arXiv subjects
Publications and source records attributed to Karl-Theodor Sturm.
We prove the Bochner-Weitzenboeck formula for the (nonlinear) Laplacian on general Finsler manifolds and derive Li-Yau type gradient estimates as well as parabolic Harnack inequalities. Moreover, we deduce Bakry-Emery gradient estimates. All these estimates depend on lower bounds for the weighted flag Ricci tensor.
We present two approaches to the heat flow on a Finsler manifold $(M,F)$: either as gradient flow on $L^2(M,m)$ for the energy; or as gradient flow on the reverse $L^2$-Wasserstein space $\mathcal{P}_2(M)$ of probability measures on $M$ for the relative entropy. Both approaches depend on the choice of a measure $m$ on $M$ and then lead to the same nonlinear evolution semigroup. We prove $\mathcal{C}^{1,α}$-regularity for solutions to the (nonlinear) heat equation on the Finsler space $(M,F,m)$. Typically, solutions to the heat equation will not be $\mathcal{C}^2$. Moreover, we derive pointwise comparison results a la Cheeger-Yau and integrated upper Gaussian estimates a la Davies.
Based on a study of the coupling by reflection of diffusion processes, a new monotonicity in time of a time-dependent transportation cost between heat distribution is shown under Bakry-Emery's curvature-dimension condition on a Riemannian manifold. The cost function comes from the total variation between heat distributions on spaceforms. As a corollary, we obtain a comparison theorem for the total variation between heat distributions. In addition, we show that our monotonicity is stable under the Gromov-Hausdorff convergence of the underlying space under a uniform curvature-dimension and diameter bound.
We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is different from the usual convexity along geodesics in non-Riemannian Finsler manifolds. As an application, we show that the heat flow on Minkowski normed spaces other than inner product spaces is not contractive with respect to the quadratic Wasserstein distance.
Von Renesse and the author (Ann. Prob. '09) developed a second order calculus on the Wasserstein space P([0,1]) of probability measures on the unit interval. The basic objects of interest had been Dirichlet form, semigroup and continuous Markov process, called Wasserstein diffusion. The goal of this paper is to derive approximations of these objects on the infinite dimensional space P([0, 1]) in terms of appropriate objects on finite dimensional spaces. In particular, we will approximate the Wasserstein diffusion in terms of interacting systems of Brownian motions.
Given a strictly increasing, continuous function $\vartheta:\R_+\to\R_+$, based on the cost functional $\int_{X\times X}\vartheta(d(x,y))\,d q(x,y)$, we define the $L^\vartheta$-Wasserstein distance $W_\vartheta(μ,ν)$ between probability measures $μ,ν$ on some metric space $(X,d)$. The function $\vartheta$ will be assumed to admit a representation $\vartheta=ϕ\circψ$ as a composition of a convex and a concave function $ϕ$ and $ψ$, resp. Besides convex functions and concave functions this includes all $\mathcal C^2$ functions. For such functions $\vartheta$ we extend the concept of Orlicz spaces, defining the metric space $L^\vartheta(X,m)$ of measurable functions $f: X\to\R$ such that, for instance, $$d_\vartheta(f,g)\le1\quad\Longleftrightarrow\quad \int_X\vartheta(|f(x)-g(x)|)\,dμ(x)\le1.$$
We prove generalized lower Ricci bounds for Euclidean and spherical cones over complete Riemannian manifolds. These cones are regarded as complete metric measure spaces. In general, they will be neither manifolds nor Alexandrov spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-dimension condition CD(0; n+1) and that the spherical cone over the same manifold fulfills the curvature-dimension condition CD(n; n+1). More generally, for each N > 1 we prove that the condition CD(N-1;N) for a weighted Riemannian space is equivalent to the condition CD(0;N +1) for its N-Euclidean cone as well as to the condition CD(N;N + 1) for its N-spherical cone.
We prove generalized lower Ricci bounds for Euclidean and spherical cones over compact Riemannian manifolds. These cones are regarded as complete metric measure spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-dimension condition CD(0,n+1) and that the spherical cone over the same manifold fulfills the curvature-dimension condition CD(n,n+1).
This paper is devoted to the analysis of metric measure spaces satisfying locally the curvature-dimension condition CD(K,N) introduced by the second author and also studied by Lott & Villani. We prove that the local version of CD(K,N) is equivalent to a global condition CD*(K,N), slightly weaker than the (usual, global) curvature-dimension condition. This so-called reduced curvature-dimension condition CD*(K,N) has the local-to-global property. We also prove the tensorization property for CD*(K,N). As an application we conclude that the fundamental group of a metric measure space (M,d,m) is finite whenever it satisfies locally the curvature-dimension condition CD(K,N) with positive K and finite N.
We construct the entropic measure $\mathbb{P}^β$ on compact manifolds of any dimension. It is defined as the push forward of the Dirichlet process (another random probability measure, well-known to exist on spaces of any dimension) under the {\em conjugation map} $$\Conj:\mathcal{P}(M)\to\mathcal{P}(M).$$ This conjugation map is a continuous involution. It can be regarded as the canonical extension to higher dimensional spaces of a map between probability measures on 1-dimensional spaces characterized by the fact that the distribution functions of $μ$ and $\Conj(μ)$ are inverse to each other. We also present an heuristic interpretation of the entropic measure as $$d\mathbb{P}^β(μ)=\frac{1}{Z}\exp(-β\cdot {Ent} (μ|m))\cdot d\mathbb{P}^0(μ).$$
We construct a new random probability measure on the sphere and on the unit interval which in both cases has a Gibbs structure with the relative entropy functional as Hamiltonian. It satisfies a quasi-invariance formula with respect to the action of smooth diffeomorphism of the sphere and the interval respectively. The associated integration by parts formula is used to construct two classes of diffusion processes on probability measures (on the sphere or the unit interval) by Dirichlet form methods. The first one is closely related to Malliavin's Brownian motion on the homeomorphism group. The second one is a probability valued stochastic perturbation of the heat flow, whose intrinsic metric is the quadratic Wasserstein distance. It may be regarded as the canonical diffusion process on the Wasserstein space.