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Probabilistic and spectral properties of weighted Riemannian manifolds with Kato bounded Bakry- Émery-Ricci curvature

A weighted Riemannian manifold is pair $(M,\Psi)$ given by a Riemannian manifold $M$ and a function $\Psi\in W^{2,2}_{\mathrm{loc}}(M)$, the weight function. In typical situations that we have in mind, $\Psi$ is indeed not smooth: this happens, for example, if one takes $\Psi$ to be the groundstate of a molecule with $m$ electrons, and $M$ the Euclidean $\mathbb{R}^{3m}$.

Such a pair canonically induces:

• the weighted Laplacian $\Delta_{\Psi}\geq 0$ in the weighted $L^2$-space $L^2(M,\Psi)$,
• the weighted heat semigroup $e^{t\Delta_{\Psi}}$ in $L^2(M,\Psi)$, $t>0$, whose integral kernel $e^{t\Delta_{\Psi}}(x,y)$, $t>0$, $x,y\in M$, is called the weighted heat kernel,
• a diffusion process, the weighted Brownian motion, whose transition density is induced by $e^{t\Delta_{\Psi}}(x,y)$.

A central geometric object in this context is the Bakry-Émery Ricci curvature, given by $$\mathrm{Ric}_{\Psi}=\mathrm{Ric}+2\nabla^2 \Psi$$ whose study under minimal local regularity assumptions on $\Psi$ is one of the main objectives of this project. Note that in the above mentioned molecular case the Bakry-Émery Ricci curvature becomes the symmetric matrix $\mathrm{Ric}_{\Psi}=2\nabla^2 \Psi$, which carries local singularities, but (as we have shown in our previous work) neverthess has a variable lower bound in the so called Kato class of $(M,\Psi)$.

Some of the main goals of this project are:

• to study parabolicity and stochastic completeness of the underlying diffusion on $(M,\Psi)$ under variable (Kato or Dynkin type) lower bounds $\mathrm{Ric}_{\Psi}$, in particular dealing with all technical issues arising from the nonsmoothness of $\Psi$.
• to use the unitary equivalence of a Schrödinger operator $\Delta+V$ to some $\Delta_{\Psi}$ via the ground state transform, in order to derive eigenvalue estimates for molecular Schrödinger operators using probabilistic and geometric methods for $\Delta_{\Psi}$. As our previous considerations indicate, these results are closely connected with Harnack inequalities for Schrödinger operators on Riemannian manifolds.
•  to use the above unitary equivalence in order to transfer scattering problems for Schrödinger operators (where the potentials are scattered) to two-Hilbert-space scattering problems for weighted Laplacians (where the weight functions are scattered); use geometric and probabilistic methods to the study the latter.
• to characterize variable lower bounds for $\mathrm{Ric}_{\Psi}$ in terms of the existence of appropriate couplings of weighted Brownian motions; obtain similar characterizations in terms of weighted Brownian bridges.