Xing Huang, Shao-Qin Zhang
The existence and uniqueness of the mild solution for a class of functional SPDEs with multiplicative noise and a locally Dini continuous drift are proved. In addition, under a reasonable condition the solution is non-explosive. Moreover, Harnack inequalities are derived for the associated semigroup under certain global conditions, which is new even in the case without delay.
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Asymptotic log-Harnack inequality for path-distribution dependent SDEs with infinite memory and Dini drift
针对具有无限记忆和Dini漂移的路径分布依赖随机微分方程，建立了渐近对数哈纳克不等式，该结果在分布无关情形下同样新颖。
Functional SPDE with Multiplicative Noise and Dini Drift
证明了一类带乘性噪声和Dini连续漂移的半线性泛函SPDE的温和解的存在唯一性和非爆炸性。在有限维有界时滞情形下，得到了log-Harnack不等式和$L^2$ -梯度估计.由于马尔可夫半群与方程的泛函（段）解相联系，需要对时滞时间区间内解的道路空间进行分析。
The existence and uniqueness of the mild solutions for a class of degenerate functional SPDEs are obtained, where the drift is assumed to be Hölder-Dini continuous. Moreover, the non-explosion of the solution is proved under some reasonable conditions. In addition, the Harnack is derived by the coupling by change of measure. Finally, the shift Harnack inequality is obtained for the equations without delay, which is new even in the non-degenerate case.
Shift Harnack Inequality and Integration by Part Formula for Semilinear SPDE
建立了半线性时滞随机微分方程和一类覆盖超耗散Naiver-Stokes/Burges方程的随机半线性发展方程的移位Harnack和分部积分公式.对于时滞随机方程的情形，给出了道路空间的一个推广。
Harnack Inequality for Semilinear spdes with multiplicative noise
利用一种新的近似方法，建立了希尔伯特空间中一类带乘性噪声的半线性随机微分方程的无量纲Harnack不等式.利用这些不等式研究了半群的强Feller性质和不变测度的一些性质。
Log-Harnack Inequality for Mild Solutions of SPDEs with Strongly Multiplicative Noise
由于技术上的原因，已有的关于带乘性噪声的SPDE的Harnack型不等式的结果只适用于噪声项中的系数是固定有界算子的Hilbert-Schmidt扰动的情况.本文研究了一类带强乘性噪声的半线性随机偏微分方程，其系数甚至可以是无界的，因而不可能是Hilbert-Schmidt的。给出了梯度估计、log-Harnack不等式及其应用。应用于时空白噪声驱动的随机反应扩散方程。
We mainly investigate the log-Harnack inequality for the reflected stochastic partial differential equation driven by multiplicative noises based on the gradient estimate of the associated Markov semigroup. To do it, the penalization method and the comparison principle for stochastic partial differential equations are adopted. As its applications, we also establish the entropy-cost inequality and study some important estimates of the transition density relative to its invariant measure.
Asymptotic Log-Harnack Inequality for Monotone SPDE with Multiplicative Noise
本研究利用测度变换的渐近耦合方法，推导了带退化乘性噪声的非线性单调SPDE的渐近对数-Harnack不等式，证明了对应马尔可夫半群的相关性质与唯一遍历不变测度的存在性，可应用于高度退化的扩散过程。
Asymptotic Log-Harnack Inequality for Degenerate SPDEs with Reflection
本文研究了退化随机偏微分方程的渐近对数Harnack不等式，揭示了其在热核估计、不变测度唯一性及渐近强Feller性质等方面的应用。
The dimension free Harnack inequality is established for the distribution dependent stochastic Hamiltonian system, where the drift is Lipschitz continuous in the measure variable under the distance induced by the Hölder-Dini continuous functions, which are $β(β>\frac{2}{3})$-Hölder continuous on the degenerate component and square root of Dini continuous on the non-degenerate one. The results extend the existing ones in which the drift is Lipschitz continuous in the measure variable under $L^2$-Wasserstein distance.
Bismut Formulae and Applications for Functional SPDEs
利用Malliavin微积分，建立了一类带加性或乘性噪声的半线性泛函随机偏微分方程的显式导数公式。作为应用，得到了相应的分段过程半群的梯度估计和Harnack不等式。关键词：Bismut公式，Malliavin演算，梯度估计，Harnack不等式，泛函SPDE
In this paper, the distribution dependent stochastic differential equation in a separable Hilbert space with a Dini continuous drift is investigated. The existence and uniqueness of weak and strong solutions are obtained. Moreover, some regularity results as well as gradient estimates and log-Harnack inequality are derived for the associated semigroup. In addition, dimensional free Harnack inequality with power and shift Harnack inequality are also proved when the noise is additive. All of the results extend the ones in the distribution independent situation.
We consider stochastic partial differential equations under minimal assumptions: the coefficients are merely bounded and measurable and satisfy the stochastic parabolicity condition. In particular, the diffusion term is allowed to be scaling-critical. We derive local supremum estimates with a stochastic adaptation of De Giorgi's iteration and establish a weak Harnack inequality for the solutions. The latter is then used to obtain pointwise almost sure continuity.
Gradient Estimates and Applications for SDEs in Hilbert Space with Multiplicative Noise and Dini Continuous Drift
考虑可分希尔伯特空间中的随机发展方程，该方程具有良好的乘性噪声和局部Dini连续漂移。我们证明了，对于任何初始数据的方程有一个唯一的（可能是爆炸性的）温和的解决方案。在保证解不爆炸的合理条件下，证明了相应的马尔可夫半群的强Feller性质。在一定的整体条件下，得到了相应半群的梯度估计和log-Harnack不等式，这些结果在有限维中也是新的。
The asymptotic log-Harnack inequality is established for several different models of stochastic differential systems with infinite memory: non-degenerate SDEs, Neutral SDEs, semi-linear SPDEs, and stochastic Hamiltonian systems. As applications, the following properties are derived for the associated segment Markov semigroups: asymptotic heat kernel estimate; uniqueness of the invariant probability measure; asymptotic gradient estimate and hence, asymptotically strong Feller property; and asymptotic irreducibilty.
By using the local dimension-free Harnack inequality established on incomplete Riemannian manifolds, integrability conditions on the coefficients are presented for SDEs to imply the non-explosion of solutions as well as the existence, uniqueness and regularity estimates of invariant probability measures. These conditions include a class of drifts unbounded on compact domains such that the usual Lyapunov conditions can not be verified. The main results are extended to second order differential operators on Hilbert spaces and semi-linear SPDEs.
In this article, we study a class of semilinear stochastic partial differential equations driven by an additive space time white noise. We establish Harnack inequalities for the semigroup associated with the solution by using coupling method, which implies the strong Feller property. Our results generalize the results of Zhang [Potential Analysis 33 (2010), no. 2, 137-151.] and can be applied to some types of SPDE such as reaction-diffusion equation and transport-diffusion equation perturbed by space time white noise.
Log-Harnack Inequality and Bismut Formula for McKean-Vlasov SDEs with Singularities in all Variables
本文在漂移项于时间-空间可积且分布连续性弱于 Dini 的条件下，为所有变量均具有奇异性的 McKean-Vlasov 随机微分方程建立了 log-Harnack 不等式和 Bismut 公式，显著推广了现有结果。
Integration by Parts Formula and Shift Harnack Inequality for Stochastic Equations
引入一种新的耦合论证，建立了Driver的分部分部积分法公式和移位Harnack不等式。与已知的耦合方法不同，其中具有不同起点的两个边缘过程被构造为尽可能快地一起移动，对于新型耦合，两个边缘过程从相同的点开始，但是它们的差异旨在在给定时间达到固定量。除了分部积分法公式外，新的耦合方法也能有效地推广平移Harnack不等式。与已知的Harnack不等式不同，在Harnack不等式中，参考函数在不同点处的值被比较，在移位Harnack不等式中，参考函数而不是初始点被移位。给出了分部积分法和移位Harnack不等式的若干应用。一般的结果说明了一些具体的模型，包括随机哈密顿系统，其中相关的扩散过程可以是高度退化，延迟的SDES，和半线性SPDES。
Harnack inequality for parabolic equations in double-divergence form with singular lower order coefficients
本文研究了双散度型二阶抛物方程非负解的Harnack不等式，设定主系数满足Dini平均振荡条件且低阶系数属于Morrey类，推进了相关方程及Fokker-Planck-Kolmogorov方程的理论基础。
In this paper, existence and uniqueness are proved for path-dependent McKean-Vlasov type SDEs with integrability conditions. Gradient estimates and Harnack type inequalities are derived in the case that the coefficients are Dini continuous in the space variable. These generalize the corresponding results derived for classical functional SDEs with singular coefficients.
We give a new, two-step approach to prove existence of finite invariant measures for a given Markovian semigroup. First, we identify a convenient auxiliary measure and then we prove conditions equivalent to the existence of an invariant finite measure which is absolutely continuous with respect to it. As applications, we give a short proof for the result of Lasota and Szarek on invariant measures and we obtain a unifying generalization of different versions for Harris ergodic theorem which provides an answer to an open question of Tweedie. We show that for a nonlinear SPDE on a Gelfand triple,…
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