Alexis Devulder
We study a one-dimensional diffusion process in a drifted Brownian potential. We characterize the upper functions of its hitting times in the sense of Paul Lévy, and determine the lower limits in terms of an iterated logarithm law.
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We study Brownian motion in a drifted Brownian potential in the subexponential regime. We prove that the annealed probability of deviating below the almost sure speed has a polynomial rate of decay and compute the exponent in this power law. This provides a continuous-time analogue of what Dembo, Peres and Zeitouni proved for the transient random walk in random environment. Our method takes a completely different route, making use of Lamperti's representation together with an iteration scheme.
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.
Fractional equations governing the distribution of reflecting drifted Brownian motions are presented. The equations are expressed in terms of tempered Riemann--Liouville type derivatives. For these operators a Marchaud-type form is obtained and a Riesz tempered fractional derivative is examined, together with its Fourier transform.
For drifted Brownian motion $X(t)= x - μt + B_t \ (μ>0)$ starting from $x>0,$ we study the joint distribution of the first-passage time below zero, $τ(x),$ and the first-passage area, $A(x),$ swept out by $X$ till the time $τ(x).$ In particular, we establish differential equations with boundary conditions for the joint moments $E[τ(x)^m A(x)^n],$ and we present an algorithm to find recursively them, for any $m$ and $n.$ Finally, the expected value of the time average of $X$ till the time $τ(x)$ is obtained.
Consider a negatively drifted one dimensional Brownian motion starting at positive initial position, its first hitting time to 0 has the inverse Gaussian law. Moreover, conditionally on this hitting time, the Brownian motion up to that time has the law of a 3- dimensional Bessel bridge. In this paper, we give a generalization of this result to a family of Brownian motions with interacting drifts, indexed by the vertices of a conductance network. The hitting times are equal in law to the inverse of a random potential that appears in the analysis of a self-interacting process called the Vertex R…
We study an inverse first-passage-time problem for Wiener process $X(t)$ subject to hold and jump from a boundary $c.$ Let be given a threshold $S>X(0) \ge c,$ and a distribution function $F$ on $[0, + \infty ).$ The problem consists in finding the distribution of the holding time at $c$ and the distribution of jumps from $c,$ so that the first-passage time of $X(t)$ through $S$ has distribution $F.$
We consider a one-dimensional diffusion process $X$ in a $(-κ/2)$-drifted Brownian potential for $κ\neq 0$. We are interested in the maximum of its local time, and study its almost sure asymptotic behaviour, which is proved to be different from the behaviour of the maximum local time of the transient random walk in random environment. We also obtain the convergence in law of the maximum local time of $X$ under the annealed law after suitable renormalization when $κ\geq 1$. Moreover, we characterize all the upper and lower classes for the hitting times of $X$, in the sense of Paul Lévy, and pro…
We study the 'Up the River' problem formulated by Aldous (2002), where a unit drift is distributed among a finite collection of Brownian particles on $ \mathbb{R}_+ $, which are annihilated once they reach the origin. Starting $ K $ particles at $ x=1 $, we prove a conjecture of Aldous (2002) that the 'push-the-laggard' strategy of distributing the drift asymptotically (as $ K\to\infty $) maximizes the total number of surviving particles, with approximately $ \frac{4}{\sqrtπ} K^{1/2} $ surviving particles. We further establish the hydrodynamic limit of the particle density, in terms of a two-p…
We study a model of diffusion in a brownian potential. This model was firstly introduced by T. Brox (1986) as a continuous time analogue of random walk in random environment. We estimate the deviations of this process above or under its typical behavior. Our results rely on different tools such as a representation introduced by Y. Hu, Z. Shi and M. Yor, Kotani's lemma, introduced at first by K. Kawazu and H. Tanaka (1997), and a decomposition of hitting times developed in a recent article by A. Fribergh, N. Gantert and S. Popov (2008). Our results are in agreement with their results in the dis…
We provide a deep connection between elastic drifted Brownian motions and inverses to tempered subordinators. Based on this connection, we establish a link between multiplicative functionals and dynamical boundary conditions given in terms of non-local equations in time. Indeed, we show that the multiplicative functional associated to the elastic Brownian motion with drift is equivalent to a multiplicative functional associated with fractional boundary conditions of tempered type. By exploiting such connection we write some functionals in terms of a simple (positive and non-decreasing) process…
We show that the slopes between h-extrema of the drifted 1D Brownian motion form a stationary alternating marked point process, extending the result of J. Neveu and J. Pitman for the non drifted case. Our analysis covers the results on the statistics of h-extrema obtained by P. Le Doussal, C. Monthus and D. Fisher via a Renormalization Group analysis and gives a complete description of the slope between h-extrema covering the origin by means of the Palm--Khinchin theory. Moreover, we analyze the behavior of the Brownian motion near its h-extrema.
An exact expression for the distribution of the area swept out by a drifted Brownian motion till its first-passage time is derived. A study of the asymptotic behaviour confirms earlier conjectures and clarifies their range of validity. The analysis also leads to a simple closed-form solution for the moments of the Airy distribution.
In this paper we study the drifted Brownian meander, that is a Brownian motion starting from $ u $ and subject to the condition that $ \min_{ 0\leq z \leq t} B(z)> v $ with $ u > v $. The limiting process for $ u \downarrow v $ is analyzed and the sufficient conditions for its construction are given. We also study the distribution of the maximum of the meander with drift and the related first-passage times. The representation of the meander endowed with a drift is provided and extends the well-known result of the driftless case. The last part concerns the drifted excursion process the distribu…
A Quasi-Stationary Distribution (QSD)for a Markov process with an almost surely hit absorbing state is a time-invariant initial distribution for the process conditioned on not being absorbed by any given time. An initial distribution for the process is in the domain of attraction of some QSD $ν$ if the distribution of the process a time $t$, conditioned not to be absorbed by time $t$ converges to $ν$. In this work study mostly Brownian motion with constant drift on the half line $[0,\infty)$ absorbed at $0$. Previous work by Martinez et al. identifies all QSDs and provides a nearly complete ch…
We consider a diffusion process $X$ in a random potential $\V$ of the form $\V_x = §_x -δx$ where $δ$ is a positive drift and $§$ is a strictly stable process of index $α\in (1,2)$ with positive jumps. Then the diffusion is transient and $X_t / \log^αt$ converges in law towards an exponential distribution. This behaviour contrasts with the case where $\V$ is a drifted Brownian motion and provides an example of a transient diffusion in a random potential which is as "slow" as in the recurrent setting.
In this paper we study the sojourn time on the positive half-line up to time $ t $ of a drifted Brownian motion with starting point $ u $ and subject to the condition that $ \min_{ 0\leq z \leq l} B(z)> v $, with $ u > v $. This process is a drifted Brownian meander up to time $ l $ and then evolves as a free Brownian motion. We also consider the sojourn time of a bridge-type process, where we add the additional condition to return to the initial level at the end of the time interval. We analyze the weak limit of the occupation functional as $ u \downarrow v $. We obtain explicit distributiona…
Persistence of the Wiener Sausage: Sampling Stability and a Law of Large Numbers for Drifted Planar Brownian Motion DRAFT -CURRENTLY UNDER REVIEW
研究漂移平面布朗运动范围生成的偏移过滤的持续同调，提出采样定理和大数定律，揭示拓扑复杂性增长的强度度量。
Generalized Counting Process with Random Drift and Different Brownian Clocks
本文引入了具有确定性和随机漂移的广义计数过程（GCP），推导了其概率律和分数阶微分方程，并研究了在多种布朗时钟及不完全伽马函数相关从属过程下的时间变更GCP及其分数阶积分。
Study of direct and inverse first-exit problems for drifted Brownian motion with Poissonian resetting
本文研究带泊松重置的漂移布朗运动从区间首次退出的直接与逆问题，涵盖首次退出时间与面积的统计性质，并给出逆问题解的显式示例。
The perturbed GUE corners ensemble is the joint distribution of eigenvalues of all principal submatrices of a matrix $G+\mathrm{diag}(\mathbf{a})$, where $G$ is the random matrix from the Gaussian Unitary Ensemble (GUE), and $\mathrm{diag}(\mathbf{a})$ is a fixed diagonal matrix. We introduce Markov transitions based on exponential jumps of eigenvalues, and show that their successive application is equivalent in distribution to a deterministic shift of the matrix. This result also leads to a new distributional symmetry for a family of reflected Brownian motions with drifts coming from an arith…
We present a self-contained and modern survey of some existing quasi-sure results via the connection to the Brownian sheet. Among other things, we prove that quasi-every continuous function: (i) satisfies the local law of the iterated logarithm; (ii) has Levy's modulus of continuity for Brownian motion; (iii) is nowhere differentiable; and (iv) has a nontrivial quadratic variation. We also present a hint of how to extend (iii) to obtain a quasi-sure refinement of the M. Csorgo--P. Revesz modulus of continuity for almost every continuous function along the lines suggested by M. Fukushima.
