We establish universality for the largest singular values of products of random matrices with right unitarily invariant distributions, in a regime where the number of matrix factors and size of the matrices tend to infinity simultaneously. The behavior of the largest log singular values coincides with the large N limit of Dyson Brownian motion with a characteristic drift vector consisting of equally spaced coordinates, which matches the large N limit of the largest log singular values of Brownian motion on GL(N, C). Our method utilizes the formalism of multivariate Bessel generating functions, also known as spherical transforms, to obtain and analyze combinatorial expressions for observables of these processes.
We first study the convergence of solutions of a system of F-KPP equations related to irreducible multitype branching Brownian motions with Heaviside-type initial conditions to traveling wave solutions. Then we apply this convergence result to prove that the extremal processes of irreducible multitype branching Brownian motions converges weakly to a cluster point process.
In a previous paper, the authors proved a conjecture of Lalley and Sellke that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a Gumbel distribution. The result is extended here to the entire system of particles that are extremal, i.e. close to the maximum. Namely, it is proved that the distribution of extremal particles under time-average converges to a Poisson cluster process.
We derive asymptotics for the probability of the origin to be an extremal point of a random walk in R^n. We show that in order for the probability to be roughly 1/2, the number of steps of the random walk should be between e^{c n / log n}$ and e^{C n log n}. As a result, we attain a bound for the ?pi/2-covering time of a spherical brownian motion.
We construct and describe the extremal process for variable speed branching Brownian motion, studied recently by Fang and Zeitouni, for the case of piecewise constant speeds; in fact for simplicity we concentrate on the case when the speed is $σ_1$ for $s\leq bt$ and $σ_2$ when $bt\leq s\leq t$. In the case $σ_1>σ_2$, the process is the concatenation of two BBM extremal processes, as expected. In the case $σ_1<σ_2$, a new family of cluster point processes arises, that are similar, but distinctively different from the BBM process. Our proofs follow the strategy of Arguin, Bovier, and Kistler.
We prove the convergence of the extremal processes for variable speed branching Brownian motions where the "speed functions", that describe the time-inhomogeneous variance, lie strictly below their concave hull and satisfy a certain weak regularity condition. These limiting objects are universal in the sense that they only depend on the slope of the speed function at $0$ and the final time $t$. The proof is based on previous results for two-speed BBM obtained in a recent paper of ours and uses Gaussian comparison arguments to extend these to the general case.
We prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson cluster process, where the positions of the clusters is a Poisson process with exponential density. The law of the individual clusters is characterized as branching Brownian motions conditioned to perform "unusually large displacements", and its existence is proved. The proof combines three main ingredients. First, the results of Bramson on the convergence of solutions of the Kolmogorov-Petrovsky-Piscouno…
We extend the results of Arguin et al and A\"ıdékon et al on the convergence of the extremal process of branching Brownian motion by adding an extra dimension that encodes the "location" of the particle in the underlying Galton-Watson tree. We show that the limit is a cluster point process on $\mathbb{R}_+\times \mathbb{R}$ where each cluster is the atom of a Poisson point process on $\mathbb{R}_+\times \mathbb{R}$ with a random intensity measure $Z(dz) \times Ce^{-\sqrt 2x}dx$, where the random measure is explicitly constructed from the derivative martingale. This work is motivated by an anal…
Recently Ren et al. [Stoch. Proc. Appl., 137 (2021)] have proved that the extremal process of the super-Brownian motion converges in distribution in the limit of large times. Their techniques rely heavily on the study of the convergence of solutions to the Kolmogorov-Petrovsky-Piscounov equation along the lines of [M. Bramson, Mem. Amer. Math. Soc., 44 (1983)]. In this paper we take a different approach. Our approach is based on the skeleton decomposition of super-Brownian motion. The skeleton may be interpreted as immortal particles that determine the large time behaviour of the process. We e…
