In this paper,we investigate the strong Feller property of stochastic differential equations(SDEs)with super-linear drift and Hölder diffusion coefficients.By utilizing the Girsanov theorem,coupling method,trunca...In this paper,we investigate the strong Feller property of stochastic differential equations(SDEs)with super-linear drift and Hölder diffusion coefficients.By utilizing the Girsanov theorem,coupling method,truncation method and the Yamada-Watanabe approximation technique,we derived the strong Feller property of the solution.展开更多
We obtain the H?lder continuity and joint H?lder continuity in space and time for the random field solution to the parabolic Anderson equation ■ in d-dimensional space, where ■ is a mean zero Gaussian noise with tem...We obtain the H?lder continuity and joint H?lder continuity in space and time for the random field solution to the parabolic Anderson equation ■ in d-dimensional space, where ■ is a mean zero Gaussian noise with temporal covariance γ0 and spatial covariance given by a spectral density μ(ξ). We assume that ■ and ■ , where αi, i = 1, · · ·, d(or α) can take negative value.展开更多
The Hǒlder continuity is proved for the gradient of the solution Jo the one-sided obstacle problem of the following variational inequality in the case 1<p<2
In [1] we construct a unique bounded H■lder continuous viscosity solution for the nonlinear PDEs with the evolution p-Laplacian equation and its anisotropic version as typical examples. In this part, we investigate t...In [1] we construct a unique bounded H■lder continuous viscosity solution for the nonlinear PDEs with the evolution p-Laplacian equation and its anisotropic version as typical examples. In this part, we investigate the Lipschitz continuity of the free boundary of viscosity solution and its asymptotic spherical symmetricity, however,this result does not include the anisotropic case.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
The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Capu...The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)^(-σ)u) = f, 0 < σ <1/2.This paper poses the problem over {t ∈ R^+, x ∈ R^n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.展开更多
基金Supported by the National Natural Science Foundation of China(11926322)the Fundamental Research Funds for the Central Universities of South-Central MinZu University(CZY22013,3212023sycxjj001)。
文摘In this paper,we investigate the strong Feller property of stochastic differential equations(SDEs)with super-linear drift and Hölder diffusion coefficients.By utilizing the Girsanov theorem,coupling method,truncation method and the Yamada-Watanabe approximation technique,we derived the strong Feller property of the solution.
基金supported by an NSERC granta startup fund of University of Albertasupported by Martin Hairer’s Leverhulme Trust leadership award
文摘We obtain the H?lder continuity and joint H?lder continuity in space and time for the random field solution to the parabolic Anderson equation ■ in d-dimensional space, where ■ is a mean zero Gaussian noise with temporal covariance γ0 and spatial covariance given by a spectral density μ(ξ). We assume that ■ and ■ , where αi, i = 1, · · ·, d(or α) can take negative value.
基金in part by Zhongshan University Science Research Fund
文摘The Hǒlder continuity is proved for the gradient of the solution Jo the one-sided obstacle problem of the following variational inequality in the case 1<p<2
文摘In [1] we construct a unique bounded H■lder continuous viscosity solution for the nonlinear PDEs with the evolution p-Laplacian equation and its anisotropic version as typical examples. In this part, we investigate the Lipschitz continuity of the free boundary of viscosity solution and its asymptotic spherical symmetricity, however,this result does not include the anisotropic case.
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supported by NSG grant DMS-1303632NSF grant DMS-1500871,NSF grant DMS-1209420
文摘The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)^(-σ)u) = f, 0 < σ <1/2.This paper poses the problem over {t ∈ R^+, x ∈ R^n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.