A Quasi-sure Non-degeneracy Property for the Brownian Rough Path
本文研究粗糙路径理论中的签名唯一性相关问题，证明拟必然意义下布朗运动签名路径不自交，进而推出布朗粗糙路径无树状片段、样本路径可由签名在重参数化下唯一确定的结论。
We establish an almost sure scaling limit theorem for super-Brownian motion on $\mathbb{R}^d$ associated with the semi-linear equation $u_t = {1/2}Δu +βu-αu^2$, where $α$ and $β$ are positive constants. In this case, the spectral theoretical assumptions that required in Chen et al (2008) are not satisfied. An example is given to show that the main results also hold for some sub-domains in $\mathbb{R}^d$.
In this note we consider a branching Brownian motion (BBM) on $\mathbb{R}$ in which a particle at spatial position $y$ splits into two at rate $βy^2$, where $β>0$ is a constant. This is a critical breeding rate for BBM in the sense that the expected population size blows up in finite time while the population size remains finite, almost surely, for all time. We find an asymptotic for the almost sure rate of growth of the population.
Almost sure path localisation for the derivative martingale of branching Brownian motion
本研究针对分支布朗运动导数鞅，建立几乎必然路径局部化结果，明确其极限由与极值粒子距离为s^(1/2)的细管内的粒子决定，刻画了贡献极限的粒子范围。
We give a proof of a result on the growth of the number of particles along chosen paths in a branching Brownian motion. The work follows the approach of classical large deviations results, in which paths in $C[0,1]$ are rescaled onto $C[0,T]$ for large $T$. The methods used are probabilistic and take advantage of modern spine techniques.
Exact maximum likelihood inference for drifted multi-sub-fractional Brownian motion at discrete observation
本文针对离散观测的漂移多子分数布朗运动，证明了完整有限样本似然理论的存在，给出显式极大似然估计量、精确水平的区间与检验，并验证了强相合性、渐近正态性及模拟效果。
We study the asymptotic behaviour of the maximum local time L*(t) of the Brox's process, the diffusion in Brownian environment. Shi proved that the maximum speed of L*(t) is surprisingly, at least t log(log(log t)) whereas in the discrete case it is t. We show here that t log(log(log t)) is the proper rate and we prove that for the minimum speed the rate is the same as in the discrete case namely t/log(log(log t)).
We give new results on the growth of the number of particles in a dyadic branching Brownian motion which follow within a fixed distance of a path $f:[0,\infty)\to \mathbb{R}$. We show that it is possible to count the number of particles without rescaling the paths. Our results reveal that the number of particles along certain paths can oscillate dramatically. The methods used are entirely probabilistic, taking advantage of the spine technique developed by, amongst others, Lyons et al, Kyprianou, and Hardy & Harris.
Persistent Homology of the Wiener Sausage II: A Central Limit Theorem for Drifted Planar Brownian Motion
本文研究漂移平面布朗运动的维纳萨姆萨的持久同调，证明了平滑贝蒂曲线函数的中心极限定理，并推导了有限维高斯极限。
Let $B$ be a bifractional Brownian motion with parameters $H\in (0, 1)$ and $K\in(0,1]$. For any $n\geq1$, set $Z_n =\sum_{i=0}^{n-1}\big[n^{2HK}(B_{(i+1)/n}-B_{i/n})^2-\E((B_{i+1}-B_{i})^2)\big]$. We use the Malliavin calculus and the so-called Stein's method on Wiener chaos introduced by Nourdin and Peccati \cite{NP09} to derive, in the case when $0<HK\leq3/4$, Berry-Esséen-type bounds for the Kolmogorov distance between the law of the correct renormalization $V_n$ of $Z_n$ and the standard normal law. Finally, we study almost sure central limit theorems for the sequence $V_n$.
For $d \in \{1,2,3\}$, let $(B^d_t;~ t \geq 0)$ be a $d$-dimensional standard Brownian motion. We study the $d$-Brownian span set $Span(d):=\{t-s;~ B^d_s=B^d_t~\mbox{for some}~0 \leq s \leq t\}$. We prove that almost surely the random set $Span(d)$ is $σ$-compact and dense in $\mathbb{R}_{+}$. In addition, we show that $Span(1)=\mathbb{R}_{+}$ almost surely; the Lebesgue measure of $Span(2)$ is $0$ almost surely and its Hausdorff dimension is $1$ almost surely; and the Hausdorff dimension of $Span(3)$ is $\frac{1}{2}$ almost surely. We also list a number of conjectures and open problems.
In this paper by calculating carefully the capacities (defined by high order Sobolev norms on the Wiener space) for some functions of Brownian motion, we show that the dyadic approximations of the sample paths of the Brownian motion converge in the $p$-variation distance to the Brownian motion except for a slim set (i.e. except for a zero subset with respect to the capacity on the Wiener space of any order). This presents a way for studying quasi-sure properties of Wiener functionals by means of the rough path analysis.
Martin boundary of a degenerate Reflected Brownian Motion in a wedge
本文研究退化反射布朗运动在楔形中的马丁边界，通过计算格林函数的拉普拉斯变换并推导其渐进行为，确定了最小马丁边界及相应的调和函数。
Fractional Brownian Motion: Local Modulus of Continuity with Refined Almost Sure Upper Bound and First Exit Time from One-sided Barrier
本文基于最优速率小波级数表示，推导出分数布朗运动局部连续模的改进几乎必然上界及其 $p$ 阶矩上界，填补了迭代对数律的空白，并据此给出了从正值边界首次退出时间的改进上尾概率渐近估计。
Special times for the point set process of the Brownian Net
本文研究了布朗网点集过程的特殊时间，证明此类随机时间集合几乎 surely 具有 Hausdorff 维度 1/2。
We study the set of points $\mathcal{D}_{n,m}$ around which two independent Brownian motions wind at least $n$ (resp. $m$) times. We prove that its area is asymptotically equivalent, in $L^p$ and almost surely, to $\frac{\ell(\mathbb{R}^2)}{4π^2 n m}$, where $\ell$ is the intersection measure of the two trajectories. We also prove that the properly scaled Lebesgue measure carried by $\mathcal{D}_{n,m}$ converges almost surely weakly toward $\ell$.
In this paper, we consider the explicit bound for the second-order approximation of the quadratic variation of a general fractional Gaussian process $(G_t)_{t\ge 0}$. The second order mixed partial derivative of the covariance function $ R(t,\, s)=\mathbb{E}[G_t G_s]$ can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other of which is bounded by $(ts)^{H-1}$ up to a constant factor. This condition is valid for a class of continuous Gaussian processes that fails to be self-similar or have stationary increments. %Some examples include the su…
Many years ago, Griego, Heath and Ruiz-Moncayo proved that it is possible to define realizations of a sequence of uniform transform processes that converges almost surely to the standard Brownian motion, uniformly on the unit time interval. In this paper we extend their results to the multi parameter case. We begin constructing a family of processes, starting from a set of independent standard Poisson processes, that has realizations that converge almost surely to the Brownian sheet, uniformly on the unit square. At the end the extension to the $d$-parameter Wiener processes is presented.
