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On the long time behavior of the TCP window size process
 Stochastic Processes and their Applications 120 (2010
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An introduction to stochastic PDEs
 Lecture notes, 2009. URL http://arxiv.org/abs/0907.4178
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2010: A simple framework to justify linear response theory
 Nonlinearity
"... The use of linear response theory for forced dissipative stochastic dynamical systems through the fluctuation dissipation theorem is an attractive way to study climate change systematically among other applications. Here, a mathematically rigorous justification of linear response theory for forced d ..."
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The use of linear response theory for forced dissipative stochastic dynamical systems through the fluctuation dissipation theorem is an attractive way to study climate change systematically among other applications. Here, a mathematically rigorous justification of linear response theory for forced dissipative stochastic dynamical systems is developed. The main results are formulated in an abstract setting and apply to suitable systems, in finite and infinite dimensions, that are of interest in climate change science and other applications. 1
Geometric ergodicity of a beadspring pair with stochastic Stokes forcing
, 2009
"... We consider a simple model for the fluctuating hydrodynamics of a flexible polymer in dilute solution, demonstrating geometric ergodicity for a pair of particles that interact with each other through a nonlinear spring potential while being advected by a stochastic Stokes fluid velocity field. This ..."
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We consider a simple model for the fluctuating hydrodynamics of a flexible polymer in dilute solution, demonstrating geometric ergodicity for a pair of particles that interact with each other through a nonlinear spring potential while being advected by a stochastic Stokes fluid velocity field. This is a generalization of previous models which have used linear spring forces as well as whiteintime fluid velocity fields. We follow previous work combining control theoretic arguments, Lyapunov functions, and hypoelliptic diffusion theory to prove exponential convergence via a Harris chain argument. To this, we add the possibility of excluding certain “bad ” sets in phase space in which the assumptions are violated but from which the systems leaves with a controllable probability. This allows for the treatment of singular drifts, such as those derived from the LennardJones potential, which is an novel feature of this work. 1
How hot can a heat bath get?
, 2008
"... We study a model of two interacting Hamiltonian particles subject to a common potential in contact with two Langevin heat reservoirs: one at finite and one at infinite temperature. This is a toy model for ‘extreme ’ nonequilibrium statistical mechanics. We provide a full picture of the longtime be ..."
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We study a model of two interacting Hamiltonian particles subject to a common potential in contact with two Langevin heat reservoirs: one at finite and one at infinite temperature. This is a toy model for ‘extreme ’ nonequilibrium statistical mechanics. We provide a full picture of the longtime behaviour of such a system, including the existence / nonexistence of a nonequilibrium steady state, the precise tail behaviour of the energy in such a state, as well as the speed of convergence toward the steady state. Despite its apparent simplicity, this model exhibits a surprisingly rich variety of long time behaviours, depending on the parameter regime: if the surrounding potential is ‘too stiff’, then no stationary state can exist. In the softer regimes, the tails of the energy in the stationary state can be either algebraic, fractional exponential, or exponential. Correspondingly, the speed of convergence to the stationary state can be either algebraic, stretched exponential, or exponential. Regarding both types of claims, we obtain matching upper and lower bounds.
Propagating Lyapunov functions to prove noise–induced stabilization
"... We investigate an example of noiseinduced stabilization in the plane that was also considered in (Gawedzki, Herzog, Wehr 2010) and (Birrell, Herzog, Wehr 2011). We show that despite the deterministic system not being globally stable, the addition of additive noise in the vertical direction leads to ..."
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We investigate an example of noiseinduced stabilization in the plane that was also considered in (Gawedzki, Herzog, Wehr 2010) and (Birrell, Herzog, Wehr 2011). We show that despite the deterministic system not being globally stable, the addition of additive noise in the vertical direction leads to a unique invariant probability measure to which the system converges at a uniform, exponential rate. These facts are established primarily through the construction of a Lyapunov function which we generate as the solution to a sequence of Poisson equations. Unlike a number of other works, however, our Lyapunov function is constructed in a systematic way, and we present a metaalgorithm we hope will be applicable to other problems. We conclude by proving positivity properties of the transition density by using Malliavin calculus via some unusually explicit calculations.
unknown title
, 2009
"... Asymptotic coupling and a weak form of Harris ’ theorem with applications to stochastic delay equations ..."
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Asymptotic coupling and a weak form of Harris ’ theorem with applications to stochastic delay equations
unknown title
, 2009
"... Asymptotic coupling and a weak form of Harris ’ theorem with applications to stochastic delay equations ..."
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Asymptotic coupling and a weak form of Harris ’ theorem with applications to stochastic delay equations
THÈSE Pour l’obtention du grade de DOCTEUR DE L’UNIVERSITÉ PARISEST ÉCOLE DOCTORALE MATHEMATIQUES ET SCIENCES ET TECHNOLOGIE DE L’INFORMATION ET DE LA COMMUNICATION DISCIPLINE: MATHÉMATIQUES
, 2013
"... Comportement asymptotique de processus avec sauts et applications pour des modèles avec branchement Directeur de thèse: Djalil CHAFAÏ Soutenue le vendredi 14 juin 2013 Devant le jury composé de ..."
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Comportement asymptotique de processus avec sauts et applications pour des modèles avec branchement Directeur de thèse: Djalil CHAFAÏ Soutenue le vendredi 14 juin 2013 Devant le jury composé de