论文标题
Lipschitz函数的亚代桥的密度在公制的Sobolev空间和应用于Wasserstein Sobolev空间的应用
Density of subalgebras of Lipschitz functions in metric Sobolev spaces and applications to Wasserstein Sobolev spaces
论文作者
论文摘要
我们证明了lipschitz合适子代理的密度密度的一般标准，在metric-sobolev space $ h^{1，p}（x，x，x，\ shssf {d}，\ mathfrak {m mathfrak {m}）$与正面和有限的borel borel borel量子指标$ \ m rabfak $ \ mathfrak $ \ mathfrak rape an $（x，\ mathsf {d}）$。然后，我们为Wasserstein Sobolev空间$ H^{1,2}（\ Mathcal {p} _2（\ Mathbb {M}），W_ {2}，W_ {2}，\ Mathfrak {m}）$ potition $ potition $ potion $ potion $ potion $ protial potifra，我们在WASSERSTEIN SOBOLEV空间中提供相关应用Kantorovich-Rubinstein-Wasserstein space $（\ Mathcal {p} _2（\ Mathbb {m}），W_ {2}）$概率措施在有限的欧几里得空间中，是一个完整的Riemannian歧管，一个完整的Riemannian歧管，或可分离我们将证明这种Sobolev空间始终是希尔伯特式的，独立于选择参考度量$ \ Mathfrak {M} $，因此由此产生的Cheeger Energy是Dirichlet形式。我们最终将为$ \ mathfrak {M} $ - Wasserstein渐变的相应概念提供明确的表征，显示了有用的计算规则及其与切线束的一致性以及从Dirichlet形式继承的$γ$ -calculus。
We prove a general criterion for the density in energy of suitable subalgebras of Lipschitz functions in the metric-Sobolev space $H^{1,p}(X,\mathsf{d},\mathfrak{m})$ associated with a positive and finite Borel measure $\mathfrak{m}$ in a separable and complete metric space $(X,\mathsf{d})$. We then provide a relevant application to the case of the algebra of cylinder functions in the Wasserstein Sobolev space $H^{1,2}(\mathcal{P}_2(\mathbb{M}),W_{2},\mathfrak{m})$ arising from a positive and finite Borel measure $\mathfrak{m}$ on the Kantorovich-Rubinstein-Wasserstein space $(\mathcal{P}_2(\mathbb{M}),W_{2})$ of probability measures in a finite dimensional Euclidean space, a complete Riemannian manifold, or a separable Hilbert space $\mathbb{M}$. We will show that such a Sobolev space is always Hilbertian, independently of the choice of the reference measure $\mathfrak{m}$ so that the resulting Cheeger energy is a Dirichlet form. We will eventually provide an explicit characterization for the corresponding notion of $\mathfrak{m}$-Wasserstein gradient, showing useful calculus rules and its consistency with the tangent bundle and the $Γ$-calculus inherited from the Dirichlet form.
