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Frontiers of Mathematics in China

ISSN 1673-3452

ISSN 1673-3576(Online)

CN 11-5739/O1

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Front. Math. China    2019, Vol. 14 Issue (6) : 1317-1338    https://doi.org/10.1007/s11464-019-0807-3
RESEARCH ARTICLE
Nash inequality for diffusion processes associated with Dirichlet distributions
Feng-Yu WANG1,2(), Weiwei ZHANG3
1. Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
2. Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, UK
3. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
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Abstract

For any N 2 and α=(α1,...,αN+1)(0,)N+1, let μα(N) be the Dirichlet distribution with parameter α on the set Δ(N):={x[0,1]N:1iNxi1}. The multivariate Dirichlet diffusion is associated with the Dirichlet form

Eα(N)(f,f):=n=1NΔ(N)(11iNxi)xn(nf)2(x)μα(N)(dx)

with Domain D(Eα(N)) being the closure of C1(Δ(N)). We prove the Nash inequality

μα(N)(f2)CEα(N)(f,f)p/(p+1)μα(N)(|f|)2/(p+1),fD(Eα(N)),μα(N)(f)=0,

for some constant C>0 and p=(αN+11)++i=1N1(2αi), where the constant p is sharp when max1iNαi1/2 and αN+11. This Nash inequality also holds for the corresponding Fleming-Viot process.

Keywords Dirichlet distribution      Nash inequality      super Poincaré inequality      diffusion process     
Corresponding Author(s): Feng-Yu WANG   
Issue Date: 07 January 2020
 Cite this article:   
Feng-Yu WANG,Weiwei ZHANG. Nash inequality for diffusion processes associated with Dirichlet distributions[J]. Front. Math. China, 2019, 14(6): 1317-1338.
 URL:  
https://academic.hep.com.cn/fmc/EN/10.1007/s11464-019-0807-3
https://academic.hep.com.cn/fmc/EN/Y2019/V14/I6/1317
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