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Let $u (x, y)$ be a harmonic function in the half-plane $R_{+}^{n + 1}$ , $n \geq 2$ . We define a family of functionals $D (u; r), - \infty > r > \infty$ , that are analogs of the family of local times associated to the process $u (x_{t}, y_{t})$ where $(x_{t}, y_{t})$ is Brownian motion in $R_{+}^{n + 1}$ . We show that $D (u) = {sup}_{r} D (u; r)$ is bounded in $L^{p}$ if and only if $u (x, y)$ belongs to $H^{p}$ , an equivalence already proved by Barlow and Yor for the supremum of the local times. Our proof relies on the theory of singular integrals due to Caldéron and Zygmund, rather than the stochastic calculus.

The dimension of the frontier of planar Brownian motion.

Lawler, G. (1996)

Electronic Communications in Probability [electronic only]

The distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group

A. Hulanicki (1976)

Studia Mathematica

The distribution of local times of a brownian bridge

Jim Pitman (1999)

Séminaire de probabilités de Strasbourg

The distribution of time spent by a standard excursion above a given level, with applications to ring polymers near a discontinuity in potential.

Jansons, Kalvis M. (1997)

Electronic Communications in Probability [electronic only]

The Dyson Brownian Minor Process

Mark Adler, Eric Nordenstam, Pierre Van Moerbeke (2014)

Annales de l’institut Fourier

Consider an $n \times n$ Hermitean matrix valued stochastic process ${H_{t}}_{t \geq 0}$ where the elements evolve according to Ornstein-Uhlenbeck processes. It is well known that the eigenvalues perform a so called Dyson Brownian motion, that is they behave as Ornstein-Uhlenbeck processes conditioned never to intersect.In this paper we study not only the eigenvalues of the full matrix, but also the eigenvalues of all the principal minors. That is, the eigenvalues of the $k \times k$ minors in the upper left corner of $H_{t}$ . Projecting this...

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Tail estimates for the Brownian excursion area and other Brownian areas.

Techniques biharmoniques pour l'étude du mouvement brownien de P. Lévy à trois paramètres

Temps locaux d'intersection et points multiples des processus de Lévy

The Asymptotic Behaviour of Certain Addititve Functionals of Brownian Motion.

The best estimation of a ratio inequality for continuous martingales

The branching process in a brownian excursion

The brownian burglar : conditioning brownian motion by its local time process

The Brunn-Minkowski Inequality in Gauss Space.

The chance of a long lifetime for Brownian motion in a horn-shaped domain.

The construction of brownian motion on the Sierpinski carpet

The density of the area integral in $ℝ_{+}^{n + 1}$

The dimension of the frontier of planar Brownian motion.

The distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group

The distribution of local times of a brownian bridge

The distribution of time spent by a standard excursion above a given level, with applications to ring polymers near a discontinuity in potential.

The Dyson Brownian Minor Process

The energy representation of Sobolev-Lie groups

The exact asymptotic of the time to collision.

The existence of a multiple spider martingale in the natural filtration of a certain diffusion in the plane

The exit place of Brownian motion in an unbounded domain.

Currently displaying 1 – 20 of 49