Probabilistic approach for granular media equations in the non-uniformly convex case |
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| Authors: | P Cattiaux A Guillin F Malrieu |
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| Institution: | (1) école Polytechnique, CMAP, Route de Saclay, 91128 Palaiseau Cedex, France;(2) Université Paris X Nanterre, Equipe MODAL’X, UFR SEGMI, 200 avenue de la République, 92001 Nanterre Cedex, France;(3) CEREMADE, UMR CNRS 7534, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France;(4) IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France |
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| Abstract: | We use here a particle system to prove both a convergence result (with convergence rate) and a deviation inequality for solutions
of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. The
proof of convergence is simpler than the one in Carrillo–McCann–Villani (Rev. Mat. Iberoamericana 19:971–1018, 2003; Arch.
Rat. Mech. Anal. 179:217–263, 2006). All the results complete former results of Malrieu (Ann. Appl. Probab. 13:540–560, 2003)
in the uniformly convex case. The main tool is an uniform propagation of chaos property and a direct control in Wasserstein
distance of solutions starting with different initial measures. The deviation inequality is obtained via a T
1 transportation cost inequality replacing the logarithmic Sobolev inequality which is no more clearly dimension free.
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| Keywords: | Granular media equation Transportation cost inequality Logarithmic Sobolev Inequalities Concentration inequalities |
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