Nash inequality for diffusion processes associated with Dirichlet distributions

Abstract

For any N\ge \mathrm{2} and \alpha ={(\alpha }_1,\mathrm{...},{\alpha }_{N+1})\in {(0,\infty )}^{N+1}, let {\mu }_{\alpha }^{(N)} be the Dirichlet distribution with parameter α on the set {\mathrm{\Delta }}^{(N)}:={\{x\in [0,1]}^N:{\displaystyle {\sum }_{\mathrm{1}\le i\le N}}{x}_i\le \mathrm{1}\}. The multivariate Dirichlet diffusion is associated with the Dirichlet form

{\mathcal{E}}_{\alpha }^{(N)}(f,f):={\displaystyle \sum _{n=\mathrm{1}}^N}{\displaystyle {\int }_{{\mathrm{\Delta }}^{(N)}}}(\mathrm{1}-{\displaystyle \sum _{\mathrm{1}\le i\le N}}{x}_i)x_n{({\partial }_{n}f)}^{\mathrm{2}}(x){\mu }_{\alpha }^{(N)}(\mathrm{d}x)

with Domain \mathcal{D}({\mathcal{E}}_{\alpha }^{(N)}) being the closure of {C^1(\mathrm{\Delta }}^{(N)}). We prove the Nash inequality

{\mu }_{\alpha }^{(N)}(f^{\mathrm{2}})\le C{\mathcal{E}}_{\alpha }^{(N)}{(f,f)}^{p/(p+\mathrm{1})}{\mu }_{\alpha }^{(N)}{(|f|)}^{\mathrm{2}/(p+\mathrm{1})},\mathrm{\hspace{0.17em}}f\in \mathcal{D}({\mathcal{E}}_{\alpha }^{(N)}),{\mu }_{\alpha }^{(N)}(f)=0,

for some constant C>0 and p={({\alpha }_{N+\mathrm{1}}-\mathrm{1})}^{+}+{\displaystyle {\sum }_{i=\mathrm{1}}^N\mathrm{1}}\vee (\mathrm{2}{\alpha }_i), where the constant p is sharp when {\max }_{\mathrm{1}\le i\le N}\mathrm{\hspace{0.17em}}{\alpha }_i\le 1/2 and {\alpha }_{N+\mathrm{1}}\ge \mathrm{1}. This Nash inequality also holds for the corresponding Fleming-Viot process.

Keywords

Dirichlet distribution / Nash inequality / super Poincaré inequality / diffusion process

References

[1] Bakosi J, Ristorcelli J R. A stochastic diffusion process for the Dirichlet distribution. Int J Stoch Anal, 2013, Article ID: 842981 (7 pp)

[2] Bakry D, Gentil I, Ledoux M. Analysis and Geometry of Markov Diffusion Operators. Berlin: Springer, 2014

[3] Connor R J, Mosimann J E. Concepts of independence for proportions with a generalization of the Dirichlet distribution. J. Amer Statist Assoc, 1969, 64: 194–206

[4] Davies E B, Simon B. Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J Funct Anal, 1984, 59: 335–395

[5] Epstein C L, Mazzeo R. Wright-Fisher diffusion in one dimension. SIAM J Math Anal, 2010, 42: 568–608

[6] Feng S, Miclo L. Wang F-Y. Poincare inequality for Dirichlet distributions and infinite- dimensional generalizations. ALEA Lat Am J Probab Math Stat, 2017, 14: 361–380

[7] Feng S, Wang F-Y. A class of infinite-dimensional diffusion processes with connection to population genetics. J Appl Probab, 2007, 44: 938–949

[8] Feng S, Wang F-Y. Harnack inequality and applications for infinite-dimensional GEM processes. Potential Anal, 2016, 44: 137–153

[9] Gross L. Logarithmic Sobolev inequalities and contractivity properties of semigroups. In: Dell’Antonio G, Mosco U, eds. Dirichlet Forms. Lecture Notes in Math, Vol 1563. Berlin: Springer, 1993, 54–88

[10] Jacobsen M. Examples of multivariate diffusions: time-reversibility; a Cox-Ingersoll- Ross type process. Department of Theoretical Statistics, Univ of Copenhagen, 2001, Preprint

[11] Johnson N. L. An approximation to the multinomial distribution, some properties and applications. Biometrika, 1960, 47: 93–102

[12] Miclo L. About projections of logarithmic Sobolev inequalities. In: Azéma J, Émery M, Ledoux M, Yor M, eds. Séminaire de Probabilités XXXVI. Lecture Notes in Math, Vol 1801. Berlin: Springer, 2003, 201–221

[13] Miclo L. Sur l’inégalité de Sobolev logarithmique des opérateurs de Laguerre à petit paramètre. In: Azéma J, Émery M, Ledoux M, Yor M, eds. Séminaire de Probabilités XXXVI. Lecture Notes in Math, Vol 1801. Berlin: Springer, 2003, 222–229

[14] Mosimann J E. On the compound multinomial distribution, the multivariate- distribution, and correlations among proportions. Biometrika, 1962, 49: 65–82

[15] Shimakura N. Equations différentielles provenant de la génetique des populations. Tôhoka Math J, 1977, 29: 287–318

[16] Stannat W. On validity of the log-Sobolev inequality for symmetric Fleming-Viot operators. Ann Probab, 2000, 28: 667–684

[17] Wang F-Y. Functional inequalities for empty essential spectrum. J Funct Anal, 2000, 170: 219–245

[18] Wang F-Y. Functional inequalities, semigroup properties and spectrum estimates. Infin Dimens Anal Quantum Probab Relat Top, 2000, 3: 263–295

[19] Wang F-Y. Functional Inequalities, Markov Semigroups and Spectral Theory. Beijing: Science Press, 2005

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