非对称扭曲布朗运动：强解、强Feller性质和非爆炸结果
Non-symmetric distorted Brownian motion: strong solutions, strong Feller property and non-explosion results
摘要：
利用加权空间中的椭圆正则性结果、随机分析和非对称Dirichlet型理论，我们首先证明了非对称扭曲布朗运动对$\mathbb{R}^d$的某个定义域$E$中的任意起始点的弱存在性，其中$E$被明确地给出为密度的唯一连续形式对其不变测度的严格正性的点.非对称扭曲布朗运动是奇异扩散，即通常具有无界和不连续漂移的扩散。一旦证明了弱解的存在性，我们就从\cite{KR}的一个结果得到，所构造的弱解在爆炸时刻之前确实是强、弱和路径唯一的.由于我们的方法，我们可以使用的狄利克雷形式的理论来证明进一步的性质的解决方案。更准确地说，我们得到了新的非爆炸准则。最后给出了与一类Muckenhoupt权和相应的无散度扰动相关的非对称扭曲布朗运动的存在性和非爆炸性结果.
英文摘要：
Using elliptic regularity results in weighted spaces, stochastic calculus and the theory of non-symmetric Dirichlet forms, we first show weak existence of non-symmetric distorted Brownian motion for any starting point in some domain $E$ of $\mathbb{R}^d$, where $E$ is explicitly given as the points of strict positivity of the unique continuous version of the density to its invariant measure. Non-symmetric distorted Brownian motion is a singular diffusion, i.e. a diffusion that typically has an unbounded and discontinuous drift. Once having shown weak existence, we obtain from a result of \cite{KR} that the constructed weak solution is indeed strong and weakly as well as pathwise unique up to its explosion time. As a consequence of our approach, we can use the theory of Dirichlet forms to prove further properties of the solutions. More precisely, we obtain new non-explosion criteria for them. We finally present concrete existence and non-explosion results for non-symmetric distorted Brownian motion related to a class of Muckenhoupt weights and corresponding divergence free perturbations.
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