The first exit time of Brownian motion from a parabolic domain (original) (raw)
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Brownian motion with a parabolic drift and airy functions
Probability Theory and Related Fields, 1989
: t ~ R} be two-sided Brownian motion, originating from zero, and let V(a) be defined by V(a)= sup {t ~IR: W(t)-(t-a) 2 is maximal}. Then {V(a):aslR} is a Markovian jump process, running through the locations of maxima of two-sided Brownian motion with respect to the parabolas fa(t)=(t--a) 2. We give an analytic expression for the infinitesimal generators of the processes {(a+t, V(a+t)): t>=O}, a~IR, in terms of Airy functions in Theorem 4.1. This makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators (i.e. estimators derived under order restrictions). An example of this is given in , where the asymptotic distribution of the (standardized) L 1-distance between a decreasing density and the Grenander maximum likelihood estimator of this density is determined. On our way to Theorem 4.1 we derive some other results. For example, we give an analytic expression for the joint density of the maximum and the location of the maximum of the process { W(t) -ct 2 : t ~ IR}, where c is an aribrary positive constant. We also determine the Laplace transform of the integral over a Brownian excursion. These last results also have recently been derived by several other authors, using a variety of methods. * This paper was awarded the Rollo Davidson prize 1985 (Cambridge, UK) Theorem 2.1. Let, for c>0, s, xelR, Q}S.x) be the probability measure on the Boret a-field of C([s, oo);1R), corresponding to the process {X(t): t > s}, where X(t) = W(t) -ct 2 and { W(t) : t >= s} is Brownian motion, starting at x + cs 2 at time s. Let the first passage time % of the process X be defined by (2.1), where, as usual, we define ra = 0% if {t > S : X(t) = a} = 0. Then
Constrained Brownian motion: fluctuations away from circular and parabolic
2016
Motivated by the polynuclear growth model, we consider a Brownian bridge b(t) with b(±T) = 0 conditioned to stay above the semicircle cT (t) =√ T 2 − t2. In the limit of large T, the fluctuation scale of b(t) − cT (t) is T 1/3 and its time-correlation scale is T 2/3. We prove that, in the sense of weak convergence of path measures, the conditioned Brownian bridge, when properly rescaled, converges to a stationary diffusion process with a drift explicitly given in terms of Airy functions. The dependence on the reference point t = τT, τ ∈ (−1,1), is only through the second derivative of cT (t) at t = τT. We also prove a corresponding result where instead of the semicircle the barrier is a parabola of height T γ, γ> 1/2. The fluctuation scale is then T (2−γ)/3. More general conditioning shapes are briefly discussed.
A characterization ofh-Brownian motion by its exit distributions
Probability Theory and Related Fields, 1992
Let X h be an h-Brownian motion in the unit ball D c R d with h harmonic, such that the representing measure of h is not singular with respect to the surface measure on 0D. If Y is a continuous strong Markov process in D with the same killing distributions as X h, then Yis a time change of X h. Similar results hold in simply connected domains in C provided with either the Martin or the Euclidean boundary.
Constrained Brownian motion: Fluctuations away from circular and parabolic barriers
The Annals of Probability, 2005
Motivated by the polynuclear growth model, we consider a Brownian bridge b(t) with b(±T ) = 0 conditioned to stay above the semicircle cT (t) = √ T 2 − t 2 . In the limit of large T , the fluctuation scale of b(t) − cT (t) is T 1/3 and its time-correlation scale is T 2/3 . We prove that, in the sense of weak convergence of path measures, the conditioned Brownian bridge, when properly rescaled, converges to a stationary diffusion process with a drift explicitly given in terms of Airy functions. The dependence on the reference point t = τ T , τ ∈ (−1, 1), is only through the second derivative of cT (t) at t = τ T . We also prove a corresponding result where instead of the semicircle the barrier is a parabola of height T γ , γ > 1/2. The fluctuation scale is then T (2−γ)/3 . More general conditioning shapes are briefly discussed.
2021
We prove a number of results relating exit times of planar Brownian with the geometric properties of the domains in question. Included are proofs of the conformal invariance of moduli of rectangles and annuli using Brownian motion; similarly probabilistic proofs of some recent results of Karafyllia on harmonic measure on starlike domains; examples of domains and their complements which are simultaneously large when measured by the moments of exit time of Brownian motion, and examples of domains and their complements which are simultaneously small; and proofs of several identities involving the Cauchy distribution using the optional stopping theorem.
Limiting behaviors of the Brownian motions on hyperbolic spaces
Colloquium Mathematicum, 2009
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.
2020
A fundamental question in rough path theory is whether the expected signature of a geometric rough path completely determines the law of signature. One sufficient condition is that the expected signature has infinite radius of convergence, which is satisfied by various stochastic processes on a fixed time interval, including the Brownian motion. In contrast, for the Brownian motion stopped upon the first exit time from a bounded domain Ω, it is only known that the radius of convergence for the expected signature on sufficiently regular Ω is strictly positive everywhere, and that the radius of convergence is finite at some point when Ω is the 2-dimensional unit disc ([1]). In this paper, we prove that on any bounded C-domain Ω ⊂ R with 2 ≤ d ≤ 8, the expected signature of the stopped Brownian motion has finite radius of convergence everywhere. A key ingredient of our proof is the introduction of a “domain-averaging hyperbolic development” (see Definition 4.1), which allows us to symm...
Central limit theorems for parabolic stochastic partial differential equations
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
Let {u(t , x)} t≥0,x∈R d denote the solution of a d-dimensional nonlinear stochastic heat equation that is driven by a Gaussian noise, white in time with a homogeneous spatial covariance that is a finite Borel measure f and satisfies Dalang's condition. We prove two general functional central limit theorems for occupation fields of the form N −d´R d g(u(t , x))ψ(x/N) dx as N → ∞, where g runs over the class of Lipschitz functions on R d and ψ ∈ L 2 (R d). The proof uses Poincaré-type inequalities, Malliavin calculus, compactness arguments, and Paul Lévy's classical characterization of Brownian motion as the only mean zero, continuous Lévy process. Our result generalizes central limit theorems of Huang et al [15, 16] valid when g(u) = u and ψ = 1 [0,1] d. Résumé Soit {u(t , x)} t≥0,x∈R d la solution d'uneéquation de la chaleur stochastique non-linéaire ddimensionnelle, perturbée par un bruit gaussien, blanc en temps et avec une covariance homogène en espace donnée par une mesure de Borel finie qui satisfait la condition de Dalang. Nous démontrons deux théorèmes de la limite centrale fonctionnels pour des champs d'occupation de la forme N −d´R d g(u(t , x))ψ(x/N) dx quand N → ∞, où g est une function lipschitzienne sur R d et ψ ∈ L 2 (R d). La preuve utilise des inegalités de Poincaré, le calcul de Malliavin, des arguments de compacité et la caractérisation du mouvement brownien comme le seul processus de Lévy continu avec moyenne zéro. Notre résultat généralise les théorèmes de la limite centrale de Huang et al [15, 16] qui sont valables pour g(u) = u et ψ = 1 [0,1] d .