Conservativeness of non-symmetric diffusion processes generated by perturbed divergence forms

Conservativeness and uniqueness of invariant measures related to non-symmetric divergence type operators

We present conservativeness criteria for sub-Markovian semigroups generated by divergence type operators with specified infinitesimally invariant measures. The conservativeness criteria in this article are derived by $L^1$-uniqueness and imply that a given infinitesimally invariant measure becomes an invariant measure. We explore further conditions on the coefficients of the partial differential operators that ensure the uniqueness of the invariant measure beyond the case where the corresponding semigroups are recurrent. A main observation is that for conservativeness and uniqueness of invaria…

Conservativeness criteria for generalized Dirichlet forms

We develop sufficient analytic conditions for conservativeness of non-sectorial perturbations of symmetric Dirichlet forms which can be represented through a carré du champ on a locally compact separable metric space. These form an important subclass of generalized Dirichlet forms which were introduced in \cite{St1}. In case there exists an associated strong Feller process, the analytic conditions imply conservativeness, i.e. non-explosion of the associated process in the classical probabilistic sense. As an application of our general results on locally compact separable metric state spaces, w…

受输运噪声扰动的三维Navier-Stokes方程柯西问题的全局存在性与非唯一性

Global existence and non-uniqueness for the Cauchy problem associated to 3D Navier-Stokes equations perturbed by transport noise

本文证明了受Stratonovich输运噪声扰动的三维Navier-Stokes方程柯西问题存在全局的概率强解析弱解，且解不唯一；可指定任意无散度平方可积初值或以高概率任意大的停时前动能，解具有Sobolev正则性但非Leray-Hopf。

Dirichlet过程的扰动Kullback-Leibler偏差界

Perturbed Kullback-Leibler Deviation Bounds for Dirichlet Processes

本研究针对Dirichlet过程，通过引入受控扰动的方法，依托超加性技术和约简为Beta分布两种证明思路，得到了采用KL散度构建的改进型非渐近偏差界，拓展了Beta分布的同类不等式。

带边界黎曼流形上的时空散度引理与扩散过程的最优非可逆提升

Space-time divergence lemmas and optimal non-reversible lifts of diffusions on Riemannian manifolds with boundary

本文针对带边界黎曼流形上的可逆扩散过程，建立带显式常数的时空散度引理，证明含哈密顿流、Ornstein-Uhlenbeck噪声的非可逆提升在参数平衡时可达最优平方根级弛豫时间缩短效果。

闭式响应演算与Liouville量子引力度量的联合密度

Closed Response Calculus and Joint Densities for the Liouville Quantum Gravity Metric

本文提出一种将底层概率空间的一阶响应转化为可观测分布上闭式微分演算的一般准则，并应用于次临界Liouville量子引力度量，证明对数距离比向量具有Lebesgue密度。