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J. F. Le Gall

Bio: J. F. Le Gall is an academic researcher from Pierre-and-Marie-Curie University. The author has contributed to research in topics: Brownian motion & Hypoelliptic operator. The author has an hindex of 9, co-authored 13 publications receiving 407 citations.

Papers
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Journal ArticleDOI
TL;DR: In this article, intersection properties of multi-dimensional random walks were studied and a central limit theorem for the range of a two-dimensional recurrent random walk was proved. But the results were only applied to the case of two independent Brownian motions.
Abstract: We study intersection properties of multi-dimensional random walks. LetX andY be two independent random walks with values in ℤd (d≦3), satisfying suitable moment assumptions, and letIn denote the number of common points to the paths ofX andY up to timen. The sequence (In), suitably normalized, is shown to converge in distribution towards the “intersection local time” of two independent Brownian motions. Results are applied to the proof of a central limit theorem for the range of a two-dimensional recurrent random walk, thus answering a question raised by N. C. Jain and W. E. Pruitt.

111 citations

01 Jan 1986
TL;DR: On considere le processus obtenu en ajoutant a un mouvement brownien reflechi un multiple de son temps local en 0.
Abstract: On considere le processus obtenu en ajoutant a un mouvement brownien reflechi un multiple de son temps local en 0. La famille des temps locaux de ce processus suit la loi du carre d'un processus de Bessel de dimension d dependant de la constante multiplicative choisie

48 citations

Journal ArticleDOI
TL;DR: In this paper, an asymptotic expansion of the integral of the set K to the surrounding medium ℝ3−K is given, which can be interpreted as the expected value of the volume of the Wiener sausage associated with K and a d-dimensional Brownian motion.
Abstract: We consider the following heat conduction problem. Let K be a compact set in Euclidean space ℝ3. Suppose that K is held at the temperature 1, while the surrounding medium is at the temperature 0 at time 0. Following Spitzer we investigate the asymptotic behaviour of the integral E K (t) which represents the total energy flow in time t from the set K to the surrounding medium ℝ3−K. An asymptotic expansion is given for E K (t) which refines a theorem due to Spitzer. This expansion also verifies and improves a formal calculation of Kac. Similar results are proved in higher dimensions. Up to the constant m(K), the quantity E K (t) can be interpreted as the expected value of the volume of the Wiener sausage associated with K and a d-dimensional Brownian motion. This point of view both plays a major role in the proofs and leads to a probabilistic interpretation of the different terms of the expansion.

40 citations

Book ChapterDOI
01 Jan 1987
TL;DR: In this paper, the conditions générales d'utilisation (http://www.numdam.org/conditions) of the agreement with the séminaire de probabilités (Strasbourg) are discussed.
Abstract: © Springer-Verlag, Berlin Heidelberg New York, 1987, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

39 citations


Cited by
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Journal ArticleDOI
TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

5,689 citations

BookDOI
01 Jan 2006
TL;DR: In this paper, the Brownian forest and the additive coalescent were constructed for random walks and random forests, respectively, and the Bessel process was used for random mappings.
Abstract: Preliminaries.- Bell polynomials, composite structures and Gibbs partitions.- Exchangeable random partitions.- Sequential constructions of random partitions.- Poisson constructions of random partitions.- Coagulation and fragmentation processes.- Random walks and random forests.- The Brownian forest.- Brownian local times, branching and Bessel processes.- Brownian bridge asymptotics for random mappings.- Random forests and the additive coalescent.