The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka--Rosen--Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-i…
Let (B_t : t > 0) be a planar Brownian motion and define gauge functions $ϕ_α(s)=log(1/s)^{-α}$ for $α>0$. If $α<1$ we show that almost surely there exists a point x in the plane such that $H^{ϕ_α}({t > 0 : B_t=x})>0$, but if $α>1$ almost surely $H^{ϕ_α} ({t > 0 : B_t=x})=0$ simultaneously for all $x\in R^2$. This resolves a longstanding open problem posed by S.,J. Taylor in 1986.
Fine regularity of stochastic processes is usually measured in a local way by local Hölder exponents and in a global way by fractal dimensions. Following a previous work of Adler, we connect these two concepts for multiparameter Gaussian random fields. More precisely, we prove that almost surely the Hausdorff dimensions of the range and the graph in any ball $B(t_0,ρ)$ are bounded from above using the local Hölder exponent at $t_0$. We define the deterministic local sub-exponent of Gaussian processes, which allows to obtain an almost sure lower bound for these dimensions. Moreover, the Hausdor…
We construct a wavelet-based almost sure uniform approximation of fractional Brownian motion (fBm) B_t^(H), t in [0, 1], of Hurst index H in (0, 1). Our results show that by Haar wavelets which merely have one vanishing moment, an almost sure uniform expansion of fBm of H in (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an fBm efficiently.
We consider the additive martingale $W_t(λ)$ and the derivative martingale $\partial W_t(λ)$ for one-dimensional supercritical super-Brownian motions with general branching mechanism. In the critical case $λ=λ_0$, we prove that $\sqrt{t}W_t(λ_0)$ converges in probability to a positive limit, which is a constant multiple of the almost sure limit $\partial W_\infty(λ_0)$ of the derivative martingale $\partial W_t(λ_0)$. We also prove that, on the survival event, $\limsup_{t\to\infty}\sqrt{t}W_t(λ_0)=\infty$ almost surely.
Let β_k(n) be the number of self-intersections of order k, appropriately renormalized, for a mean zero random walk X_n in Z^2 with 2+δmoments. On a suitable probability space we can construct X_n and a planar Brownian motion W_t such that for each k\geq 2, |β_k(n)-γ_k(n)|=O(n^{-a}), a.s. for some a>0 where γ_k(n) is the renormalized self-intersection local time of order k at time 1 for the Brownian motion W_{nt}/\sqrt n.
Zeros of the Brownian Sheet
本文回答了 Khoshnevisan 的开放问题，证明了几乎必然不存在足够大秩的投影能改变 Brownian sheet 零点的 Hausdorff 维数，且不满足条件的投影将零点集映射为整个射影空间。
A well-known result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a set-valued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost surely a locally finite set at all positive times, and a finite set almost surely if the initial value is compact: the key to both of these facts is the observation that, because of the topology of the real line and the continuity of Brownian sample paths, at the tim…
Any fixed cylinder is hit almost surely by a 3-dimensional Brownian motion, but is there a random cylinder that is in the complement? We answer this for cylinders, and then replacing a cylinder with a more general set.
We study a system of coalescing random walks on the integer lattice $\mathbb{Z}^{d}$ in which the walk is oriented in the $d$-th direction and follows certain specified rules. We first study the geometry of the paths and show that, almost surely, the paths from a graph consisting of just one tree for dimentions $d=2,3$ and infinitely many disjoint trees for dimensions $d\geq 4$. Also, there is no bi-infinite path in the graph almost surely for $d\geq 2$. Subsequently, we prove that for $d=2$ the diffusive scaling of this system converges in distribution to the Brownian web.
We investigate an almost sure limit theorem (ASCLT) for sequences of random variables having the form of a ratio of two terms such that the numerator satisfies the ASCLT and the denominator is a positive term which converges almost surely to 1. This result leads to the ASCLT for least square estimators for Ornstein-Uhlenbeck process driven by fractional Brownian motion.
