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Invariance principles for random walks conditioned to stay positive

Francesco CaravennaLoïc Chaumont — 2008

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

Let { be a random walk in the domain of attraction of a stable law 𝒴 , i.e. there exists a sequence of positive real numbers ( ) such that / converges in law to 𝒴 . Our main result is that the rescaled process ( / , ≥0), when conditioned to stay positive, converges in law (in the functional sense) towards the corresponding stable Lévy process conditioned to stay positive. Under some additional assumptions,...

Large scale behavior of semiflexible heteropolymers

Francesco CaravennaGiambattista GiacominMassimiliano Gubinelli — 2010

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

We consider a general discrete model for heterogeneous semiflexible polymer chains. Both the thermal noise and the inhomogeneous character of the chain (the ) are modeled in terms of random rotations. We focus on the regime, i.e., the analysis is performed for a given realization of the disorder. Semiflexible models differ substantially from random walks on short scales, but on large scales a brownian behavior emerges. By exploiting techniques from tensor analysis and non-commutative Fourier analysis,...

The discrete-time parabolic Anderson model with heavy-tailed potential

Francesco CaravennaPhilippe CarmonaNicolas Pétrélis — 2012

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

We consider a discrete-time version of the parabolic Anderson model. This may be described as a model for a directed ( 1 + d ) -dimensional polymer interacting with a random potential, which is constant in the deterministic direction and i.i.d. in the d orthogonal directions. The potential at each site is a positive random variable with a polynomial tail at infinity. We show that, as the size of the system diverges, the polymer extremity is localized almost surely at one single point which grows ballistically....

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