This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition.Motivated by the works of Thomann ...This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition.Motivated by the works of Thomann and Tzvetkov(Nonlinearity,23(2010),2771–2791),Tzvetkov(Probab.Theory Relat.Fields,146(2010),4679–4714),Burq,Thomann and Tzvetkov(Ann.Fac.Sci.Toulouse Math.,27(2018),527–597),the authors first construct the Borel measure of Gibbs type in the Sobolev spaces with lower regularity,and then establish the existence of global solution to the equation with the helps of Prokhorov compactness theorem,Skorokhod convergence theorem and Gibbs measure.展开更多
In this article, we consider the continuous gas in a bounded domain ∧ of R^+ or R^d described by a Gibbsian probability measure μη∧ associated with a pair interaction φ, the inverse temperature β, the activity...In this article, we consider the continuous gas in a bounded domain ∧ of R^+ or R^d described by a Gibbsian probability measure μη∧ associated with a pair interaction φ, the inverse temperature β, the activity z 〉 0, and the boundary condition η. Define F ∫ωf(s)wA(ds). Applying the generalized Ito's formula for forward-backward martingales (see Klein et M. [5]), we obtain convex concentration inequalities for F with respect to the Gibbs measure μη∧. On the other hand, by FKG inequality on the Poisson space, we also give a new simple argument for the stochastic domination for the Gibbs measure.展开更多
We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials.We study the existence and uniqueness and the compatibility of geometric Gibbs measures associ...We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials.We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials.We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures.We prove that with this complex Banach manifold structure,the space is complete and,moreover,is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures.There is a maximum metric on the space,which is incomplete.We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same.We prove that a geometric Gibbs measure is an equilibrium state,and the in mum of the metric entropy function on the space is zero.展开更多
The characteristics of cookie-cutter sets in R^d are investigated. A Bowen's formula for the Hausdorff dimension of a cookie-cutter set in terms of the pressure function is derived. The existence of self-similar m...The characteristics of cookie-cutter sets in R^d are investigated. A Bowen's formula for the Hausdorff dimension of a cookie-cutter set in terms of the pressure function is derived. The existence of self-similar measures, conformal measures and Gibbs measures on cookie-cutter sets is proved. The dimension spectrum of each of these measures is analyzed. In addition, the locally uniformly a-dimensional condition and the fractal Plancherel Theorem for these measures are shown. Finally, the existence of order-two density for the Hausdorff measure of a cookie-cutter set is proved.展开更多
基金supported by the National Natural Science Foundation of China(Nos.11901302,11401180)the Natural Science Foundation from Jiangsu province BK20171029the Academic Discipline Project of Shanghai Dianji University(No.16JCXK02)。
文摘This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition.Motivated by the works of Thomann and Tzvetkov(Nonlinearity,23(2010),2771–2791),Tzvetkov(Probab.Theory Relat.Fields,146(2010),4679–4714),Burq,Thomann and Tzvetkov(Ann.Fac.Sci.Toulouse Math.,27(2018),527–597),the authors first construct the Borel measure of Gibbs type in the Sobolev spaces with lower regularity,and then establish the existence of global solution to the equation with the helps of Prokhorov compactness theorem,Skorokhod convergence theorem and Gibbs measure.
文摘In this article, we consider the continuous gas in a bounded domain ∧ of R^+ or R^d described by a Gibbsian probability measure μη∧ associated with a pair interaction φ, the inverse temperature β, the activity z 〉 0, and the boundary condition η. Define F ∫ωf(s)wA(ds). Applying the generalized Ito's formula for forward-backward martingales (see Klein et M. [5]), we obtain convex concentration inequalities for F with respect to the Gibbs measure μη∧. On the other hand, by FKG inequality on the Poisson space, we also give a new simple argument for the stochastic domination for the Gibbs measure.
基金This work was supported by National Science Foundation of USA(Grant No.DMS-1747905)the Simons Foundation(Grant No.523341)+1 种基金Professional Sta Congress of the City University of New York Enhanced Award(Grant No.62777-0050)National Natural Science Foundation of China(Grant No.11571122).
文摘We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials.We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials.We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures.We prove that with this complex Banach manifold structure,the space is complete and,moreover,is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures.There is a maximum metric on the space,which is incomplete.We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same.We prove that a geometric Gibbs measure is an equilibrium state,and the in mum of the metric entropy function on the space is zero.
基金supported by the National Natural Science Foundation of China.
文摘The characteristics of cookie-cutter sets in R^d are investigated. A Bowen's formula for the Hausdorff dimension of a cookie-cutter set in terms of the pressure function is derived. The existence of self-similar measures, conformal measures and Gibbs measures on cookie-cutter sets is proved. The dimension spectrum of each of these measures is analyzed. In addition, the locally uniformly a-dimensional condition and the fractal Plancherel Theorem for these measures are shown. Finally, the existence of order-two density for the Hausdorff measure of a cookie-cutter set is proved.