Olga V. Aryasova, Andrey Yu. Pilipenko
We consider a Brownian motion on the plane with semipermeable membranes on n rays that have a common endpoint in the origin. We obtain the necessary and sufficient conditions for the process to reach the origin and we show that the probability of hitting the origin is equal to zero or one.
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Local perturbations of a Brownian motion are considered. As a limit we obtain a non-Markov process that behaves as a reflected Brownian motion on the positive half line until its local time at zero reaches some exponential level, then changes a sign and behaves as a reflected Brownian motion on the negative half line until some stopping time, etc.
We consider two depending Wiener processes which have membranes at zero with different permeability coefficients. Starting from different points, the processes almost surely do not meet at any fixed point except that where membranes are situated. The necessary and sufficient conditions for the meeting of the processes are found. It is shown that the probability of meeting is equal to zero or one.
A multidimensional Brownian motion with partial reflection on a hyperplane $S$ in the direction $qN+α$, where $N$ is the conormal vector to the hyperplane and $q\in [-1,1], α\in S$ are given parametres, is constructed and this construction is based on both analytic and probabilistic approaches. The joint distribution of $d-$dimensional analogy to skew Brownian motion and its local time on the hyperplane is obtained.
Applying the technique of characteristic functions developped for one-dimensional regular surfaces (curves) with compact support, we obtain the distribution of hitting probabilities for a wide class of finite membranes on square lattice. Then we generalize it to multi-dimensional finite membranes on hypercubic lattice. Basing on these distributions, we explicitly construct the Brownian self-transport operator which governs the Laplacian transfer. In order to verify the accuracy of the distribution of hitting probabilities, numerical analysis is carried out for some particular membranes.
We study Markov chains on $\mathbb Z^m$, $m\geq 2$, that behave like a standard symmetric random walk outside of the hyperplane (membrane) $H=\{0\}\times \mathbb Z^{m-1}$. The transition probabilities on the membrane $H$ are periodic and also depend on the incoming direction to $H$, what makes the membrane $H$ two-sided. Moreover, sliding along the membrane is allowed. We show that the natural scaling limit of such Markov chains is a $m$-dimensional diffusion whose first coordinate is a skew Brownian motion and the other $m-1$ coordinates is a Brownian motion with a singular drift controlled b…
A particle moves randomly over the integer points of the real line. Jumps of the particle outside the membrane (a fixed "locally perturbating set") are i.i.d., have zero mean and finite variance, whereas jumps of the particle from the membrane have other distributions with finite means which may be different for different points of the membrane; furthermore, these jumps are mutually independent and independent of the jumps outside the membrane. Assuming that the particle cannot jump over the membrane we prove that the weak scaling limit of the particle position is a skew Brownian motion with p…
For a continuous function $f \in \mathcal{C}([0,1])$, define the Vervaat transform $V(f)(t):=f(τ(f)+t \mod1)+f(1)1_{\{t+τ(f) \geq 1\}}-f(τ(f))$, where $τ(f)$ corresponds to the first time at which the minimum of $f$ is attained. Motivated by recent study of quantile transforms for random walks and Brownian motion, we study the Vervaat transform of Brownian motion and Brownian bridges with arbitary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decompositions and study its semimartingale properties. …
On the convex hull of a planar Brownian bridge with a random Gaussian endpoint
研究一类各向同性平面高斯过程，计算其凸包的期望周长和面积，得到关于参数σ的闭式公式，并推广到多个独立过程的情形。
Functional limit theorems for perturbed random walks
本文研究扰动随机游走的标度极限，在膜外跳跃为有限方差零均值时给出反射布朗运动等极限，并探讨稳定律吸引域情形下的可能极限。
In a recent work J. Pitman and W. Tang defined the Vervaat's transform for a Brownian bridge with two different endpoints and for a Brownian motion between times $0$ and $1$. They proved some path decomposition properties for these Vervaat's transforms and raised open questions on their semi-martingale decomposition. In this paper we give an alternative proof for the path decomposition and answer the open questions.
For a continuous function $f \in \mathcal{C}([0,1])$, define the Vervaat transform $V(f)(t):=f(τ(f)+t \mod1)+f(1)1_{\{t+τ(f) \geq 1\}}-f(τ(f))$, where $τ(f)$ corresponds to the first time at which the minimum of $f$ is attained. Motivated by recent study of quantile transforms of random walks and Brownian motion, we investigate the Vervaat transform of Brownian motion and Brownian bridges with arbitrary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decomposition and study its semi-martingale proper…
From snapping out Brownian motions to Walsh's spider processes on star-like graphs
该研究通过矩阵分析证明大渗透系数的弹离布朗运动可良好近似星形图$K_{1,k}$上的Walsh蜘蛛过程，揭示后者等价于图中心受半透膜迹线扰动的布朗运动，丰富了相关随机过程的近似理论。
There are no geodesic hubs in the Brownian sphere
本文定义度量空间中的 k-枢纽，并证明 Brownian 球面内不存在任何 k≥3 的测地枢纽。
We consider n non-intersecting Brownian motions with two fixed starting positions and two fixed ending positions in the large n limit. We show that in case of 'large separation' between the endpoints, the particles are asymptotically distributed in two separate groups, with no interaction between them, as one would intuitively expect. We give a rigorous proof using the Riemann-Hilbert formalism. In the case of 'critical separation' between the endpoints we are led to a model Riemann-Hilbert problem associated to the Hastings-McLeod solution of the Painleve II equation. We show that the Painlev…
We provide a simple algorithm for construction of Brownian paths approximating those of a Lévy process on a finite time interval. It requires knowledge of the Lévy process trajectory on a chosen regular grid and the law of its endpoint, or the ability to simulate from that. This algorithm is based on reordering of Brownian increments, and it can be applied in a recursive manner. We establish an upper bound on the mean squared maximal distance between the paths and determine a suitable mesh size in various asymptotic regimes. The analysis proceeds by reduction to the comonotonic coupling of inc…
Perimeter length of the convex hull of Brownian motion in the hyperbolic plane
研究双曲平面中布朗运动轨迹凸包周长的期望，通过将其与一维实值布朗运动的指数泛函期望关联，推导出小时间和大时间渐近行为。
For any starting point in $\mathbb{R}^d$, we identify the stochastic differential equation that is satisfied by distorted Brownian motion with respect to a certain discontinuous Muckenhoupt $A_2$-weight $ψ$. The discontinuities of $ψ$ typically take place on a sequence of level sets of the Euclidean norm $D_k:=\{x\in \mathbb{R}^d\, | \;\|x\|=d_k\}$, $k\in\mathbb{Z}$, where $(d_k)_{k\in\mathbb{Z}}\subset (0,\infty)$ may have accumulation points and each level set $D_k$ plays the role of a permeable membrane.
