Regularity and stability of invariant measures for diffusion processes under synthetic lower Ricci curvature bounds. (English) Zbl 1502.53068
The article in review establishes the construction of diffusion processes and studies the existence of invariant measures and the uniqueness of the densities under the assumption that \(\mu\) has the density \(\rho\) with respect to \(m\).
For a given measure space \((X,m)\) and a strongrly continuous semigroup \(\{T_t\}\) on \(L^p(X,m)\), a measure \(\mu = \rho m\) with \(\rho \in L^q(X,m)\) and \((1/p + 1/q = 1)\) is said to be an invariant measure for \(\{T_t\}\) if \[ \int_X T_t f \,d\mu = \int_X f\,d\mu,\quad \forall f \in L^p(X,m),\,\forall t \geq 0. \] Invariant measures can also be characterized in terms of the infinitesimal generator.
The existence, uniqueness, and regularity of invariant measures are fundamental questions for the theory of Markov processes and for the theory of elliptic PDEs. The article in review studies invariant measures and the related elliptic PDE on non-smooth metric measure spaces satisfying synthetic lower Ricci curvature bounds (RCD). The infinitesimal generator condidered int the article in review can be written in the following form: \[ L = \frac{1}{2} \Delta + \boldsymbol{b}. \] Here \(\Delta\) denotes the Laplacian and \(\boldsymbol{b}\) denotes a derivation operator which is the first-order differential operator.
The main result of the article is as follows: The Sobolev regularity of density \(\rho\) and the gradient estimate of \(\rho\) are obtained whenever \(\mu = \rho m\) satisfies the following equation: \[ \int_X \left(\frac{1}{2} \Delta + \boldsymbol{b}\right) (\phi) d \mu = 0, \quad \forall \mu \in \text{TestF}(X). \] An application of the Sobolev regularity is given and answers a question on the symmetrizability of the semigroup \(\{T_t\}\) on \(L^2(X,m)\) on RCD spaces with \(m(X) < \infty\): When is \(\{T_t\}\) symmetrizable? The article gives the following answer: \(\{T_t\}\) is symmetrizable by a measure \(\mu = \rho m\) with density \(\rho \in L^2(X,m)\) if and only if there is a Lipschitz continuous function \(f\) with \(f \in L^2(m)\) so that \(\boldsymbol{b}(\cdot) = \langle \nabla f, \nabla \cdot \rangle\).
For a given measure space \((X,m)\) and a strongrly continuous semigroup \(\{T_t\}\) on \(L^p(X,m)\), a measure \(\mu = \rho m\) with \(\rho \in L^q(X,m)\) and \((1/p + 1/q = 1)\) is said to be an invariant measure for \(\{T_t\}\) if \[ \int_X T_t f \,d\mu = \int_X f\,d\mu,\quad \forall f \in L^p(X,m),\,\forall t \geq 0. \] Invariant measures can also be characterized in terms of the infinitesimal generator.
The existence, uniqueness, and regularity of invariant measures are fundamental questions for the theory of Markov processes and for the theory of elliptic PDEs. The article in review studies invariant measures and the related elliptic PDE on non-smooth metric measure spaces satisfying synthetic lower Ricci curvature bounds (RCD). The infinitesimal generator condidered int the article in review can be written in the following form: \[ L = \frac{1}{2} \Delta + \boldsymbol{b}. \] Here \(\Delta\) denotes the Laplacian and \(\boldsymbol{b}\) denotes a derivation operator which is the first-order differential operator.
The main result of the article is as follows: The Sobolev regularity of density \(\rho\) and the gradient estimate of \(\rho\) are obtained whenever \(\mu = \rho m\) satisfies the following equation: \[ \int_X \left(\frac{1}{2} \Delta + \boldsymbol{b}\right) (\phi) d \mu = 0, \quad \forall \mu \in \text{TestF}(X). \] An application of the Sobolev regularity is given and answers a question on the symmetrizability of the semigroup \(\{T_t\}\) on \(L^2(X,m)\) on RCD spaces with \(m(X) < \infty\): When is \(\{T_t\}\) symmetrizable? The article gives the following answer: \(\{T_t\}\) is symmetrizable by a measure \(\mu = \rho m\) with density \(\rho \in L^2(X,m)\) if and only if there is a Lipschitz continuous function \(f\) with \(f \in L^2(m)\) so that \(\boldsymbol{b}(\cdot) = \langle \nabla f, \nabla \cdot \rangle\).
Reviewer: Loreno Heer (Zürich)
MSC:
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 35J15 | Second-order elliptic equations |
Keywords:
regularity; stability; invariant measures; lower Ricci curvature bonds; Ricci curvature bonds; diffusion processesReferences:
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