Joseph Najnudel
In this paper, we construct a family of probability measures, by penalizations of a Walsh's Brownian motion with a weight dependent on its value and its local time at a time t. We prove that this family converges to a probability measure as t tends to infinity, and we study the behaviour of this limit measure.
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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.
We present a new proof of the extended arc-sine law related to Walsh's Brownian motion, known also as Brownian spider. The main argument mimics the scaling property used previously, in particular by D. Williams in the 1-dimensional Brownian case, which can be generalized to the multivariate case. A discussion concerning the time spent positive by a skew Bessel process is also presented.
In this paper, we construct a family of probability measures, by penalizations of a Walsh Brownian motion with a weight dependent on its value and its local time at a time t. We prove that this family converges to a probability measure as t tends to infinity, and we study the behaviour of this limit measure.
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。
We define an equation on a simple graph which is an extension of Tanaka equation and the skew Brownian motion equation. We then apply the theory of transition kernels developped by Le Jan and Raimond and show that all the solutions can be classified by probability measures.
In a previous work, we have defined a Tanaka SDE related to Walsh Brownian motion which depends on kernels. It was shown that there are only one Wiener solution and only one flow of mappings solving this equation. In the terminology of Le Jan and Raimond, these are respectively the stronger and the weaker among all solutions. In this paper, we obtain these solutions as limits of discrete models.
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。
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。
Let $(Z,κ)$ be a Walsh Brownian motion with spinning measure $κ$. Suppose $μ$ is a probability measure on $\mathbb{R}^n$. We characterize all the $κ$ such that $μ$ is a stopping distribution of $(Z,κ)$. If we further restrict the solution to be integrable, we show that there would be only one choice of $κ$. We also generalize Vallois' embedding, and prove that it minimizes the expectation $\mathbb{E}[Ψ(L^Z_τ)]$ among all the admissible solutions $τ$, where $Ψ$ is a strictly convex function and $(L_t^Z)_{t \geq 0}$ is the local time of the Walsh Brownian motion at the origin.
We study a single-server Markovian queueing model with $N$ customer classes in which priority is given to the shortest queue. Under a critical load condition, we establish the diffusion limit of the workload and queue length processes in the form of a Walsh Brownian motion (WBM) living in the union of the $N$ nonnegative coordinate axes in $\mathbb{R}^N$ and a linear transformation thereof. This reveals the following asymptotic behavior. Each time that queues begin to build starting from an empty system, one of them becomes dominant in the sense that it contains nearly all the workload in the …
We construct planar semimartingales that include the Walsh Brownian motion as a special case, and derive Harrison-Shepp-type equations and a change-of-variable formula in the spirit of Freidlin-Sheu for these so-called "Walsh semimartingales". We examine the solvability of the resulting system of stochastic integral equations. In appropriate Markovian settings we study two types of connections to martingale problems, questions of uniqueness in distribution for such processes, and a few examples.
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 consider the Brownian ``spider process'', also known as Walsh Brownian motion, first introduced in the epilogue of Walsh 1978. The paper provides the best constant $C_n$ for the inequality $$ E D_τ\leq C_n \sqrt{E τ},$$ where $τ$ is the class of all adapted and integrable stopping times and $D$ denotes the diameter of the spider process measured in terms of the British rail metric. The proof relies on the explicit identification of the value function for the associated optimal stopping problem.
A classification for Brownian motions on metric graphs, that is, right continuous strong Markov processes which behave like a one-dimensional Brownian motion on the edges and feature effects like Walsh skewness, stickiness and jumps at the vertices, is obtained. The Feller property of these processes is proved, and the boundary conditions of their generators are identified as non-local Feller-Wentzell boundary conditions. By using a technique of successive revivals, a complete description of the generator is achieved for Brownian motions on star graphs.
In this paper we consider the Skorokhod embedding problem in Brownian motion. In particular, we give a solution based on the local time at zero of a variably skewed Brownian motion related to the underlying Brownian motion. Special cases of the construction include the Azema-Yor and Vallois embeddings. In turn, the construction has an interpretation in the Chacon-Walsh framework.
For a Brownian bridge from $0$ to $y$ we prove that the mean of the first exit time from interval $(-h,h), \,\, h>0,$ behaves as $O(h^2)$ when $h \downarrow 0.$ Similar behavior is seen to hold also for the 3-dimensional Bessel bridge. For Brownian bridge and 3-dimensional Bessel bridge this mean of the first exit time has a puzzling representation in terms of the Kolmogorov distribution. The result regarding the Brownian bridge is applied to prove in detail an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.
Inspired by Tsirelson proof of the non Brownian character of WalshBrownian motion ltration on three or more rays, we prove some results on aparticular coupling of solutions to the interface SDE on a star graph, recentlyintroduced. This coupling consists in two solutions which are independentgiven the driving Brownian motion. As a consequence, we deduce that if the stargraph contains 3 or more rays, the argument of the solution at a xed time isindependent of the driving Brownian motion.
Brownian motions on star graphs in the sense of Itô-McKean, that is, Walsh processes admitting a generalized boundary behavior including stickiness and jumps and having an angular distribution with finite support, are examined. Their generators are identified as Laplace operators on the graph subject to non-local Feller-Wentzell boundary conditions. A pathwise description is achieved for every admissible boundary condition: For finite jump measures, a construction of Kostrykin, Potthoff and Schrader in the continuous setting is expanded via a technique of successive killings and revivals; for …
Martingale problem for a Walsh spider process with spinning measure selected from its own local time
本文研究旋转测度和系数依赖于交汇点局部时的Walsh蜘蛛扩散过程，通过证明鞅问题适定、利用拼接程序构造解并结合局部时Kirchhoff传输条件证明唯一性，还给出了标准布朗运动分支下的显式分布律及广义SDE解的存在唯一性。
Pathwise constructions of Brownian motions which satisfy all possible boundary conditions at the vertex of single vertex graphs are given.
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.
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.
We study the fluctuations of the power variation of fractional Brownian motion in Brownian time
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.
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.
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.
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 $H\to 0$.
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.
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。
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 $[0,T]$
A kinetic model approximation of Walsh's spider process on the infinite star-like graph
本文研究了星型图副本上受界面和跳跃扰动的确定性运动过程，并证明其扩散标度在极限下收敛为 Walsh 蜘蛛过程。
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。
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.
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.
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.
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…
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, …
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…
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.
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.
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。
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。
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…
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.
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.
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.
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…
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.
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.
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.
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.
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…
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.
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.
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.
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.
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.
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.
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…
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.
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