David Márquez-Carreras

On stochastic partial differential equations with spatially correlated noise: smoothness of the law

We deal with the following general kind of stochastic partial differential equations:with null initial conditions, L a second-order partial differential operator and F a Gaussian noise, white in time and correlated in space. Firstly, we prove that the solution u(t,x) possesses a smooth density pt,x for every . We use the tools of Malliavin Calculus. Secondly, we apply this general result to two particular cases: the d-dimensional spatial heat equation, d[greater-or-equal, slanted]1, and the wave equation, d[set membership, variant]{1,2}.

Small perturbations in a hyperbolic stochastic partial differential equation

We study the existence and properties of the density for the law of the solution to a nonlinear hyperbolic stochastic partial differential equation, driven by a two-parameter white noise. We also analyze the asymptotic behavior of the density for the law of the solution to the equation obtained by perturbing the noise. Under unrestricted Hormanders-type conditions on the coefficients, we establish Varadhans estimates.

Higher Order Expansions for the Overlap of the SK Model

In this note, the Sherrington-Kirkpatrick model of interacting spins is under consideration. In the high temperature region, we give an asymptotic expansion for the expected value of some genereral polynomial of the overlap of the system when the size N grows to infinity. Some of the coefficients obtained are shown to be vanishing, while the procedure to get the nontrivial ones has to be performed by a computer program, due to the great amount of computation involved.

Expansion of the density: a Wiener-chaos approach

decomposition FC y + n-,= ,(fn), e E (0, 1]. Using Malliavin calculus, a precise description of the coefficients in the development in terms of the multiple integrals In(fA) 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 e.

GENERALIZED FRACTIONAL KINETIC EQUATIONS: ANOTHER POINT OF VIEW

In this paper we deal with generalized fractional kinetic equations driven by a Gaussian noise, white in time and correlated in space, and where the diffusion operator is the composition of the Bessel and Riesz potentials for any fractional parameters. We give results on the existence and uniqueness of solutions by means of a weak formulation and study the Hölder continuity. Moreover, we prove the existence of a smooth density associated to the solution process and study the asymptotics of this density. Finally, when the diffusion coefficient is constant, we look for its Gaussian index.

Behaviour of the density in perturbed SPDE's with spatially correlated noise

Abstract Consider the following general type of perturbed stochastic partial differential equations: Lu e t,x =eα u e t,x F (t,x)+β u e t,x , (t,x)∈ R + × R d , e>0, with null initial conditions, L a second-order partial differential operator and F a Gaussian noise, white in time and correlated in space. In a previous work we proved the existence of smooth density p t,x e (y), t>0, x∈ R d , for the law of the solution of above-mentioned equation. In this paper we study the logarithmic estimates for this density, that means to establish the behaviour of 2 e 2 log p t , x e ( y ), as e ↓0. This kind of estimates is also called Varadhan–Leandre estimates.

Generalized Stochastic Heat Equations

In this article, we study some properties about the solution of generalized stochastic heat equations driven by a Gaussian noise, white in time and correlated in space, and where the diffusion operator is the inverse of a Riesz potential for any positive fractional parameter. We prove the existence and uniqueness of solution and the Holder continuity of this solution. In time, Holder’s parameter does not depend on the fractional parameter. However, in space, Holder’s parameter has a different behavior depending on the fractional parameter. Finally, we show that the law of the solution is absolutely continuous with respect to Lebesgue’s measure and its density is infinitely differentiable.

ON THE ASYMPTOTICS OF THE DENSITY IN PERTURBED SPDE'S WITH SPATIALLY CORRELATED NOISE

We consider a general type of perturbed stochastic partial differential equations: \[ {\mathcal L} u^\varepsilon_{t, x} =\varepsilon a u^\varepsilon_{t, x} \dot F (t, x) +b u^\varepsilon_{t, x}\,,\qquad (t, x)\in {\mathbb R}_+ \times {\mathbb R}^d\,,\qquad \varepsilon > 0\,, \] with null initial conditions,

On Exponential Moments for Functionals Defined on the Loop Group

{\mathcal L}

Varadhan–Léandre estimates for a family of random vectors

a second-order partial differential operator and