Random continuum and Brownian motion

Brownian Motions on Metric Graphs II - Construction of Brownian Motions on Single Vertex Graphs

Pathwise constructions of Brownian motions which satisfy all possible boundary conditions at the vertex of single vertex graphs are given.

布朗连续随机树上的布朗运动的覆盖时间

On the cover time of Brownian motion on the Brownian continuum random tree

本文证明了布朗连续随机树上的布朗运动的覆盖时间等于局部时间处处严格为正的时刻的下确界，并利用递归自相似性和新的Ray-Knight定理，推导出临界有限方差Galton-Watson树上简单随机游走的覆盖时间在退火律下依分布收敛到该覆盖时间。

Brownian Motions on Metric Graphs: Feller Brownian Motions on Intervals Revisited

The construction of the paths of all possible Brownian motions (in the sense of Knight) on a half line or a finite interval is reviewed.

Asymptotic behavior of weighted power variations of fractional Brownian motion in Brownian time

We study the asymptotic behavior of weighted power variations of fractional Brownian motion in Brownian time Z_t:= X_{Y_t}, t >= 0, where X is a fractional Brownian motion and Y is an independent Brownian motion.

Fluctuations of the power variation of fractional Brownian motion in Brownian time

We study the fluctuations of the power variation of fractional Brownian motion in Brownian time

On the excursions of drifted Brownian motion and the successive passage times of Brownian motion

By using the law of the excursions of Brownian motion with drift, we find the distribution of the $n-$th passage time of Brownian motion through a straight line $S(t)= a + bt.$ In the special case when $b = 0,$ we extend the result to a space-time transformation of Brownian motion.

Rate of convergence of Euler approximations of solution to mixed stochastic differential equation involving Brownian motion and fractional Brownian motion

We consider a mixed stochastic differential equation involving both standard Brownian motion and fractional Brownian motion with Hurst parameter $H>1/2$. The mean-square rate of convergence of Euler approximations of solution to this equation is obtained.

A Gladyshev theorem for trifractional Brownian motion and $n$-th order fractional Brownian motion

We prove limit theorems for the weighted quadratic variation of trifractional Brownian motion and $n$-th order fractional Brownian motion. Furthermore, a sufficient condition for the $L^P$-convergence of the weighted quadratic variation for Gaussian processes is obtained as a byproduct. As an application, we give a statistical estimator for the self-similarity index of trifractional Brownian motion. These theorems extend results of Baxter, Gladyshev, and Norvaiša.

Asymptotics of the persistence exponent of integrated fractional Brownian motion and fractionally integrated Brownian motion

We consider the persistence probability for the integrated fractional Brownian motion and the fractionally integrated Brownian motion with parameter $H,$ respectively. For the integrated fractional Brownian motion, we discuss a conjecture of Molchan and Khokhlov and determine the asymptotic behavior of the persistence exponent as $H\to 0$ and $H\to 1,$ which is in accordance with the conjecture. For the fractionally integrated Brownian motion, also called Riemann-Liouville process, we find the asymptotic behavior of the persistence exponent as $H\to 0$.

Spectral representation of one-dimensional Liouville Brownian Motion and Liouville Brownian excursion

In this paper we apply the spectral theory of linear diffusions to study the one-dimensional Liouville Brownian Motion and Liouville Brownian excursions from a given point. As an application we estimate the fractal dimensions of level sets of one-dimensional Liouville Brownian motion as well as various probabilistic asymptotic behaviours of Liouville Brownian motion and Liouville Brownian excursions.

On quasisymmetric embeddings of the Brownian map and continuum trees

The Brownian map is a model of random geometry on the sphere and as such an important object in probability theory and physics. It has been linked to Liouville Quantum Gravity and much research has been devoted to it. One open question asks for a canonical embedding of the Brownian map into the sphere or other, more abstract, metric spaces. Similarly, Liouville Quantum Gravity has been shown to be "equivalent" to the Brownian map but the exact nature of the correspondence (i.e.\ embedding) is still unknown. In this article we show that any embedding of the Brownian map or continuum random tree…

由分数布朗运动和布朗运动驱动的慢-快McKean-Vlasov随机微分方程的大偏差

Large deviation for slow-fast McKean-Vlasov stochastic differential equations driven by fractional Brownian motions and Brownian motions

本文研究了由布朗运动和分数布朗运动驱动的慢-快McKean-Vlasov随机微分方程，提出乘积空间上的大偏差原理新定义，并在适当假设下利用弱收敛方法证明了该方程的LDP。

$k$-cut model for the Brownian Continuum Random Tree

To model the destruction of a resilient network, Cai, Holmgren, Devroye and Skerman introduced the $k$-cut model on a random tree, as an extension to the classic problem of cutting down random trees. Berzunza, Cai and Holmgren later proved that the total number of cuts in the $k$-cut model to isolate the root of a Galton--Watson tree with a finite-variance offspring law and conditioned to have $n$ nodes, when divided by $n^{1-1/2k}$, converges in distribution to some random variable defined on the Brownian CRT. We provide here a direct construction of the limit random variable, relying upon th…

Stratonovich's Signatures of Brownian Motion Determine Brownian Sample Paths

The signature of Brownian motion in $\mathbb{R}^{d}$ over a running time interval $[0,T]$ is the collection of all iterated Stratonovich path integrals along the Brownian motion. We show that, in dimension $d\geq 2$, almost all Brownian motion sample paths (running up to time $T$) are determined by its signature over $[0,T]$

通过 snapping-out Brownian motions 逼近斜布朗运动

Approximation of skew Brownian motion by snapping-out Brownian motions

本文证明并讨论 snapping-out Brownian motions 在渗透系数趋于无穷且比值恒定时收敛到斜布朗运动，并给出相关半群、余弦族及投影的收敛结果。

Stochastic Differential Equations Driven by Fractional Brownian Motion and Standard Brownian Motion

We prove an existence and uniqueness theorem for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter H>1/2 and a multidimensional standard Brownian motion. The proof relies on some a priori estimates, which are obtained using the methods of fractional integration, and the classical Ito stochastic calculus. The existence result is based on the Yamada-Watanabe theorem.

Fake Exponential Brownian Motion

We construct a fake exponential Brownian motion, a continuous martingale different from classical exponential Brownian motion but with the same marginal distributions, thus extending results of Albin and Oleszkiewicz for fake Brownian motions. The ideas extend to other diffusions.

Dyson's Brownian motions, intertwining and interlacing

A family of reflected Brownian motions is used to construct Dyson's process of non-colliding Brownian motions. A number of explicit formulae are given, including one for the distribution of a family of coalescing Brownian motions.

Transfer Principle for nth Order Fractional Brownian Motion with Applications to Prediction and Equivalence in Law

The $n$th order fractional Brownian motion was introduced by Perrin et al. It is the (upto a multiplicative constant) unique self-similar Gaussian process with Hurst index $H \in (n-1,n)$, having $n$th order stationary increments. We provide a transfer principle for the $n$th order fractional Brownian motion, i.e., we construct a Brownian motion from the $n$the order fractional Brownian motion and then represent the $n$the order fractional Brownian motion by using the Brownian motion in a non-anticipative way so that the filtrations of the $n$the order fractional Brownian motion and the associ…

A set-indexed fractional Brownian motion

We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Levy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no "really nice" set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, …

Lower bounds for distribution of suprema of Brownian increments and Brownian motion normalized by the corresponding modulus functions

The Lévy-Ciesielski Construction of Brownian motion is used to determine non-asymptotic estimates for the maximal deviation of increments of a Brownian motion process $(W_{t})_{t\in \left[ 0,T\right] }$ normalized by the global modulus function, for all positive $\varepsilon $ and $δ$. Additionally, uniform results over $δ$ are obtained. Using the same method, non-asymptotic estimates for the distribution function for the standard Brownian motion normalized by its local modulus of continuity are obtained. Similar results for the truncated Brownian motion are provided and play a crucial role in…

A strong uniform approximation of sub-fractional Brownian motion

Sub-fractional Brownian motion is a process analogous to fractional Brownian motion but without stationary increments. In \cite{GGL1} we proved a strong uniform approximation with a rate of convergence for fractional Brownian motion by means of transport processes. In this paper we prove a similar type of approximation for sub-fractional Brownian motion.

Symmetric weighted odd-power variations of fractional Brownian motion and applications

We prove a non-central limit theorem for the symmetric weighted odd-power variations of the fractional Brownian motion with Hurst parameter H< 1/2. As applications, we study the asymptotic behavior of the trapezoidal weighted odd-power variations of the fractional Brownian motion and the fractional Brownian motion in Brownian time Z_t:= X_{Y_t}, t >= 0, where X is a fractional Brownian motion and Y is an independent Brownian motion.

超临界分支布朗运动和超布朗运动中加法鞅与导数鞅的小值概率

Small value probabilities of additive and derivative martingales in supercritical branching Brownian motions and super Brownian motions

该研究建立了超临界分支布朗运动和超布朗运动中两类鞅小值概率的渐近结果，推广了相关分支过程的已有结论，且与已有文献结果吻合。

布朗连续统随机树的 Oliver 曲率界

Oliver Curvature Bounds for the Brownian Continuum Random Tree

本文计算布朗连续统随机树的期望 Ollivier 曲率界，证明在考虑尺度依赖性后，其 Ollivier-Ricci 曲率几乎处处被任意实数上界控制，与连续统树均为 CAT(K) 空间的经典结果相呼应。

由标准Brown运动和分数阶Brown运动驱动的条件分布依赖型随机微分方程的适定性与平均原理

Wellposedness and averaging principle for conditional distribution dependent SDEs driven by standard Brownian motions and fractional Brownian motions

本文研究了由标准Brown运动和分数阶Brown运动驱动的条件分布依赖型随机微分方程，通过不动点定理证明了解的存在唯一性，并建立了其解可由平均方程逼近的平均原理。

The Multiparameter Fractional Brownian Motion

We define and study the multiparameter fractional Brownian motion. This process is a generalization of both the classical fractional Brownian motion and the multiparameter Brownian motion, when the condition of independence is relaxed. Relations with the Lévy fractional Brownian motion and with the fractional Brownian sheet are discussed. Different notions of stationarity of the increments for a multiparameter process are studied and applied to the fractional property. Using self-similarity we present a characterization for such processes. Finally, behavior of the multiparameter fractional Bro…

Tempered Fractional Brownian Motion with Variable Index and Variable Tempering Parameter

Generalizations of tempered fractional Brownian from single index to two indices and variable index or tempered multifractional Brownian motion are studied. Tempered fractional Brownian motion and tempered multifractional Brownian motion with variable tempering parameter are considered.

Coalescence of skew Brownian motions

We prove that two skew Brownian motions with the same skewness parameter (different from 0) and driven by the same Brownian motion coalesce a.s.

Some explicit distributions for Brownian motion indexed by the Brownian tree

We derive several explicit distributions of functionals of Brownian motion indexed by the Brownian tree. In particular, we give a direct proof of a result of Bousquet-Mélou and Janson identifying the distribution of the density at 0 of the integrated super-Brownian excursion.

Singularity of Generalized Grey Brownian Motion and Time-Changed Brownian Motion

The generalized grey Brownian motion is a time continuous self-similar with stationary increments stochastic process whose one dimensional distributions are the fundamental solutions of a stretched time fractional differential equation. Moreover, the distribution of the time-changed Brownian motion by an inverse stable process solves the same equation, hence both processes have the same one dimensional distribution. In this paper we show the mutual singularity of the probability measures on the path space which are induced by generalized grey Brownian motion and the time-changed Brownian motio…

Continuum limits of random matrices and the Brownian carousel

We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general beta siblings converge to Sine_beta, a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane. The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We show that the gap probability of Sine_beta is continuous in the gap size and $β$, and compute its asympto…

与布朗路径相关的凸下鞅树及最小生成树的连续极限

Convex minorant trees associated with Brownian paths and the continuum limit of the minimum spanning tree

本研究构造完全图最小生成树的标度极限即布朗抛物树，证明其豪斯多夫维数几乎必然为3，提出凸下鞅树框架并建立相关碎裂与表示的耦合。

Escape rate of the Brownian motions on hyperbolic spaces

We discuss the escape rate of the Brownian motion on a hyperbolic space. We point out that the escape rate is determined by using the Brownian expression of the radial part and a generalized Kolmogorov's test for the one dimensional Brownian motion.

Bessel processes, the Brownian snake and super-Brownian motion

We prove that, both for the Brownian snake and for super-Brownian motion in dimension one, the historical path corresponding to the minimal spatial position is a Bessel process of dimension -5. We also discuss a spine decomposition for the Brownian snake conditioned on the minimizing path.

Large deviations for flows of interacting Brownian motions

We establish the large deviation principle (LDP) for stochastic flows of interacting Brownian motions. In particular, we consider smoothly correlated flows, coalescing flows and Brownian motion stopped at a hitting moment.

Snapping out Walsh's Brownian motion and related stiff problem

Firstly, we shall introduce the so-called snapping out Walsh's Brownian motion and present its relation with Walsh's Brownian motion. Then the stiff problem related to Walsh's Brownian motion will be described and we shall build a phase transition for it. The snapping out Walsh's Brownian motion corresponds to the so-called semi-permeable pattern of this stiff problem.

Unitary Brownian motions are linearizable

Brownian motions in the infinite-dimensional group of all unitary operators are studied under strong continuity assumption rather than norm continuity. Every such motion can be described in terms of a countable collection of independent one-dimensional Brownian motions. The proof involves continuous tensor products and continuous quantum measurements. A by-product: a Brownian motion in a separable F-space (not locally convex) is a Gaussian process.

G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion

The present paper is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the view of rough path theory. As the starting point, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the funda…

Game-theoretic Brownian motion

This paper suggests a perfect-information game, along the lines of Levy's characterization of Brownian motion, that formalizes the process of Brownian motion in game-theoretic probability. This is perhaps the simplest situation where probability emerges in a non-stochastic environment.

Stability theorems for stochastic differential equations driven by G-Brownian motion

In this paper, stability theorems for stochastic differential equations and backward stochastic differential equations driven by G-Brownian motion are obtained. We show the existence and uniqueness of solutions to forward-backward stochastic differential equations driven by G-Brownian motion. Stability theorem for forward-backward stochastic differential equations driven by G-Brownian motion is also presented.

On the sub-mixed fractional Brownian motion

Let ${S_t^H, t \geq 0} $ be a linear combination of a Brownian motion and of an independent sub-fractional Brownian motion with Hurst index $0 < H < 1$. Its main properties are studied and it is shown that $S^H $ can be considered as an intermediate process between a sub-fractional Brownian motion and a mixed fractional Brownian motion. Finally, we determine the values of $H$ for which $S^H$ is not a semi-martingale.

Weak-consistent dynamic correlation estimators for Brownian motion pairs and for Geometric Brownian motion pairs

Estimating dynamic correlation between a pair of time series is of importance in many applications. We present new estimators for the dynamic correlation between a pair of correlated Brownian motions and separately for dynamic correlation between a pair of correlated Geometric Brownian motions. We show that, as the sample size increases, all estimators presented in this paper converge in probability to the underlying true dynamic correlation.

On a Cameron--Martin Type Quasi-Invariance Theorem and Applications to Subordinate Brownian Motion

We present a Cameron--Martin type quasi-invariance theorem for subordinate Brownian motion. As applications, we establish an integration by parts formula and construct a gradient operator on the path space of subordinate Brownian motion, and we obtain some canonical Dirichlet forms. These findings extend the corresponding classical results for Brownian motion.

The probability that Brownian motion almost covers a line

Lower and upper estimates are given for the probability that the epsilon-enlargement of planar Brownian motion to time 1 (the epsilon sausage) contains a unit line segment. The estimates imply that Brownian motion to time 1 itself contains no line segment.

Growth-fragmentation processes in Brownian motion indexed by the Brownian tree

We consider the model of Brownian motion indexed by the Brownian tree. For every $r\geq 0$ and every connected component of the set of points where Brownian motion is greater than $r$, we define the boundary size of this component, and we then show that the collection of these boundary sizes evolves when $r$ varies like a well-identified growth-fragmentation process. We then prove that the same growth-fragmentation process appears when slicing a Brownian disk at height $r$ and considering the perimeters of the resulting connected components.

双分数布朗运动的极大不等式

Maximal inequalities for bifractional Brownian motion

本文研究双分数布朗运动的极大不等式问题，核心方法是借助高斯过程比较定理，得出若干相关极大不等式。

超布朗运动局部时间的随机微分方程

A stochastic differential equation for local times of super-Brownian motion

本文证明了超布朗运动或由Brownian tree索引的Brownian motion的局部时间满足显式随机微分方程，其证明结合了Brownian snake的excursion theory与超过程理论工具。

Girsanov formula for $G$-Brownian motion: the degenerate case

In this paper, we prove the Girsanov formula for $G$-Brownian motion without the non-degenerate condition. The proof is based on the perturbation method in the nonlinear setting by constructing a product space of the $G$-expectation space and a linear space that contains a standard Brownian motion. The estimates for exponential martingale of $G$-Brownian motion are important for our arguments.

On the multi-dimensional skew Brownian motion

We provide a new, concise proof of weak existence and uniqueness of solutions to the stochastic differential equation for the multidimensional skew Brownian motion. We also present an application to Brownian particles with skew-elastic collisions.

Fractional Brownian motion in a nutshell

This is an extended version of the lecture notes to a mini-course devoted to fractional Brownian motion and delivered to the participants of 7th Jagna International Workshop.

A decomposition of the bifractional Brownian motion and some applications

In this paper we show a decomposition of the bifractional Brownian motion with parameters H,K into the sum of a fractional Brownian motion with Hurst parameter HK plus a stochastic process with absolutely continuous trajectories. Some applications of this decomposition are discussed.

Simultaneous boundary hitting by coupled reflected Brownian motions

(i) Uncountably many synchronized reflected Brownian motions can hit the boundary of a $C^2$ domain at the same time. (ii) Measures associated to local times of two synchronized reflected Brownian motions are mutually singular until the time when the normal vectors at the reflection locations become identical. (iii) Mirror coupled reflected Brownian motions can simultaneously hit opposite sides of a wedge at different distances from the origin.

Some points on Vervaat's transform of Brownian bridges and Brownian motion

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.