Results 1  10
of
171
Ergodicity of the 2D NavierStokes equations with degenerate forcing, preprint
"... The stochastic 2D NavierStokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization ..."
Abstract

Cited by 48 (12 self)
 Add to MetaCart
The stochastic 2D NavierStokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in L 2 0(T 2). Unlike in previous works, this class is independent of the viscosity and the strength of the noise. The two main tools of our analysis are the asymptotic strong Feller property, introduced in this work, and an approximate integration by parts formula. The first, when combined with a weak type of irreducibility, is shown to ensure that the dynamics is ergodic. The second is used to show that the first holds under a Hörmandertype condition. This requires some interesting nonadapted stochastic analysis. 1
Nonequilibrium statistical mechanics of strongly anharmonic chains of oscillators
 Comm. Math. Phys
"... We study the model of a strongly nonlinear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, ReyBellet [EPR99a, EPR99b] to potentials with ..."
Abstract

Cited by 44 (11 self)
 Add to MetaCart
We study the model of a strongly nonlinear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, ReyBellet [EPR99a, EPR99b] to potentials with essentially arbitrary growth at infinity. This extension is possible by introducing a stronger version of Hörmander’s theorem for Kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains.
Isotropic hypoellipticity and trend to the equilibrium for the FokkerPlanck equation with high degree potential
, 2002
"... ..."
Uniqueness of the Invariant Measure for a Stochastic PDE Driven by Degenerate Noise
, 2001
"... We consider the stochastic GinzburgLandau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The lowlying frequencies are then only connected to this forcing through the nonlinear (cubic) term of the GinzburgLandau equation. Under these assumpti ..."
Abstract

Cited by 40 (11 self)
 Add to MetaCart
We consider the stochastic GinzburgLandau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The lowlying frequencies are then only connected to this forcing through the nonlinear (cubic) term of the GinzburgLandau equation. Under these assumptions, we show that the stochastic PDE has a unique invariant measure. The techniques of proof combine a controllability argument for the lowlying frequencies with an infinite dimensional version of the Malliavin calculus to show positivity and regularity of the invariant measure. This then implies the uniqueness of that measure. Contents 1 Introduction 2 2 Some Preliminaries on the Dynamics 5 3 Controllability 6 4 Strong Feller Property and Proof of Theorem 1.1 9 5 Regularity of the Cutoff Process 11 5.1 Splitting and Interpolation Spaces . . . . . . . . . . . . . . . . . . . 12 5.2 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.3 Smoothing Properties of the...
Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance
 NONLINEAR PROBLEMS IN MATHEMATICAL PHYSICS AND RELATED TOPICS VOL. II IN HONOR OF PROFESSOR O.A. LADYZHENSKAYA”. INTERNATIONAL MATHEMATICAL SERIES
, 2002
"... This paper contains a survey on a series of papers by the authors, dealing with linear and non linear Kolmogorovtype operators, arising in diffusion theory, probability and finance. Some new results, about existence for Cauchy problems, regularity properties and pointwise estimates of solutions, ar ..."
Abstract

Cited by 18 (14 self)
 Add to MetaCart
This paper contains a survey on a series of papers by the authors, dealing with linear and non linear Kolmogorovtype operators, arising in diffusion theory, probability and finance. Some new results, about existence for Cauchy problems, regularity properties and pointwise estimates of solutions, are also announced.
Smoothing effect for the nonlinear Vlasov–Poisson–Fokker–Planck equation
 J. Differential Equations
, 1995
"... We study a smoothing effect of a nonlinear degenerate parabolic problem, the VlasovPoissonFokkerPlanck system, in three dimensions. We prove that a solution gives rise actually to very smooth macroscopic density and force field for positive time. This is obtained by analyzing the effect of the F ..."
Abstract

