Title :

Convergence to Equilibrium in the Free Fokker-Planck Equation With a Double-Well Potential

Author(s) :

Donati-Martin, Catherine [Auteur]
Laboratoire de Mathématiques de Versailles [LMV]
Groux, Benjamin [Auteur]
Laboratoire de Mathématiques de Versailles [LMV]
Maida, Mylene [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Keyword(s) :

Equation des milieux granulaires
Equation de Fokker-Planck
Comportement en temps long
Potentiel à double puits
Probabilités libres
Mesure d'équilibre
Matrices aléatoires

English keyword(s) :

Random matrices
Fokker-Planck equation
Granular media equation
Long-time behaviour
Double-well potential
Free probability
Equilibrium measure

HAL domain(s) :

Mathématiques [math]/Probabilités [math.PR]
Mathématiques [math]/Equations aux dérivées partielles [math.AP]

English abstract : [en]

We consider the one-dimensional free Fokker-Planck equation $\frac{\partial \mu_t}{\partial t} = \frac{\partial}{\partial x} \left[ \mu_t \left( \frac12 V' - H\mu_t \right) \right]$, where $H$ denotes the Hilbert transform ...
Show more >We consider the one-dimensional free Fokker-Planck equation $\frac{\partial \mu_t}{\partial t} = \frac{\partial}{\partial x} \left[ \mu_t \left( \frac12 V' - H\mu_t \right) \right]$, where $H$ denotes the Hilbert transform and $V$ is a particular double-well quartic potential, namely $V(x) = \frac14 x^4 + \frac{c}{2} x^2$, with $-2 \le c < 0$. We prove that the solution $(\mu_t)_{t \ge 0}$ of this PDE converges to the equilibrium measure $\mu_V$ as $t$ goes to infinity, which provides a first result of convergence in a non-convex setting. The proof involves free probability and complex analysis techniques.Show less >

Language :

Anglais

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL