For any N≥2 andα=(α1…,αN+1)∈(0,∞)^N+1,letμa^(N)be the Dirichlet distribution with parameterαon the set△(N):={x^μa∈[0,1]^N:∑1≤i≤N^xi≤1}.The multivariate Dirichlct diffusion is associated with the Dirich...For any N≥2 andα=(α1…,αN+1)∈(0,∞)^N+1,letμa^(N)be the Dirichlet distribution with parameterαon the set△(N):={x^μa∈[0,1]^N:∑1≤i≤N^xi≤1}.The multivariate Dirichlct diffusion is associated with the Dirichlet formεa^(N)(f,f):=∑n=i^N∫△(N)(1-∑1≤i≤N^xi)xn(Эnf)^2(x)μα^(N)(dx)with Domain D(εa^(N))being the closure of C^1(△^(N)).We prove the Nash inequalityμa^(N)(f^2)≤Cεa^(N)(f,f)^p/(p+1)μa^(N)(|f|)^2/(p+1),f∈D(εa^(N)),μa^(N)(f)=0 for some constant C>0 and p=(aN+1-1)++∑i^N=11∨(2ai),where the constant p is sharp when max1≤i≤N ai≤1/2 and aN+1≥1.This Nash inequality also holds for the corresponding Fleming-Viot process.展开更多
This paper deals with the Nash inequalities and the related ones for general symmetric forms which can be very much unbounded. Some sufficient conditions in terms of the isoperimetric inequalities and some necessary c...This paper deals with the Nash inequalities and the related ones for general symmetric forms which can be very much unbounded. Some sufficient conditions in terms of the isoperimetric inequalities and some necessary conditions for the inequalities are presented. The resulting conditions can be sharp qualitatively as illustrated by some examples. It turns out that for a probability measure, the Nash inequalities are much stronger than the Poincare and the logarithmic Sobolev inequalities in the present context.展开更多
In this paper. we give characterizations of Nash inequalities for birth-death process and diffusion process on the line. As a by-product. we prove that for these processes. transience implies that the semigroups P(t) ...In this paper. we give characterizations of Nash inequalities for birth-death process and diffusion process on the line. As a by-product. we prove that for these processes. transience implies that the semigroups P(t) decay as ‖P(t)‖_(1--x)≤Ct^(-1). Sufficient conditions for general Msrkov chains are also obtained.展开更多
In author's one previous paper, the same topic was studied for onedimensional diffusions. As a continuation, this paper studies the discrete case, that is thebirth-death processes. The explicit criteria for the in...In author's one previous paper, the same topic was studied for onedimensional diffusions. As a continuation, this paper studies the discrete case, that is thebirth-death processes. The explicit criteria for the inequalities, the variational formulas andexplicit bounds of the corresponding constants in the inequalities are presented. As typicalapplications, the Nash inequalities and logarithmic Sobolev inequalities are examined.展开更多
Criteria for the super-Poincare inequality and the weak-Pincare inequality about ergodic birth-death processes are presented. Our work further completes ten criteria for birth-death processes presented in Table 1.4 ...Criteria for the super-Poincare inequality and the weak-Pincare inequality about ergodic birth-death processes are presented. Our work further completes ten criteria for birth-death processes presented in Table 1.4 （p. 15） of Prof. Mu-Fa Chen＇s book ＂Eigenvalues, Inequalities and Ergodic展开更多
This paper studies the Hardy-type inequalities on the intervals （may be infinite） with two weights, either vaaishing at two endpoiats of the interval or having mean zero. For the first type of inequalities, in terms...This paper studies the Hardy-type inequalities on the intervals （may be infinite） with two weights, either vaaishing at two endpoiats of the interval or having mean zero. For the first type of inequalities, in terms of new isoperimetric constants, the factor of upper and lower bounds becomes smaller than the known ones. The second type of the inequalities is motivated from probability theory and is new ia the analytic context. The proofs are now rather elementary. Similar improvements are made for Nash inequality, Sobolev-type inequality, and the logarithmic Sobolev inequality on the intervals.展开更多
基金The authors would like to thank the referees for helpful comments on an earlier version of the paper.This work was supported in part by the National Natural Science Foundation of China(Grant Nos.11771326,11726627,11831014).
文摘For any N≥2 andα=(α1…,αN+1)∈(0,∞)^N+1,letμa^(N)be the Dirichlet distribution with parameterαon the set△(N):={x^μa∈[0,1]^N:∑1≤i≤N^xi≤1}.The multivariate Dirichlct diffusion is associated with the Dirichlet formεa^(N)(f,f):=∑n=i^N∫△(N)(1-∑1≤i≤N^xi)xn(Эnf)^2(x)μα^(N)(dx)with Domain D(εa^(N))being the closure of C^1(△^(N)).We prove the Nash inequalityμa^(N)(f^2)≤Cεa^(N)(f,f)^p/(p+1)μa^(N)(|f|)^2/(p+1),f∈D(εa^(N)),μa^(N)(f)=0 for some constant C>0 and p=(aN+1-1)++∑i^N=11∨(2ai),where the constant p is sharp when max1≤i≤N ai≤1/2 and aN+1≥1.This Nash inequality also holds for the corresponding Fleming-Viot process.
基金Research supported in part by NSFC (No. 19631060), Math. Tian Yuan Found., Qiu Shi Sci. & Tech. Found., RFDP and MCME
文摘This paper deals with the Nash inequalities and the related ones for general symmetric forms which can be very much unbounded. Some sufficient conditions in terms of the isoperimetric inequalities and some necessary conditions for the inequalities are presented. The resulting conditions can be sharp qualitatively as illustrated by some examples. It turns out that for a probability measure, the Nash inequalities are much stronger than the Poincare and the logarithmic Sobolev inequalities in the present context.
基金Research supported in part by RFDP (No 96002704)NSFC (No 19771008) Fok Ying-Tung Youth Foundation
文摘In this paper. we give characterizations of Nash inequalities for birth-death process and diffusion process on the line. As a by-product. we prove that for these processes. transience implies that the semigroups P(t) decay as ‖P(t)‖_(1--x)≤Ct^(-1). Sufficient conditions for general Msrkov chains are also obtained.
文摘In author's one previous paper, the same topic was studied for onedimensional diffusions. As a continuation, this paper studies the discrete case, that is thebirth-death processes. The explicit criteria for the inequalities, the variational formulas andexplicit bounds of the corresponding constants in the inequalities are presented. As typicalapplications, the Nash inequalities and logarithmic Sobolev inequalities are examined.
基金Supported by Natural Science Foundation of Fujian Province(Grant No.2010J05002)
文摘Criteria for the super-Poincare inequality and the weak-Pincare inequality about ergodic birth-death processes are presented. Our work further completes ten criteria for birth-death processes presented in Table 1.4 （p. 15） of Prof. Mu-Fa Chen＇s book ＂Eigenvalues, Inequalities and Ergodic
基金Supported by NSFC(GrantNo.11131003)the"985"project from the Ministry of Education in China
文摘This paper studies the Hardy-type inequalities on the intervals （may be infinite） with two weights, either vaaishing at two endpoiats of the interval or having mean zero. For the first type of inequalities, in terms of new isoperimetric constants, the factor of upper and lower bounds becomes smaller than the known ones. The second type of the inequalities is motivated from probability theory and is new ia the analytic context. The proofs are now rather elementary. Similar improvements are made for Nash inequality, Sobolev-type inequality, and the logarithmic Sobolev inequality on the intervals.