Cited by 18 (1 self)
 Add to MetaCart
We study a smoothing effect of a nonlinear degenerate parabolic problem, the VlasovPoissonFokkerPlanck system, in three dimensions. We prove that a solution gives rise actually to very smooth macroscopic density and force field for positive time. This is obtained by analyzing the effect of the FokkerPlanck kernel on the force term in the Vlasov equation and using classical convolution inequalities. Résumé Nous étudions l’effet régularisant dans un problème parabolique dégénéré nonlinéaire, le système de VlasovPoissonFokkerPlanck, en trois dimensions. Nous montrons qu’une solution faible conduit en fait à une densité macroscopique et un champ de force très réguliers en temps positif. Ceci est obtenu en analysant l’effet du noyau de FokkerPlanck sur le terme de force dans l’équation de Vlasov et en utilisant des inégalités de convolution. Keywords FokkerPlanck operator – VlasovPoisson equation – Convolution inequalities – Weighted estimates – Regularizing estimates – Velocity averages
Hypoelliptic Regularity in Kinetic Equations
, 2003
"... We establish new regularity estimates, in terms of Sobolev spaces, of the solution f to a kinetic equation. The righthand side can contain partial derivatives in time, space and velocity, as in classical averaging, and f is assumed to have a certain amount of regularity in velocity. The result ..."
Abstract

Cited by 17 (1 self)
 Add to MetaCart
We establish new regularity estimates, in terms of Sobolev spaces, of the solution f to a kinetic equation. The righthand side can contain partial derivatives in time, space and velocity, as in classical averaging, and f is assumed to have a certain amount of regularity in velocity. The result is that f is also regular in time and space, and this is related to a commutator identity introduced by Hormander for hypoelliptic operators.
On the complete model with stochastic volatility by Hobson and Rogers
, 2004
"... In the complete model with stochastic volatility by Hobson and Rogers, preference independent options prices are solutions to degenerate PDEs obtained by including additional state variables describing the dependence on past prices of the underlying. In this note, we aim to emphasize the mathemat ..."
Abstract

Cited by 16 (13 self)
 Add to MetaCart
In the complete model with stochastic volatility by Hobson and Rogers, preference independent options prices are solutions to degenerate PDEs obtained by including additional state variables describing the dependence on past prices of the underlying. In this note, we aim to emphasize the mathematical tractability of the model by presenting analytical and numerical results comparable with the known ones in the classical BlackScholes environment.
Regularity Properties of Viscosity Solutions of a NonHörmander Degenerate Equation
 J. Math. Pures Appl
, 2001
"... We study the interior regularity properties of the solutions of a nonlinear degenerate equation arising in mathematical finance. We set the problem in the framework of Hrmander type operators without assuming any hypothesis on the degeneracy of the associated Lie algebra. We prove that the viscosity ..."
Abstract

Cited by 16 (11 self)
 Add to MetaCart
We study the interior regularity properties of the solutions of a nonlinear degenerate equation arising in mathematical finance. We set the problem in the framework of Hrmander type operators without assuming any hypothesis on the degeneracy of the associated Lie algebra. We prove that the viscosity solutions are indeed classical solutions. 2001 ditions scientifiques et mdicales Elsevier SAS Keywords: Nonlinear degenerate Kolmogorov equation, Interior regularity, Hrmander operators RSUM.  Nous tudions la rgularit intrieure des solutions de viscosit d'une quation non linaire du second ordre dgnre que l'on rencontre en finance mathmatique. Nous tudions le problme par la thorie des oprateurs de Hrmander sans aucune hypothse sur la dgnerescence de l'algbre de Lie engendre. Nous montrons que la solution de viscosit est une solution classique. 2001 ditions scientifiques et mdicales Elsevier SAS 1.
On the Regularity of Solutions to a Nonlinear Ultraparabolic Equation Arising in Mathematical Finance
 in mathematical finance, Differential Integral Equations 14 (6
, 2001
"... We consider the following nonlinear degenerate parabolic equation which arises in some recent problems of mathematical finance:... ..."
Abstract

Cited by 16 (12 self)
 Add to MetaCart
We consider the following nonlinear degenerate parabolic equation which arises in some recent problems of mathematical finance:...