1,371 citations

Book
01 Jul 1999
TL;DR: In this article, the authors present an overview of the history of continuous-state branching processes and superprocesses and their connections with statistical mechanics and interacting particle systems, including the connections with partial differential equations.
Abstract: I An Overview.- I.1 Galton-Watson processes and continuous-state branching processes.- I.2 Spatial branching processes and superprocesses.- I.3 Quadratic branching and the Brownian snake.- I.4 Some connections with partial differential equations.- I.5 More general branching mechanisms.- I.6 Connections with statistical mechanics and interacting particle systems.- II Continuous-state Branching Processes and Superprocesses.- II.1 Continuous-state branching processes.- II.2 Superprocesses.- II.3 Some properties of superprocesses.- II.4 Calculations of moments.- III The Genealogy of Brownian Excursions.- III.1 The Ito excursion measure.- III.2 Binary trees.- III.3 The tree associated with an excursion.- III.4 The law of the tree associated with an excursion.- III.5 The normalized excursion and Aldous' continuum random tree.- IV The Brownian Snake and Quadratic Superprocesses.- IV.1 The Brownian snake.- IV.2 Finite-dimensional marginals of the Brownian snake.- IV.3 The connection with superprocesses.- IV.4 The case of continuous spatial motion.- IV.5 Some sample path properties.- IV.6 Integrated super-Brownian excursion.- V Exit Measures and the Nonlinear Dirichlet Problem.- V.1 The construction of the exit measure.- V.2 The Laplace functional of the exit measure.- V.3 The probabilistic solution of the nonlinear Dirichlet problem.- V.4 Moments of the exit measure.- VI Polar Sets and Solutions with Boundary Blow-up.- VI.1 Solutions with boundary blow-up.- VI.2 Polar sets.- VI.3 Wiener's test for the Brownian snake.- VI.4 Uniqueness of the solution with boundary blow-up.- VII The Probabilistic Representation of Positive Solutions.- VII.1 Singular solutions and boundary polar sets.- VII.2 Some properties of the exit measure from the unit disk.- VII.3 The representation theorem.- VII.4 Further developments.- VIII Levy Processes and the Genealogy of General Continuous-state Branching Processes.- VIII.1 The discrete setting.- VIII.2 Levy processes.- VIII.3 The height process.- VIII.4 The exploration process.- VIII.5 Proof of Theorem 2.- Bibliographical Notes.

404 citations

Journal ArticleDOI
TL;DR: In this paper, general formulae are obtained for size-biased sampling from a Poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function.
Abstract: Some general formulae are obtained for size-biased sampling from a Poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function. These formulae explain why in certain cases (gamma and stable) the size-biased permutation of the normalized jumps of a subordinator can be represented by a stickbreaking (residual allocation) scheme defined by independent beta random variables. An application is made to length biased sampling of excursions of a Markov process away from a recurrent point of its statespace, with emphasis on the Brownian and Bessel cases when the associated inverse local time is a stable subordinator. Results in this case are linked to generalizations of the arcsine law for the fraction of time spent positive by Brownian motion.

392 citations

01 Jan 2014
TL;DR: (1 < p ≤ ∞) [LS87f] (2) [HR88a].
Abstract: (1 < p ≤ ∞) [LS87f]. (2) [HR88a]. (2m− 2) [KL88]. (A0, A1)θ1 [Xu87a]. (α, β) [Pie88a, Fin88a]. (d ≥ 1) [Wsc85a]. (λ) [DM85b]. (Z/2) [Car86b]. (nα) [Sch85h]. (φ)2 [BM89c]. (τ − λ)u = f [Wei87r]. (x, t) [Lum87, Lum89]. (X1 −X3, X2 −X3) [SW87]. 0 [Caz88, Kas86, Pro87]. 0 < p < 1 [Cle87]. 1 [Bak85a, DD85, Drm87, Eli88, FT88a, Gek86d, HN88, Kos86a, LT89, Pet89a, Pro87, Tan87, vdG89]. 1/4 [KS86e]. 1 ≤ q < 2 [Gue86]. 2 [BPPS87, Cam88, Cat85a, ES87b, Gan85e, Gol86a, HRL89g, Hei85, Hua86, Kan89, KB86, Li86, LT89, Mil87b, Mur85a, Qui85b, SP89, Shi85, Spe86, Wal85b, Wan86]. 2m− 2 [Kos88b]. 2m− 3 [Kos88b]. 2m− 4 [Kos88b]. 2× 2 [Vog88]. 3 [Aso89, BPPS87, BW85c, BG88b, Che86d, Fis86, Gab85, Gu87a, HLM85b, Kam89b, Kir89c, Lev85c, Mil85c, Néd86, Pet86, Ron86, Sch85b, ST88, Tur88b, Wan86, Wen85, tDP89, vdW86]. 4 [Bau88a, Don85a, FKV88, Kha88, Kir89l, SS86, Seg85b, Wal85b]. 5 [Ito89, Kir89e, SV85]. 5(4) [Cas86]. 5819539783680 [KSX87]. 6 [PH89, Žub88].

171 citations