We consider the Anderson polymer partition function $$ u(t):=\mathbb{E}^X\Bigl[e^{\int_0^t \mathrm{d}B^{X(s)}_s}\Bigr]\,, $$ where $\{B^{x}_t\,;\, t\geq0\}_{x\in\mathbb{Z}^d}$ is a family of independent fractional Brownian motions all with Hurst parameter $H\in(0,1)$, and $\{X(t)\}_{t\in \mathbb{R}^{\geq 0}}$ is a continuous-time simple symmetric random walk on $\mathbb{Z}^d$ with jump rate $κ$ and started from the origin. $\mathbb{E}^X$ is the expectation with respect to this random walk. We prove that when $H\leq 1/2$, the function $u(t)$ almost surely grows asymptotically like $e^{l t}$, wh…
For a symmetric random walk in $Z^2$ with $2+δ$ moments, we represent $|\mathcal{R}(n)|$, the cardinality of the range, in terms of an expansion involving the renormalized intersection local times of a Brownian motion. We show that for each $k\geq 1$ \[ (\log n)^k [ \frac{1}{n} |\mathcal{R}(n)| +\sum_{j=1}^k (-1)^j (\textstyle{\frac1{2π}}\log n +c_X)^{-j} γ_{j,n}]\to 0, \qquad a.s. \] where $W_t$ is a Brownian motion, $W^{(n)}_t=W_{nt}/\sqrt n$, $γ_{j,n}$ is the renormalized intersection local time at time 1 for $W^{(n)}$, and $c_X$ is a constant depending on the distribution of the random wal…
We construct a family of processes, from a single Poisson process, that converges in law to a complex Brownian motion. Moreover, we find realizations of these processes that converge almost surely to the complex Brownian motion, uniformly on the unit time interval. Finally the rate of convergence is derived.
Using the explicit representations of the Brownian motions on the hyperbolic spaces, we show that their almost sure convergence and the central limit theorems for the radial components as time tends to infinity are easily obtained. We also give a straightforward strategy to obtain the explicit expressions for the limit distributions or the Poisson kernels.
A boundary Harnack principle and its application to analyticity of 3D Brownian intersection exponents
本文证明移除三维布朗运动轨迹后的区域几乎必然满足边界Harnack原理，并借此证明三维布朗运动的相交指数具有解析性。
We consider a (one-dimensional) branching Brownian motion process with a general offspring distribution having at least two moments, and in which all particles have a drift towards the origin where they are immediately absorbed. It is well-known that if and only if the branching rate is sufficiently large, then the population survives forever with positive probability. We show that throughout this super-critical regime, the number of particles inside any fixed set normalized by the mean population size converges to an explicit limit, almost surely and in $L^1$. As a consequence, we get that al…
We introduce the notion of a stationary random manifold and develop the basic entropy theory for it. Examples include manifolds admitting a compact quotient under isometries and generic leaves of a compact foliation. We prove that the entropy of an ergodic stationary random manifold is zero if and only if the manifold satisfies the Liouville property almost surely, and is positive if and only if it admits an infinite dimensional space of bounded harmonic functions almost surely. Upper and lower bounds for the entropy are provided in terms of the linear drift of Brownian motion and average volu…
We prove an almost sure invariance principle (approximation by d-dimensional Brownian motion) for vector-valued Holder observables of large classes of nonuniformly hyperbolic dynamical systems. These systems include Axiom~A diffeomorphisms and flows as well as systems modelled by Young towers with moderate tail decay rates. In particular, the position variable of the planar periodic Lorentz gas with finite horizon approximates a 2-dimensional Brownian motion.
On the differentiability of the local time of the ($1+β$)-stable super-Brownian motion
本文针对0<β<1的(1+β)-稳定超布朗运动，在d=1且初始测度无原子时，证明其局部时在固定确定性点处几乎处处可微，但无法在所有空间点同时可微，强化了前人的相关结果。
We consider branching Brownian motion on the real line with absorption at zero, in which particles move according to independent Brownian motions with the critical drift of $-\sqrt{2}$. Kesten (1978) showed that almost surely this process eventually dies out. Here we obtain upper and lower bounds on the probability that the process survives until some large time $t$. These bounds improve upon results of Kesten (1978), and partially confirm nonrigorous predictions of Derrida and Simon (2007).
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