α-time fractional Brownian motion: PDE
          connections and local times∗

Erkan Nane; Dongsheng Wu; Yimin Xiao

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α-time fractional brownian motion: PDE connections and local times

Erkan Nane, Dongsheng Wu, Yimin Xiao (2012)

ESAIM: Probability and Statistics

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For 0 < ≤ 2 and 0 < < 1, an -time fractional Brownian motion is an iterated process = {() = (()) ≥ 0} obtained by taking a fractional Brownian motion {() ∈ ℝ} with Hurst index 0 < < 1 and replacing the time parameter with a strictly -stable Lévy process {() ≥ 0} in ℝ independent of {() ∈ R}. It is shown that such processes have natural connections to partial differential equations and, when is a stable subordinator, can arise as scaling limit...

Density of paths of iterated Lévy transforms of brownian motion

Marc Malric (2012)

ESAIM: Probability and Statistics

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The Lévy transform of a Brownian motion is the Brownian motion given by = sgn()d; call the Brownian motion obtained from by iterating times this transformation. We establish that almost surely, the sequence of paths ( → ) is dense in Wiener space, for the topology of uniform convergence on compact time intervals.

Density of paths of iterated Lévy transforms of Brownian motion

Marc Malric (2012)

ESAIM: Probability and Statistics

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Wiener integral for the coordinate process under the σ-finite measure unifying brownian penalisations

Kouji Yano (2011)

ESAIM: Probability and Statistics

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Wiener integral for the coordinate process is defined under the -finite measure unifying Brownian penalisations, which has been introduced by [Najnudel , 345 (2007) 459–466] and [Najnudel , 19. Mathematical Society of Japan, Tokyo (2009)]. Its decomposition before and after last exit time from 0 is studied. This study prepares for the author's recent study [K. Yano, 258 (2010) 3492–3516] of Cameron-Martin formula for the -finite measure.

Densité des orbites des trajectoires browniennes sous l’action de la transformation de Lévy

Jean Brossard, Christophe Leuridan (2012)

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

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Let be a measurable transformation of a probability space $(E, ℰ, π)$ , preserving the measure. Let be a random variable with law . Call (⋅, ⋅) a regular version of the conditional law of given (). Fix $B \in ℰ$ . We first prove that if is reachable from -almost every point for a Markov chain of kernel , then the -orbit of -almost every point visits . We then apply this result to the Lévy transform, which transforms the Brownian motion into the Brownian motion || − , where is the local time at 0...

Pointwise constrained radially increasing minimizers in the quasi-scalar calculus of variations

Luís Balsa Bicho, António Ornelas (2014)

ESAIM: Control, Optimisation and Calculus of Variations

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We prove of vector minimizers () = (||) to multiple integrals ∫ ((), |()|) on a ⊂ ℝ, among the Sobolev functions (·) in + (, ℝ), using a : ℝ×ℝ → [0,∞] with (·) and . Besides such basic hypotheses, (·,·) is assumed to satisfy also...

Hydrodynamic limit of a d-dimensional exclusion process with conductances

Fábio Júlio Valentim (2012)

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

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Fix a polynomial of the form () = + ∑2≤≤ =1 with (1) gt; 0. We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes on $𝕋^{d}$ , with conductances given by special class of functions, is described by the unique weak solution of the non-linear parabolic partial differential equation = ∑ ...

Means in complete manifolds: uniqueness and approximation

Marc Arnaudon, Laurent Miclo (2014)

ESAIM: Probability and Statistics

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Let be a complete Riemannian manifold, ∈ ℕ and ≥ 1. We prove that almost everywhere on = ( ,, ) ∈ for Lebesgue measure in , the measure $μ (x) = N \sum_{k = 1}^{N} x_{k}$ μ ( x ) = 1 N ∑ k = 1 N δ x k has a unique–mean (). As a consequence, if = ( ,, ) is a -valued random variable with absolutely continuous law, then almost surely (()) has a unique –mean. In particular if ( ...

Hereditary properties of words

József Balogh, Béla Bollobás (2010)

RAIRO - Theoretical Informatics and Applications

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Let be a hereditary property of words, , an infinite class of finite words such that every subword (block) of a word belonging to is also in . Extending the classical Morse-Hedlund theorem, we show that either contains at least words of length for every or, for some , it contains at most words of length for every . More importantly, we prove the following quantitative extension of this result: if has words of length then, for every , it contains at most ⌈( + 1)/2⌉⌈( + 1)/2⌈...