We prove that uniform random quadrangulations of the sphere with $n$ faces, endowed with the usual graph distance and renormalized by $n^{-1/4}$, converge as $n\to\infty$ in distribution for the Gromov-Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called {\em Brownian map}, which was introduced by Marckert & Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of {\em geodesic stars} in the map, whic…
On Portenko's approximation of skew Brownian motion
本文通过算子半群理论回顾 Portenko 定理，并将其推广到星图上的 Walsh 过程，揭示了倾斜布朗运动边界条件的参数转换机制。
We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces M^{p, q}_s and Wiener amalgam spaces W^{p, q}_s. We show that the periodic Brownian motion belongs locally in time to M^{p, q}_s (T) and W^{p, q}_s (T) for (s-1)q < -1, and the condition on the indices is optimal. Moreover, with the Wiener measure μon T, we show that (M^{p, q}_s (T), μ) and (W^{p, q}_s (T), μ) form abstract Wiener spaces for the same range of indices, yielding large deviation estimates. We also establish the endpoint regularity of the periodic Brownian motion wi…
Walsh's Brownian Motion and Donsker Scaling Limits of Perturbed Random Walks
本文研究了受扰随机游走的邓斯克缩放极限，证明其收敛于具有特定权重的瓦尔什布朗运动，并展示了其在膜上的振荡斜布朗运动特性。
Boundary behavior and optimal pointwise Hölder exponent of the total local time of $(1+β)$-stable super-Brownian motion
针对带(1+β)-稳定分支机制的一维超布朗运动的总局部时，证明其正局部时点集几乎surely是有界开区间，端点处的逐点Hölder指数为1+2/β，明确了端点处Hölder条件的成立阈值。
Persistence probabilities of spherical fractional Brownian motion
本文计算了球面分数布朗运动持续概率的衰减率，发现其与欧几里得情形相似，并作为副产品推广了具有代数端点奇点函数的级数表示系数研究。
We present a generalization of multiple orthogonal polynomials of type I and type II, which we call multiple orthogonal polynomials of mixed type. Some basic properties are formulated, and a Riemann-Hilbert problem for the multiple orthogonal polynomials of mixed type is given. We derive a Christoffel-Darboux formula for these polynomials using the solution of the Riemann-Hilbert problem. The main motivation for studying these polynomials comes from a model of non-intersecting one-dimensional Brownian motions with a given number of starting points and endpoints. The correlation kernel for the …
Geodesics in the Brownian map: Strong confluence and geometric structure
研究Brownian map中测地线的强汇合现象与几何结构，证明了测地线相交与数量的精确界限，分类了点对间测地线配置，并证实测地线骨架维数为1。
We consider n one-dimensional Brownian motions, such that n/2 Brownian motions start at time t=0 in the starting point a and end at time t=1 in the endpoint b and the other n/2 Brownian motions start at time t=0 at the point -a and end at time t=1 in the point -b, conditioned that the n Brownian paths do not intersect in the whole time interval (0,1). The correlation functions of the positions of the non-intersecting Brownian motions have a determinantal form with a kernel that is expressed in terms of multiple Hermite polynomials of mixed type. We analyze this kernel in the large n limit for …
We consider non-colliding Brownian motions with two starting points and two endpoints. The points are chosen so that the two groups of Brownian motions just touch each other, a situation that is referred to as a tacnode. The extended kernel for the determinantal point process at the tacnode point is computed using new methods and given in a different form from that obtained for a single time in previous work by Delvaux, Kuijlaars and Zhang. The form of the extended kernel is also different from that obtained for the extended tacnode kernel in another model by Adler, Ferrari and van Moerbeke. W…
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。
Optimal Liquidation with Support and Resistance Levels under Multi-Skew Brownian Motion
本文研究多偏斜布朗运动下带支撑与阻力位的最优清算问题，提出三种几何形态的闭式准则，并揭示向阻力位卖出为结论及参数识别的不对称性。
Radon-Nikodym derivative of inhomogeneous Brownian last passage percolation
本文研究了非齐次布朗最后渗流穿透（LPP）空间增量法则的 Radon-Nikodym 导数，证明了其 $L^p$ 范数的阶及齐次初始数据下的最优增长，并将框架应用于 KPZ 不动点的玩具模型。
We study the law of a hypoelliptic Brownian motion on an infinite-dimensional Heisenberg group based on an abstract Wiener space. We show that the endpoint distribution, which can be seen as a heat kernel measure, is absolutely continuous with respect to a certain product of Gaussian and Lebesgue measures, that the heat kernel is quasi-invariant under translation by the Cameron-Martin subgroup, and that the Radon-Nikodym derivative is Malliavin smooth.
How long does a diffusing molecule spend in a close vicinity of a confining boundary or a catalytic surface? This quantity is determined by the boundary local time, which plays thus a crucial role in the description of various surface-mediated phenomena such as heterogeneous catalysis, permeation through semi-permeable membranes, or surface relaxation in nuclear magnetic resonance. In this paper, we obtain the probability distribution of the boundary local time in terms of the spectral properties of the Dirichlet-to-Neumann operator. We investigate the short-time and long-time asymptotic behav…
Deterministic--Distance Couplings of Brownian Motions on Radially Isoparametric Manifolds
本文在径向等参流形上建立统一几何框架，推导出控制布朗运动粒子间距离确定性演化的漂移窗口不等式，证明其作为耦合存在性判据的充分必要性，并将常曲率分类推广到一般径向等参流形。
Interval-Constrained Brownian Paths: Exact Interpolation and Extrapolation
本文研究区间约束布朗运动与布朗桥的精确插值和外推，提出统一采样套件，含边界端点并自动切换密度表示，提升接受率，并扩展至多时刻骨架路径模拟。
Diffusion through semipermeable structures arises in a wide range of processes in the physical and life sciences. Examples at the microscopic level range from artificial membranes for reverse osmosis to lipid bilayers regulating molecular transport in biological cells to chemical and electrical gap junctions. There are also macroscopic analogs such as animal migration in heterogeneous landscapes. It has recently been shown that one-dimensional diffusion through a barrier with constant permeability $κ_0$ is equivalent to snapping out Brownian motion (BM). The latter sews together successive rou…
Large deviations of Dyson Brownian motion on the circle and multiradial SLE(0+)
本文证明了圆上Dyson布朗运动在耦合参数β趋于无穷时的有限时间大偏差原理，刻画了其零能量与有限能量系统的长时间行为，并将结果推广到κ趋于0+时多径向SLE的有限时间大偏差。
Brownian Motion with a Pulse: A Biostatistician's Guide to Diffusions, Bridges, Functional PCA, and First-Passage Models
本文教程系统介绍布朗运动在生物统计中的应用，涵盖构造、路径性质、马尔可夫性、Karhunen-Loeve展开、函数主成分分析、反射原理、局部时、随机微分方程、布朗桥及经验过程极限，并通过纵向生物标志物、退化建模等实例展示其应用。
We consider directed last passage percolation on $\mathbb{Z}^2$ with exponential passage times on the vertices. A topic of great interest is the coupling structure of the weights of geodesics as the endpoints are varied spatially and temporally. A particular specialization is when one considers geodesics to points varying in the time direction starting from a given initial data. This paper considers the flat initial condition which corresponds to line-to-point last passage times. Settling a conjecture by Ferrari and Spohn (SIGMA, 2016), we show that for the passage times from the line $x+y=0$ …
For a Brownian directed polymer in a Gaussian random environment, with $q(t,\cdot)$ denoting the quenched endpoint density and \[ Q_n(t,x_1,\ldots,x_n)=\mathbf{E}[q(t,x_1)\ldots q(t,x_n)], \] we derive a hierarchical PDE system satisfied by $\{Q_n\}_{n\geq1}$. We present two applications of the system: (i) we compute the generator of $\{μ_t(dx)=q(t,x)dx\}_{t\geq0}$ for some special functionals, where $\{μ_t(dx)\}_{t\geq0}$ is viewed as a Markov process taking values in the space of probability measures; (ii) in the high temperature regime with $d\geq 3$, we prove a quantitative central limit t…
Renewal theory for Brownian motion across a stochastically gated interface
本文提出了一种基于再生理论的布朗运动模型，用于描述粒子通过随机门控界面的扩散过程，并通过数学推导展示了该模型在不同极限条件下的应用。
We consider the partition function $Z_{\ell}(\vec x,0\vert \vec y,t)$ of $\ell$ non-intersecting continuous directed polymers of length $t$ in dimension $1+1$, in a white noise environment, starting from positions $\vec x$ and terminating at positions $\vec y$. When $\ell=1$, it is well known that for fixed $x$, the field $\log Z_1(x,0\vert y,t)$ solves the Kardar-Parisi-Zhang equation and admits the Brownian motion as a stationary measure. In particular, as $t$ goes to infinity, $Z_1(x,0\vert y,t)/Z_1(x,0\vert 0,t) $ converges to the exponential of a Brownian motion $B(y)$. In this article, w…
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