WEAK POINCARÉ INEQUALITIES FOR CONVERGENCE RATE OF DEGENERATE DIFFUSION PROCESSES By

For a contraction $C_0$-semigroup on a separable Hilbert space, the decay rate is estimated by using the weak Poincar\'e inequalities for the symmetric and anti-symmetric part of the generator. As applications, non-exponential convergence rate is characterized for a class of degenerate diffusion processes, so that the study of hypocoercivity is extended. Concrete examples are presented.

[1]  M. Röckner.  L p -analysis of finite and infinite dimensional diffusion operators , 1999 .

[2]  Axel Klar,et al.  Exponential Rate of Convergence to Equilibrium for a Model Describing Fiber Lay-Down Processes , 2012, 1201.2156.

[3]  R. Schilling,et al.  Functional inequalities and subordination: stability of Nash and Poincaré inequalities , 2011, 1105.3082.

[4]  A. Bendikov,et al.  Nash type inequalities for fractional powers of non-negative self-adjoint operators , 2004, math/0403174.

[5]  M. Röckner,et al.  Weak Poincaré Inequalities and L2-Convergence Rates of Markov Semigroups , 2001 .

[6]  Feng-Yu Wang,et al.  Derivative formula and applications for degenerate diffusion semigroups , 2011, 1107.0096.

[7]  C. Mouhot,et al.  Hypocoercivity for kinetic equations with linear relaxation terms , 2008, 0810.3493.

[8]  Elliptic regularity and essential self-adjointness of Dirichlet operators on $\mathbb {R}^n$ , 1997 .

[9]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[10]  R. Feldman.  Construction , 2004, SP-110: Hyperbolic Paraboloid Shells.

[11]  Wilhelm Stannat,et al.  The theory of generalized Dirichlet forms and its applications in analysis and stochastics , 1999 .

[12]  Renjun Duan,et al.  Hypocoercivity of linear degenerately dissipative kinetic equations , 2009, 0912.1733.

[13]  J. Goldstein.  Semigroups of Linear Operators and Applications , 1985 .

[14]  Feng-Yu Wang.  Hypercontractivity and applications for stochastic Hamiltonian systems , 2017 .

[15]  F. Nier,et al.  Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians , 2005 .

[16]  Fabrice Baudoin,et al.  Bakry-Emery meet Villani , 2013, 1308.4938.

[17]  M. Röckner,et al.  Markov processes associated with Lp-resolvents and applications to stochastic differential equations on Hilbert space , 2006 .

[18]  C. Mouhot,et al.  HYPOCOERCIVITY FOR LINEAR KINETIC EQUATIONS CONSERVING MASS , 2010, 1005.1495.

[19]  Martin Grothaus,et al.  Hypocoercivity for Kolmogorov backward evolution equations and applications , 2012, 1207.5447.

[20]  N. Wielens.  The essential self-adjointness of generalized Schrödinger operators , 1985 .

[21]  E. Davies,et al.  SEMIGROUPS OF LINEAR OPERATORS AND APPLICATIONS (Oxford Mathematical Monographs) , 1986 .

[22]  M. Grothaus,et al.  Construction, ergodicity and rate of convergence of N-particle Langevin dynamics with singular potentials , 2010 .

[23]  A. Guillin,et al.  Degenerate Fokker–Planck equations: Bismut formula, gradient estimate and Harnack inequality , 2011, 1103.2817.

[24]  N. Krylov.  Elliptic regularity and essential self-adjointness of Dirichlet operators on R , 1996 .

[25]  Gerald Trutnau.  Stochastic calculus of generalized Dirichlet forms and applications to stochastic differential equations in infinite dimensions , 2000 .

[26]  R. Schilling,et al.  Subgeometric rates of convergence for Markov processes under subordination , 2015, Advances in Applied Probability.