Algebraic convergence of diffusion processes on Rn with radial diffusion and drift coefficients

Abstract

We consider the diffusion process X_t on {ℝ}^n with radial diffusion and drift coefficients. We prove that once the one-dimensional diffusion |X_t| has algebraic L²-convergence, so does X_t. And some classical examples are discussed.

Keywords

Diffusion processes / algebraic convergence / classical coupling / coupling by reflection / spherically invariant

References

[1] Bobkov S G. Spectral gap and concentration for some spherically symmetric probability measures. In: Geometric Aspects of Functional Analysis. Lecture Notes in Math, Vol 1807. Berlin: Springer, 2003, 37-43

[2] Bobkov S G, Ledoux M. Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann Probab, 2009, 37: 403-427

[3] Bonnefont M, Joulin A, Ma Y T. Spectral gap for spherically symmetric log-concave probability measures, and beyond. Preprint, 2014

[4] Chen M F. From Markov Chains to Non-equilibrium Particle Systems. Singapore: World Scientific, 1992

[5] Chen M F. Analytic proof of dual variational formula for the first eigenvalue in dimension one. Sci Sin A, 1999, 42(8): 805-815

[6] Chen M F, Li S F. Coupling Methods for Multidimensional diffusion processes. Ann Probab, 1989, 17(1): 151-177

[7] Chen M F, Wang Y Z. Algebraic convergence of Markov chains. Ann Appl Probab, 2003, 13(2): 604-627

[8] Deuschel J D. Algebraic L²-decay of attractive critical processes on the lattice. Ann Probab, 1994, 22(1): 264-283

[9] Djellout H, Wu L M. Lipschitzian norm estimate of one-dimensional Poisson equations and applications. Ann Inst Henri Poincaré Probab Stat, 2011, 47(2): 450-465

[10] Ikeda N, Watanabe S. Stochastic Differential Equations and Diffusion Processes. Tokyo: Kodansha Ltd, 1981

[11] Liggett T M. L² Rates of convergence for attractive reversible nearest particle systems: the critical case. Ann Probab, 1991, 19(3): 935-959

[12] Lindvall T, Rogers L C G. Coupling of multidimensional diffusions by reflection. Ann Probab, 1986, 14: 860-872

[13] Rockner M, Wang F-Y. Weak Poincaré inequalities and L²-convergence rates of Markov semigroup. J Funct Analysis, 2011, 185(2): 564-603

[14] Wang Y Z. Algebraic convergence of diffusion processes on a real line. J Beijing Normal Univ, 2003, 39: 448-456

[15] Wang Y Z. Algebraic convergence of diffusion processes on ℝn. Acta Math Sinica (Chin Ser), 2004, 47(5): 1001-1012 (in Chinese)

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg