, local time and occupation time of this diffusion is obtained explicitly. Furthermore, we give a construction of two Brownian motions with drift and scaling whose difference is an oscillating sticky Brownian motion, up to a multiplicative constant.
Pricing and hedging for a sticky diffusion
本文提出含粘性几何布朗运动风险资产的金融市场模型，证明仅当利率为零时无套利，推导定价方程并证明标准Black-Scholes可复制收益在粘性模型中仍可复制，最后通过数值实验评估离散时间对冲与误判粘性的影响。
Fractional Boundary Value Problems and elastic sticky Brownian motions, II: Non-local dynamic boundary conditions on smooth domains
本文研究了分数边值问题中的非局部动态边界条件，通过时间变换描述分数粘性条件和相关边界行为，揭示了布朗运动在边界上无限均时停留的陷阱效应。
Sticky-threshold diffusions, local time approximation and parameter estimation
该研究针对带奇异阈值或边界的一维扩散过程，构造收敛到局部时的高频路径泛函，结合占据时间逼近得到扩散系数存在跳跃不连续性时仍有效的粘性与偏度参数一致估计量，解决了多奇异特征共存下的参数估计问题。
We consider a diffusion in $\mathbb{R}^n$ whose coordinates each behave as one-dimensional Brownian motions, that behave independently when apart, but have a sticky interaction when they meet. The diffusion in $\mathbb{R}^n$ can be viewed as the $n$-point motion of a stochastic flow of kernels. We derive the Kolmogorov backwards equation and show that for a specific choice of interaction it can be solved exactly with the Bethe ansatz. We then use our formulae to study the behaviour of the flow of kernels for the exactly solvable choice of interaction.
Sticky Brownian motions on star graphs
本文在星图上构造随机过程，用奇异测度刻画陷阱并定义非局部算子，引入非局部动态条件，通过半马尔可夫过程获得全图运动的概率表示，并可推广至一般图结构。
We give a Dirichlet form approach for the construction of distorted Brownian motion in a bounded domain $Ω$ of $\mathbb{R}^d$, $d \geq 1$, with boundary $Γ$, where the behavior at the boundary is sticky. The construction covers the case of a static boundary behavior as well as the case of a diffusion on the hypersurface $Γ$ (for $d \geq 2)$. More precisely, we consider the state space $\overlineΩ=Ω\stackrel{.}{\cup} Γ$, the process is a diffusion process inside $Ω$, the occupation time of the process on the boundary $Γ$ is positive and the process may diffuse on $Γ$ as long as it sticks on the…
Weak approximation of stochastic differential equations with sticky boundary conditions
本文针对具有粘性边界条件的随机微分方程，构造并分析了半阶和一阶弱逼近数值格式，可用于求解相应线性抛物型偏微分方程，并通过数值实验验证了理论结果。
Sticky Brownian motion is the simplest example of a diffusion process that can spend finite time both in the interior of a domain and on its boundary. It arises in various applications such as in biology, materials science, and finance. This article spotlights the unusual behavior of sticky Brownian motions from the perspective of applied mathematics, and provides tools to efficiently simulate them. We show that a sticky Brownian motion arises naturally for a particle diffusing on $\mathbb{R}_+$ with a strong, short-ranged potential energy near the origin. This is a limit that accurately model…
We consider n-point sticky Brownian motions: a family of n diffusions that evolve as independent Brownian motions when they are apart, and interact locally so that the set of coincidence times has positive Lebesgue measure with positive probability. These diffusions can also be seen as n random motions in a random environment whose distribution is given by so-called stochastic flows of kernels. For a specific type of sticky interaction, we prove exact formulas characterizing the stochastic flow and show that in the large deviations regime, the random fluctuations of these stochastic flows are …
Close encounters of the sticky kind: Brownian motion at absorbing boundaries
本文构建了一维粘性布朗运动接触式模型，计算了粒子位置和占用时间的联合概率密度，以首次跨越随机阈值的占用时间定义吸收事件，并揭示了粘性RTP扩散极限与粘性BM的映射关系。
Sticky dispersion on the complete graph: a kinetic approach
该研究采用动力学视角，将完全图上的粘性扩散模型重新解释为新型经济物理模型，在平均场框架下推导了其长时间行为的解析与定量结果，并辅以数值验证。
Non-negative Martingale Solutions to the Stochastic Porous Medium Equation with Sticky Behavior
本文构造一维随机多孔介质方程在Dirichlet边界条件下具有零处粘性行为的非负鞅解，采用随机Faedo-Galerkin方法及弱收敛与Skorokhod表示技术。
We construct a family of semimartingales that describes the behavior of a particle system with sticky-reflecting interaction. The model is a physical improvement of the Howitt-Warren flow, an infinite system of diffusion particles on the real line that sticky-reflect from each other. But now particles have masses obeying the conservation law and the diffusion rate of each particle depends on its mass. The equation which describes the evolution of the particle system is a new type of equations in infinite-dimensional space and can be interpreted as an infinite-dimensional analog of the equation…
Boundary Approximation for Sticky Jump-Reflected Processes on the Half-Line
本文发展了一种通用理论，证明半直线上具有切换机制的随机游走弱收敛到具有广义反射的函数，并应用于存储过程、重流量极限及带延迟的扩散。
Pseudorandomness of the Sticky Random Walk
本文针对一族扩张图，肯定解决了 Golowich 和 Vadhan 关于粘性随机游走 TVD 误差依赖性的猜想，通过组合方法将误差界从 $O(λp^{O(p)})$ 加强至 $O(λ)$，并建立了广义粘性随机游走与无限扩张图族的参数化联系。
General diffusions on the star graph as time-changed Walsh Brownian motion
本研究将星图上的一般正则扩散表示为时间变换Walsh布朗运动，据此推导粘性粘合扩散的时间变换表示、占据时间公式，还拓展了Feller相关判据并证明星图Dirichlet问题解的存在唯一性及显式形式。
We study exclusion processes on the integer lattice in which particles change their velocities due to stickiness. Specifically, whenever two or more particles occupy adjacent sites, they stick together for an extended period of time, and the entire particle system is slowed down until the ``collision'' is resolved. We show that under diffusive scaling of space and time such processes converge to what one might refer to as a sticky reflected Brownian motion in the wedge. The latter behaves as a Brownian motion with constant drift vector and diffusion matrix in the interior of the wedge, and ref…
The Sticky Lévy Process as a solution to a Time Change Equation
本文提出粘性Lévy过程是由无负跳Lévy过程驱动的时间变更方程（TCE）的唯一适应解，克服了SDE在描述粘性Brownian运动时无适应解的缺陷，并提供了强近似方案。
We prove several properties of the EMCEL scheme, which is capable of approximating one-dimensional continuous strong Markov processes in distribution on the path space (the scheme is briefly recalled). Special cases include irregular stochastic differential equations and processes with sticky features. In particular, we highlight differences from the Euler scheme in the case of stochastic differential equations and discuss a certain "stabilizing" behavior of the EMCEL scheme like "smoothing and tempered growth behavior".