As a first step toward a characterization of the limiting extremal process of branching Brownian motion, we proved in a recent work [Comm. Pure Appl. Math. 64 (2011) 1647-1676] that, in the limit of large time $t$, extremal particles descend with overwhelming probability from ancestors having split either within a distance of order 1 from time 0, or within a distance of order 1 from time $t$. The result suggests that the extremal process of branching Brownian motion is a randomly shifted cluster point process. Here we put part of this picture on rigorous ground: we prove that the point process…
Branching Brownian Motion describes a system of particles which diffuse in space and split into offsprings according to a certain random mechanism. In virtue of the groundbreaking work by M. Bramson on the convergence of solutions of the Fisher-KPP equation to traveling waves, the law of the rightmost particle in the limit of large times is rather well understood. In this work, we address the full statistics of the extremal particles (first-, second-, third- etc. largest). In particular, we prove that in the large $t-$limit, such particles descend with overwhelming probability from ancestors h…
In this paper, we establish limit theorems for the supremum of the support, denoted by $M_t$, of a supercritical super-Brownian motion $\{X_t, t\ge0\}$ on $\mathbb{R}$. We prove that there exists an $m(t)$ such that $(X_t-m(t), M_t-m(t))$ converges in law, and give some large deviation results for $M_t$ as $t\to\infty$. We also prove that the limit of the extremal process $\mathcal{E}_t:=X_t-m(t)$ is a Poisson random measure with exponential intensity in which each atom is decorated by an independent copy of an auxiliary measure. These results are analogues of the results for branching Brownia…
In this article, we study the extremal processes of branching Brownian motions conditioned on having an unusually large maximum. The limiting point measures form a one-parameter family and are the decoration point measures in the extremal processes of several branching processes, including branching Brownian motions with variable speed and multitype branching Brownian motions. We give a new, alternative representation of these point measures and we show that they form a continuous family. This also yields a simple probabilistic expression for the constant that appears in the large deviation pr…
This paper derives an exact asymptotic expression for \[ \mathbb{P}_{\mathbf{x}_u}\{\exists_{t\ge0} \mathbf{X}(t)- \boldsymbolμt\in \mathcal{U} \}, \ \ {\rm as}\ \ u\to\infty, \] where $\mathbf{X}(t)=(X_1(t),\ldots,X_d(t))^\top,t\ge0$ is a correlated $d$-dimensional Brownian motion starting at the point $\mathbf{x}_u=-\boldsymbolαu$ with $\boldsymbolα\in \mathbb{R}^d$, $\boldsymbolμ \in \mathbb{R}^d$ and $\mathcal{U}=\prod_{i=1}^d [0,\infty)$. The derived asymptotics depends on the solution of an underlying multidimensional quadratic optimization problem with constraints, which leads in some c…
The convergence of properly time-scaled and normalized maxima of independent standard Brownian motions to the Brown-Resnick process is well-known in the literature. In this paper, we study the extremal functional behavior of non-Gaussian processes, namely squared Bessel processes and scalar products of Brownian motions. It is shown that maxima of independent samples of those processes converge weakly on the space of continuous functions to the Brown-Resnick process.
We consider the limiting extremal process ${\mathcal X}$ of the particles of the binary branching Brownian motion. We show that after a shift by the logarithm of the derivative martingale $Z$, the rescaled "density" of particles, which are at distance $n+x$ from a position close to the tip of ${\mathcal X}$, converges in probability to a multiple of the exponential $e^x$ as $n\to+\infty$. We also show that the fluctuations of the density, after another scaling and an additional random but explicit shift, converge to a $1$-stable random variable. Our approach uses analytic techniques and is mot…
For general $β\geq 1$, we consider Dyson Brownian motion at equilibrium and prove convergence of the extremal particles to an ensemble of continuous sample paths in the limit $N \to \infty$. For each fixed time, this ensemble is distributed as the Airy$_β$ random point field. We prove that the increments of the limiting process are locally Brownian. When $β>1$ we prove that after subtracting a Brownian motion, the sample paths are almost surely locally $r$-H{ö}lder for any $r<1-(1+β)^{-1}$. Furthermore for all $β\geq 1$ we show that the limiting process solves an SDE in a weak sense. When $β=2…
We define a family of diffeomorphism-invariant models of random connections on principal $G$-bundles over the plane, whose curvatures are concentrated on singular points. In a limit when the number of point grows whilst the singular curvature on each point diminishes, the model converges in some sense towards a Yang--Mills field. We study another regime for which we prove that the holonomy along a Brownian trajectory converges towards an explicit limit.
In this paper, we study complex valued branching Brownian motion in the so-called glassy phase, or also called phase II. In this context, we prove a limit theorem for the complex partition function hence confirming a conjecture formulated by Lacoin and the last two authors in a previous paper on complex Gaussian multiplicative chaos. We will show that the limiting partition function can be expressed as a product of a Gaussian random variable, mainly due to the windings of the phase, and a stable transform of the so called derivative martingale, mainly due to the clustering of the modulus. The …
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…
Let $Mat_{\mathbb{C}}(K,N)$ be the space of $K\times N$ complex matrices. Let $\mathbf{B}_t$ be Brownian motion on $Mat_{\mathbb{C}}(K,N)$ starting from the zero matrix and $\mathbf{M}\in Mat_{\mathbb{C}}(K,N)$. We prove that, with $K\ge N$, the $N$ eigenvalues of $\left(\mathbf{B}_t+t\mathbf{M}\right)^*\left(\mathbf{B}_t+t\mathbf{M}\right)$ form a Markov process with an explicit transition kernel. This generalizes a classical result of Rogers and Pitman for multidimensional Brownian motion with drift which corresponds to $N=1$. We then give two more descriptions for this Markov process. First…
The path W[0,t] of a Brownian motion on a d-dimensional torus T^d run for time t is a random compact subset of T^d. We study the geometric properties of the complement T^d \ W[0,t] for t large and d >= 3. In particular, we show that the largest regions in this complement have a linear scale phi = [(d log t)/(d-2)kt]^{1/(d-2)}, where k is the capacity of the unit ball. More specifically, we identify the sets E for which T^d \ W[0,t] contains a translate of phi E, and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of T^…
The logarithmic correction for the order of the maximum for two-speed branching Brownian motion changes discontinuously when approaching slopes $σ_1^2=σ_2^2=1$ which corresponds to standard branching Brownian motion. In this article we study this transition more closely by choosing $σ_1^2=1\pm t^{-α}$ and $σ_2^2=1\pm t^{-α}$. We show that the logarithmic correction for the order of the maximum now smoothly interpolates between the correction in the iid case $\frac{1}{2\sqrt 2}\ln(t),\;\frac{3}{2\sqrt 2}\ln(t)$ and $\frac{6}{2\sqrt 2}\ln(t)$ when $0<α<\frac{1}{2}$. This is due to the localisati…
We investigate in this work the spectrum of singularities of super-Brownian motion with stable branching. The main purpose is to provide a uniform description of the latter in high dimension $d\geq\tfrac{2}{γ-1}$, presenting the singularities existing at every time t and characterising as well the set of random times at which singularities of higher order appear. In lower dimensions, we give a partial description of the singularities which complement the recent study of the density by Mytnik and Wachtel.
We establish a small time large deviation principle and a Varadhan type asymptotics for Brownian motion with singular drift on $\bR^d$ with $d\geq 3$ whose infinitesimal generator is $\frac12 Δ+ μ\cdot \nabla$, where each $μ_i$ of $μ= (μ_1, \cdots, μ_d)$ is a measure in some suitable Kato class.
In this note we prove that the probability measures generated by two generalized grey Brownian motions with different parameters are singular with respect to each other. This result can be interpreted as an extension of the Feldman-Hájek dichotomy of Gaussian measures to a family of non-Gaussian measures
We study sufficient conditions which ensure that the probability measures generated by two bifractional Brownian motions on an interval [0,1] are singular with respect to each other and sufficient conditions for the probability measures generated by two trifractional Brownian motions on an interval [0,1] are singular with respect to each other.
In this note, we study the asymptotical frontier behavior of a branching reflected Brownian motion. There is essentially no difference in maximal displacement between a branching Brownian motion and its reflected counterpart. We provide two proofs of this fact, one via a soft argument on the dependance of two-sided extremal particles in a branching Brownian motion and the other based on direct computations as in Roberts. The asymptotics of minimal displacement is also given.
In this paper we study a singular stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter $H>\frac 12$. Under some assumptions on the drift, we show that there is a unique solution, which has moments of all orders. We also apply the techniques of Malliavin calculus to prove that the solution has an absolutely continuous law at any time $t>0$.
We prove a Chung-type law of the iterated logarithm for a multiparameter extension of the fractional Brownian motion which is not increment stationary. This multiparameter fractional Brownian motion behaves very differently at the origin and away from the axes, which also appears in the Hausdorff dimension of its range and in the measure of its pointwise Hölder exponents. A functional version of this Chung-type law is also provided.
We establish the Lifschitz-type singularity around the bottom of the spectrum for the integrated density of states for a class of subordinate Brownian motions in presence of the nonnegative Poissonian random potentials, possibly of infinite range, on the Sierpiński gasket. We also study the long-time behaviour for the corresponding averaged Feynman-Kac functionals.
We give short proofs of two classical results about the position of the extremal particle in a branching Brownian motion, one concerning the median position and another the almost sure behaviour.
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 …
We consider solutions of the linear heat equation in $\mathbb{R}^N$ with isolated singularities. It is assumed that the position of a singular point depends on time and is Hölder continuous with the exponent $α\in (0,1)$. We show that any isolated singularity is removable if it is weaker than a certain order depending on $α$. We also show the optimality of the removability condition by showing the existence of a solution with a nonremovable singularity. These results are applied to the case where the singular point behaves like a fractional Brownian motion with the Hurst exponent $H \in (0,1/2…
In this paper we derive a Bismut-Elworthy-Li type formula with respect to strong solutions to singular stochastic differential equations (SDE's) with additive noise given by a multi-dimensional fractional Brownian motion with Hurst parameter $H<1/2$. "Singular" here means that the drift vector field of such equations is allowed to be merely bounded and integrable. As an application we use this representation formula for the study of price sensitivities of financial claims based on a stock price model with stochastic volatility, whose dynamics is described by means of fractional Brownian motion…
We construct a class of one-dimensional diffusion processes on the particles of branching Brownian motion that are symmetric with respect to the limits of random martingale measures. These measures are associated with the extended extremal process of branching Brownian motion and are supported on a Cantor-like set. The processes are obtained via a time-change of a standard one-dimensional reflected Brownian motion on $\mathbb{R}_+$ in terms of the associated positive continuous additive functionals. The processes introduced in this paper may be regarded as an analogue of the Liouville Brownian…
We consider a one-parameter family of Grushin-type singularities on surfaces, and discuss the possible diffusions that extend Brownian motion to the singularity. This gives a quick proof and clear intuition for the fact that heat can only cross the singularity for an intermediate range of the parameter. When crossing is possible and the singularity consists of one point, we give a complete description of these diffusions, and we describe a "best" extension, which respects the isometry group of the surface and also realizes the unique symmetric one-point extension of the Brownian motion, in the…
Strong solutions for singular SDEs driven by long-range dependent fractional Brownian motion and other Volterra processes
Institute for Theoretical Physics III, Ruhr University Bochum（波鸿鲁尔大学第三理论物理研究所）；Department of Physics, Institute of Science Tokyo（东京科学大学物理系）；Department of Mathematics, Pohang University of Science and Technology（浦项科技大学数学系）
We establish the singularity with respect to Lebesgue measure as a function of time of the conditional probability that the sum of two one-dimensional Brownian motions will exit from the unit interval before time $t$, given the trajectory of the second Brownian motion up to the same time. On the way of doing so we show that if one solves the one-dimensional heat equation with zero condition on a trajectory of a one-dimensional Brownian motion, which is the lateral boundary, then for each moment of time with probability one the normal derivative of the solution is zero, provided that the diffus…
In this paper we study weak solutions for the following type of stochastic differential equation \[ dX_{t}=dW_{t}+b(t, X_{t})dt, \quad t\ge s, \quad X_{s}=x, \] where $b: [0,\infty) \times \mathbb{R}^{d} \to \mathbb{R}^{d}$ is a measurable drift, $W=(W_{t})_{t \ge 0}$ is a $d$-dimensional Brownian motion and $(s,x)\in [0,\infty) \times \mathbb{R}^{d}$ is the starting point. A solution $X=(X_t)_{t \ge s}$ for the above SDE is called a Brownian motion with time-dependent drift $b$ starting from $(s,x)$. Under the assumption that $|b|$ belongs to the forward-Kato class $\mathcal{F} \mathcal{K}_{d…
In this paper we present a new method for the construction of strong solutions of SDE's with merely integrable drift coefficients driven by a multidimensional fractional Brownian motion with Hurst parameter H < 1/2. Furthermore, we prove the rather surprising result of the higher order Frechet differentiability of stochastic flows of such SDE's in the case of a small Hurst parameter. In establishing these results we use techniques from Malliavin calculus combined with new ideas based on a "local time variational calculus". We expect that our general approach can be also applied to the study of…
