We establish Talagrand＇s T2-transportation inequalities for infinite dimensional dissipative diffusions with sharp constants, through Galerkin type＇s approximations and the known results in the finite dimensional ca...We establish Talagrand＇s T2-transportation inequalities for infinite dimensional dissipative diffusions with sharp constants, through Galerkin type＇s approximations and the known results in the finite dimensional case. Furthermore in the additive noise case we prove also logarithmic Sobolev inequalities with sharp constants. Applications to Reaction- Diffusion equations are provided.展开更多
This paper aims to analyze the weak approximation error of a fully discrete scheme for a class of semi-linear parabolic stochastic partial differential equations(SPDEs)driven by additive fractional Brownian motions wi...This paper aims to analyze the weak approximation error of a fully discrete scheme for a class of semi-linear parabolic stochastic partial differential equations(SPDEs)driven by additive fractional Brownian motions with the Hurst parameter H∈(1/2,1).The spatial approximation is performed by a spectral Galerkin method and the temporal discretization by an exponential Euler method.As far as we know,the weak error analysis for approximations of fractional noise driven SPDEs is absent in the literature.A key difficulty in the analysis is caused by the lack of the associated Kolmogorov equations.In the present work,a novel and efficient approach is presented to carry out the weak error analysis for the approximations,which does not rely on the associated Kolmogorov equations but relies on the Malliavin calculus.To the best of our knowledge,the rates of weak convergence,shown to be higher than the strong convergence rates,are revealed in the fractional noise driven SPDE setting for the first time.Numerical examples corroborate the claimed weak orders of convergence.展开更多
We establish a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises, where we suppose that the drfit term contains a gradient and satisfies certain non-Lipschitz condition....We establish a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises, where we suppose that the drfit term contains a gradient and satisfies certain non-Lipschitz condition. We prove the strong existence and uniqueness and joint Hölder continuity of the solution to the SPDEs.展开更多
We discuss stochastic functional partial differential equations and neutral partial differential equations of retarded type driven by fractional Brownian motion with Hurst parameter H 〉 1/2. Using the Girsanov transf...We discuss stochastic functional partial differential equations and neutral partial differential equations of retarded type driven by fractional Brownian motion with Hurst parameter H 〉 1/2. Using the Girsanov transformation argument, we establish the quadratic transportation inequalities for the law of the mild solution of those equations driven by fractional Brownian motion under the L2 metric and the uniform metric.展开更多
Shift Harnack inequality and integration by parts formula are established for semilinear stochastic partial differential equations and stochastic functional partial differential equations by modifying the coupling use...Shift Harnack inequality and integration by parts formula are established for semilinear stochastic partial differential equations and stochastic functional partial differential equations by modifying the coupling used by F. -Y. Wang [Ann. Probab., 2012, 42（3）： 994-1019]. Log-Harnack inequality is established for a class of stochastic evolution equations with non- Lipschitz coefficients which includes hyperdissipative Navier-Stokes/Burgers equations as examples. The integration by parts formula is extended to the path space of stochastic functional partial differential equations, then a Dirichlet form is defined and the log-Sobolev inequality is established.展开更多
In this paper,we study a discontinuous Galerkin numerical scheme for a class of elliptic stochastic partial differential equations(abbr.elliptic SPDEs)driven by space white noises with homogeneous Dirichlet boundary c...In this paper,we study a discontinuous Galerkin numerical scheme for a class of elliptic stochastic partial differential equations(abbr.elliptic SPDEs)driven by space white noises with homogeneous Dirichlet boundary conditions for two and three space dimensions.We also establish L2 error estimates for the scheme.In particular,a numerical test for d=2 is presented at the end of the article.展开更多
基金Project supported by the Yangtze Scholarship Program
文摘We establish Talagrand＇s T2-transportation inequalities for infinite dimensional dissipative diffusions with sharp constants, through Galerkin type＇s approximations and the known results in the finite dimensional case. Furthermore in the additive noise case we prove also logarithmic Sobolev inequalities with sharp constants. Applications to Reaction- Diffusion equations are provided.
基金supported by NSF of China(Grant Nos.11971488,12071488)by NSF of Hunan Province(Grant No.2020JJ2040)by the Fundamental Research Funds for the Central Universities of Central South University(Grant Nos.2017zzts318,2019zzts214).
文摘This paper aims to analyze the weak approximation error of a fully discrete scheme for a class of semi-linear parabolic stochastic partial differential equations(SPDEs)driven by additive fractional Brownian motions with the Hurst parameter H∈(1/2,1).The spatial approximation is performed by a spectral Galerkin method and the temporal discretization by an exponential Euler method.As far as we know,the weak error analysis for approximations of fractional noise driven SPDEs is absent in the literature.A key difficulty in the analysis is caused by the lack of the associated Kolmogorov equations.In the present work,a novel and efficient approach is presented to carry out the weak error analysis for the approximations,which does not rely on the associated Kolmogorov equations but relies on the Malliavin calculus.To the best of our knowledge,the rates of weak convergence,shown to be higher than the strong convergence rates,are revealed in the fractional noise driven SPDE setting for the first time.Numerical examples corroborate the claimed weak orders of convergence.
基金This work was supported in part by the National Natural Science Foundation of China(Grant Nos.11571190,11771218,11771018,12061004)the Natural Science Foundation of Ningxia(No.2020AAC03230)the Major Research Project for North Minzu University(No.ZDZX201902).
文摘We establish a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises, where we suppose that the drfit term contains a gradient and satisfies certain non-Lipschitz condition. We prove the strong existence and uniqueness and joint Hölder continuity of the solution to the SPDEs.
基金Acknowledgements The authors would like to thank the referees for helpful suggestions which allowed them to improve the presentation of this paper. This work was supported in part by the National Natural Science Foundation of China （Grant No. 11271093） and the Science Research Project of Hubei Provincial Department Of Education （No. Q20141306）.
文摘We discuss stochastic functional partial differential equations and neutral partial differential equations of retarded type driven by fractional Brownian motion with Hurst parameter H 〉 1/2. Using the Girsanov transformation argument, we establish the quadratic transportation inequalities for the law of the mild solution of those equations driven by fractional Brownian motion under the L2 metric and the uniform metric.
基金supported by the Key Laboratory of Random Complex Structures and Data Scienc, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 973 Project (2006CB8059000)Science Fund for Creative Research Groups (10721101)+1 种基金the National Science Foundation of China (10671197)the Science Foundation of Jiangsu Province (BK2006032, 06-A-038, 07-333)
文摘This paper discusses the ergodicity of a linear stochastic partial differential equation driven by Levy noise.
文摘Shift Harnack inequality and integration by parts formula are established for semilinear stochastic partial differential equations and stochastic functional partial differential equations by modifying the coupling used by F. -Y. Wang [Ann. Probab., 2012, 42（3）： 994-1019]. Log-Harnack inequality is established for a class of stochastic evolution equations with non- Lipschitz coefficients which includes hyperdissipative Navier-Stokes/Burgers equations as examples. The integration by parts formula is extended to the path space of stochastic functional partial differential equations, then a Dirichlet form is defined and the log-Sobolev inequality is established.
文摘In this paper,we study a discontinuous Galerkin numerical scheme for a class of elliptic stochastic partial differential equations(abbr.elliptic SPDEs)driven by space white noises with homogeneous Dirichlet boundary conditions for two and three space dimensions.We also establish L2 error estimates for the scheme.In particular,a numerical test for d=2 is presented at the end of the article.
