Brownian motion and generalized analytic and inner functions

Alain Bernard; Eddy A. Campbell; A. M. Davie

Displaying similar documents to “Brownian motion and generalized analytic and inner functions”

Boundary behaviour of harmonic functions in a half-space and brownian motion

D. L. Burkholder, Richard F. Gundy (1973)

Annales de l'institut Fourier

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Let $u$ be harmonic in the half-space $R_{+}^{n + 1}$ , $n \geq 2$ . We show that $u$ can have a fine limit at almost every point of the unit cubs in $R^{n} = \partial R_{+}^{n + 1}$ but fail to have a nontangential limit at any point of the cube. The method is probabilistic and utilizes the equivalence between conditional Brownian motion limits and fine limits at the boundary. In $R_{+}^{2}$ it is known that the Hardy classes $H^{p}$ , $0 < p < \infty$ , may be described in terms of the integrability of the nontangential maximal function, or, alternatively, in terms...

Superdiffusivity for brownian motion in a poissonian potential with long range correlation II: Upper bound on the volume exponent

Hubert Lacoin (2012)

Annales de l'I.H.P. Probabilités et statistiques

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This paper continues a study on trajectories of Brownian Motion in a field of soft trap whose radius distribution is unbounded. We show here that for both point-to-point and point-to-plane model the volume exponent (the exponent associated to transversal fluctuation of the trajectories) $ξ$ is strictly less than $1$ and give an explicit upper bound that depends on the parameters of the problem. In some specific cases, this upper bound matches the lower bound proved in the first part of this...

The brownian cactus I. Scaling limits of discrete cactuses

Nicolas Curien, Jean-François Le Gall, Grégory Miermont (2013)

Annales de l'I.H.P. Probabilités et statistiques

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The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space $E$ , one can associate an $ℝ$ -tree called the continuous cactus of $E$ . We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the Gromov–Hausdorff sense. Moreover, the Brownian...

An application of fine potential theory to prove a Phragmen Lindelöf theorem

Terry J. Lyons (1984)

Annales de l'institut Fourier

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We give a new proof of a Phragmén Lindelöf theorem due to W.H.J. Fuchs and valid for an arbitrary open subset $U$ of the complex plane: if $f$ is analytic on $U$ , bounded near the boundary of $U$ , and the growth of $j$ is at most polynomial then either $f$ is bounded or $U \supset {| z | > r}$ for some positive $r$ and $f$ has a simple pole.

Minkowski sums and Brownian exit times

Christer Borell (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

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If $C$ is a domain in $^{n},$ the Brownian exit time of $C$ is denoted by $T_{C} .$ Given domains $C$ and $D$ in $^{n}$ this paper gives an upper bound of the distribution function of $T_{C + D}$ when the distribution functions of $T_{C}$ and $T_{D}$ are known. The bound is sharp if $C$ and $D$ are parallel affine half-spaces. The paper also exhibits an extension of the Ehrhard inequality

Brownian motion and transient groups

Nicolas Th. Varopoulos (1983)

Annales de l'institut Fourier

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In this paper I consider $\tilde{M} \to M$ a covering of a Riemannian manifold $M$ . I prove that Green’s function exists on $\tilde{M}$ if any and only if the symmetric translation invariant random walks on the covering group $G$ are transient (under the assumption that $M$ is compact).

Optimal stopping with advanced information flow: selected examples

Yaozhong Hu, Bernt Øksendal (2008)

Banach Center Publications

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We study optimal stopping problems for some functionals of Brownian motion in the case when the decision whether or not to stop before (or at) time t is allowed to be based on the δ-advanced information $ℱ_{t + δ}$ , where $ℱ_{s}$ is the σ-algebra generated by Brownian motion up to time s, s ≥ -δ, δ > 0 being a fixed constant. Our approach involves the forward integral and the Malliavin calculus for Brownian motion.

Weak inequalities for exit times and analytic functions

D. Burkholder (1979)

Banach Center Publications

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