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Random paths with bounded local time

Itai Benjamini, Nathanaël Berestycki (2010)

Journal of the European Mathematical Society

We consider one-dimensional Brownian motion conditioned (in a suitable sense) to have a local time at every point and at every moment bounded by some fixed constant. Our main result shows that a phenomenon of entropic repulsion occurs: that is, this process is ballistic and has an asymptotic velocity approximately 4.58... as high as required by the conditioning (the exact value of this constant involves the first zero of a Bessel function). We also study the random walk case and show that the process...

Random walk local time approximated by a brownian sheet combined with an independent brownian motion

Endre Csáki, Miklós Csörgő, Antónia Földes, Pál Révész (2009)

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

Let ξ(k, n) be the local time of a simple symmetric random walk on the line. We give a strong approximation of the centered local time process ξ(k, n)−ξ(0, n) in terms of a brownian sheet and an independent Wiener process (brownian motion), time changed by an independent brownian local time. Some related results and consequences are also established.

Regular potentials of additive functionals in semidynamical systems

Nedra Belhaj Rhouma, Mounir Bezzarga (2004)

Commentationes Mathematicae Universitatis Carolinae

We consider a semidynamical system ( X , ℬ , Φ , w ) . We introduce the cone 𝔸 of continuous additive functionals defined on X and the cone 𝒫 of regular potentials. We define an order relation “ ≤ ” on 𝔸 and a specific order “ ≺ ” on 𝒫 . We will investigate the properties of 𝔸 and 𝒫 and we will establish the relationship between the two cones.

Régularité Besov-Orlicz du temps local Brownien

Yue Hu, Mohamed Mellouk (2000)

Studia Mathematica

Let ( B t , t ∈ [ 0 , 1 ] ) be a linear Brownian motion starting from 0 and denote by ( L t ( x ) , t ≥ 0 , x ∈ ℝ ) its local time. We prove that the spatial trajectories of the Brownian local time have the same Besov-Orlicz regularity as the Brownian motion itself (i.e. for all t>0, a.s. the function x → L t ( x ) belongs to the Besov-Orlicz space B M 2 , ∞ 1 / 2 with M 2 ( x ) = e | x | 2 - 1 ). Our result is optimal.

Régularité du temps local brownien dans les espaces de Besov-Orlicz

B. Boufoussi (1996)

Studia Mathematica

Let ( B t , t ≥ 0 ) be a linear Brownian motion and (L(t,x), t > 0, x ∈ ℝ) its local time. We prove that for all t > 0, the process (L(t,x), x ∈ [0,1]) belongs almost surely to the Besov-Orlicz space B M 1 , ∞ 1 / 2 with M 1 ( x ) = e | x | - 1 .

Revisiting the sample path of Brownian motion

S. James Taylor (2006)

Banach Center Publications

Brownian motion is the most studied of all stochastic processes; it is also the basis for stochastic analysis developed in the second half of the 20th century. The fine properties of the sample path of a Brownian motion have been carefully studied, starting with the fundamental work of Paul Lévy who also considered more general processes with independent increments and extended the Brownian motion results to this class. Lévy showed that a Brownian path in d (d ≥ 2) dimensions had zero Lebesgue measure;...

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