Discrete sticky couplings of functional autoregressive processes
本文通过构造离散粘性 Markov 链，为函数型自回归过程（包括 Langevin 扩散的 Euler-Maruyama 离散化）建立了关于参数 γ 一致成立的 Wasserstein 和全变差距离非渐近界，并应用于常微分方程参数的贝叶斯推断。
In this paper we study the asymptotic behavior of Brownian motion in both comb-shaped planar domains, and comb-shaped graphs. We show convergence to a limiting process when both the spacing between the teeth \emph{and} the width of the teeth vanish at the same rate. The limiting process exhibits an anomalous diffusive behavior and can be described as a Brownian motion time-changed by the local time of an independent sticky Brownian motion. In the two dimensional setting the main technical step is an oscillation estimate for a Neumann problem, which we prove here using a probabilistic argument.…
General diffusion processes as the limit of time-space Markov chains
本文证明网格值随机游动（时空马尔可夫链）的分布收敛于一般扩散过程，并给出在任意 $p$-Wasserstein 距离下基于网格单元尺寸的收敛速率及最优速率，进而建立渐近最优近似格式并通过数值实验验证。
We consider the system of sticky-reflected Brownian particles on the real line proposed in [arXiv:1711.03011]. The model is a modification of the Howitt-Warren flow but now the diffusion rate of particles is inversely proportional to the mass which they transfer. It is known that the system consists of a finite number of distinct particles for almost all times. In this paper, we show that the system also admits an infinite number of distinct particles on a dense subset of the time interval if and only if the function responsible for the splitting of particles takes an infinite number of values…
Perturbing Walsh's spider process by infinitesimal drifts
本文研究星形图上Walsh过程受半渗透软膜扰动的极限行为，发现极限过程仍为Walsh型但参数需修改，与相关论文构成配套研究。
We consider the optimal stopping problem consisting in, given a strong Markov process, a reward function and a discount rate, finding the stopping time such that the expected reward at the stopping time is maximum. The approach we follow, has two main components: the Dynkin's characterization of the value function as the smallest excessive function dominating the reward; and the Riesz representation of excessive functions in terms of the Green kernel, the main reference being Salminen 85. In the context of one-dimensional diffusions we give a complete characterization of the solution under som…
Graphon Limits of Graph Reaction--Diffusion Equations
研究图反应-扩散方程序列在图面容极限下的收敛性，揭示其非局部极限方程的解的收敛性质，并分析随机粒子过程的大数定律。
On the structure of increasing profits in a 1D general diffusion market with interest rates
本文研究了具有利率的1D一般扩散市场中增加利润的结构，通过分析确定性测度和交易策略，揭示了无套利理论与随机过程理论之间的新联系。
We introduce order-based diffusion processes as the solutions to multidimensional stochastic differential equations, with drift coefficient depending only on the ordering of the coordinates of the process and diffusion matrix proportional to the identity. These processes describe the evolution of a system of Brownian particles moving on the real line with piecewise constant drifts, and are the natural generalization of the rank-based diffusion processes introduced in stochastic portfolio theory or in the probabilistic interpretation of nonlinear evolution equations. Owing to the discontinuity …
A limit theorem for a sequence of diffusion processes on graphs is proved in a case when vary both parameters of the processes (the drift and diffusion coefficients on every edge and the asymmetry coefficients in every vertex), and configuration of graphs, where the processes are set on. The explicit formulae for the parameters of asymmetry for the vertices of the limiting graph are given in the case, when, in the pre-limiting graphs, some groups of vertices form knots contracting into a points.
We consider diffusion on discrete measure spaces as encoded by Markovian semigroups arising from weighted graphs. We study whether the graph is uniquely determined if the diffusion is given up to order isomorphism. If the graph is recurrent then the complete graph structure and the measure space are determined (up to an overall scaling). As shown by counterexamples this result is optimal. Without the recurrence assumption, the graph still turns out to be determined in the case of normalized diffusion on graphs with standard weights and in the case of arbitrary graphs over spaces in which each …
Evolution of Conditional Entropy for Diffusion Dynamics on Graphs
引入图上热扩散的条件熵，建立连续时间马尔可夫链与信息论框架，证明其满足热力学第二定律的信息论版本，并给出完全图、路径图、循环图上的显式结果及一般网络的渐近结果与界。
Convolutional layers within graph neural networks operate by aggregating information about local neighbourhood structures; one common way to encode such substructures is through random walks. The distribution of these random walks evolves according to a diffusion equation defined using the graph Laplacian. We extend this approach by leveraging classic mathematical results about hypo-elliptic diffusions. This results in a novel tensor-valued graph operator, which we call the hypo-elliptic graph Laplacian. We provide theoretical guarantees and efficient low-rank approximation algorithms. In part…
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