Expansion of the density: a Wiener-chaos approach

We prove a Taylor expansion of the density pε(y) of a Wiener functional Fε with Wiener-chaos decomposition Fε=y+∑∞n=1εnIn(fn), ε∈(0,1]. Using Malliavin calculus, a precise description of the coefficients in the development in terms of the multiple integrals In(fn) is provided. This general result is applied to the study of the density in two examples of hyperbolic stochastic partial differential equations with linear coefficients, where the driving noise has been perturbed by a coefficient ε

Cristalls de spin

L'estudi dels cristalls de spin iniciat a partir dels anys setanta ha tingut darrerament grans avenços en termes matemàtics. En aquest article presentem, de manera divulgativa, els conceptes bàsics d'aquesta àrea, detallem el seu marc matemàtic, descrivim els models clàssics que considerem més importants i expliquem algunes de les tècniques que s'utilitzen per a estudiar-los.The study of spin glasses initiated from the seventies has had lately great advances in mathematical terms. In this paper we present in an informative way the basic concepts of this area, detailing its mathematical frame, describing the classical models that we consider more important and explaining some of the techniques that are used to study them

Existence and smoothness of the density of the solution to fractional stochastic integral Volterra equations

We consider stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter H>12. We first derive supremum norm estimates for the solution and its Malliavin derivative. We then show existence and smoothness of the density under suitable nondegeneracy conditions. This extends the results in Hu and Nualart [Differential equations driven by Hölder continuous functions of order greater than 1/2, Stoch. Anal. Appl. Abel Symp. 2 (2007), pp. 399-413] and Nualart and Saussereau [Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion, Stoch. Process. Appl. 119 (2009), pp. 391-409] where stochastic differential equations driven by fractional Brownian motion are considered. The proof uses a priori estimates for deterministic differential equations driven by a function in a suitable Sobolev space

Cristalls de spin

L'estudi dels cristalls de spin iniciat a partir dels anys setanta ha tingut darrerament grans avenços en termes matemàtics. En aquest article presentem, de manera divulgativa, els conceptes bàsics d'aquesta àrea, detallem el seu marc matemàtic, descrivim els models clàssics que considerem més importants i expliquem algunes de les tècniques que s'utilitzen per a estudiar-los

A model of continuous time polymer on the lattice

In this article, we try to give a rather complete picture of the behavior of the free energy for a model of directed polymer in a random environment, in which the polymer is a simple symmetric random walk on the lattice Z d, and the environment is a collection {W(t, x);t ≥ 0, x ∈ Z d} of i.i.d. Brownian motions