Lagrangian calculus for nonsymmetric diffusion operators
We characterize lower bounds for the Bakry-Emery Ricci tensor of nonsymmetric diffusion operators by convexity of entropy on the -Wasserstein space, and define a curvature-dimension condition for general metric measure spaces together with a square integrable -form in the sense of [Giga]. This extends the Lott-Sturm-Villani approach for lower Ricci curvature bounds of metric measure spaces. In generalized smooth context, consequences are new Bishop-Gromov estimates, pre-compactness under measured Gromov-Hausdorff convergence, and a Bonnet-Myers theorem that generalizes previous results by Kuwada [Kuw13]. We show that -warped products together with lifted vector fields satisfy the curvature-dimension condition. For smooth Riemannian manifolds we derive an evolution variational inequality and contraction estimates for the dual semigroup of nonsymmetric diffusion operators. Another theorem of Kuwada [Kuw10, Kuw15] yields Bakry-Emery gradient estimates.
In this article we present a Lagrangian approach for studying possibly nonsymmetric diffusion operators. For instance, we consider operators of the form where is the Laplace-Beltrami operator of a compact smooth Riemannian manifold and is a smooth -form on . Then the Bakry-Emery -Ricci tensor associated to is defined by
for where denotes the symmetric derivation of with respect to the Levi-Civita connection of . Note is also meaningful for but we will not consider these cases.
If , is the diffusion operator of the canonical symmetric Dirichlet form associated to the smooth metric measure space with , and is understood as the Ricci curvature of . In celebrated articles by Lott, Sturm and Villani [LV09, Stu06a, Stu06b] - built on previous results in [OV00, CEMS01, vRS05] - a definition of lower Ricci curvature bounds for general metric measure spaces in terms of convexity properties of entropy functionals on the -Wasserstein space was introduced. In smooth context these definitions are equivalent to lower bounds for the Bakry-Emery tensor provided is exact.
For diffusion operators where is not necessarily exact such a geometric picture was missing. Though the operator yields a bilinear form, in general this form is not symmetric and therefore cannot arise as Dirichlet form of a metric measure space. Nevertheless, there are numerous results dealing with probabilistic, analytic and geometric properties of under lower bounds on . The results are very similar to properties that one derives for symmetric operators with lower bounded Ricci curvature, e.g. [Kuw13, Kuw15, Wan11].
In this article we derive a geometric picture associated to the diffusion operator for general -forms in the spirit of the work by Lott, Sturm and Villani. We characterize lower bounds on in terms of convexity for line integrals along -Wasserstein geodesics. Moreover, for generalized smooth metric measures spaces (Definition 2.1) we impose the following definition. For simplicity, in this introduction we assume . We will say together with a -form satisfies the curvature-dimension condition if and only if for every pair there exists an -Wasserstein geodesic such that
where and denotes the line integral of along . For the case corresponding definitions are made in Definition 3.1 and Definition 3.2. In particular, we emphasize that Definition 3.2 is also meaningful in the class of general metric measure spaces together with -integrable -forms in the sense of [Giga]. However, in this article we will only study the generalized smooth case.
We prove several geometric consequences: Generalized Bishop-Gromov estimates, pre-compactness under Gromov-Hausdorff convergence, and a generalized Bonnet-Myers Theorem. The latter generalizes a result of Kuwada in [Kuw13] - even for smooth ingredients. Then, we show that the condition is stable under -warped product constructions. This also includes so-called euclidean -cones and -suspensions.
In the last section we introduce the notion of -flows that arise naturally on generalized smooth metric measure spaces together with a -form satisfying a curvature-dimension condition. More preciely, if is the semigroup associated to the operator on a smooth Riemannian manifold such that satisfies , and if is the dual flow acting on probability measures, then it is an absolutely continuous curve in -Wasserstein space and for any probability measure , satisfies the following inequality
provided standard regularity assumption for the corresponding heat kernel (Proposition 7.2). is any absolutely continuous probability measure, and is the -Wasserstein geodesic between and . This yields a contraction estimate for that is again equivalent to the Bakry-Emery conditon for by [Kuw10]. Therefore, we obtain the following theorem.
Let be a compact smooth Riemannian manifold, and let be a smooth -form. We denote with the corresponding metric measure space, and let and be as in the section 7. Then, the following statements are equivalent.
satisfies the condition ,
For every is an -flow curve starting in ,
satisfies the contraction estimate in corollary 7.5,
satisfies the condition .
This work was partly done while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Spring 2016 semester, supported by the National Science Foundation. I want to thank the organizers of the Differental Geometry Program and MSRI for providing great environment for research.
Metric measure spaces
Let be a complete and separable metric space, and let be a locally finite Borel measure.
We call a metric measure space.
The case is excluded.
The space of constant speed geodesics is denoted with ,
and it is equipped with the topology of uniform convergence. denotes the evaluation map at time that is continuous.
The -Wasserstein space of probability measures with finite second moment is denoted with , and is the -Wasserstein distance. and denote the subset of compactly supported probability measures and the family of -absolutely continuous probability measures, respectively. A coupling or plan between probability measures and is a probability measure such that where are the projection maps. A coupling is optimal if Optimal couplings exist, and if an optimal coupling is induced by a map via where is a measurable subset of , we say is an optimal map.
A probability measure is called an optimal dynamical coupling if is an optimal coupling between its marginal distributions. Let be an -Wasserstein geodesic in . We say an optimal dynamical coupling is a lift of if for every . If is the lift of an -Wasserstein geodesic , we call itself an -Wasserstein geodesic. We say has bounded compression if there exists a constant such that for every .
We say that a metric measure space is essentially non-branching if for any optimal dynamical coupling there exists such that and for all we have that For instance a metric measure space satisfying a Riemannian curvature-dimension condition in the sense of [EKS15, Gigb] is essentially non-branching [RS14].
If we assume that for -almost every pair there exists a unique geodesic between and then by measurable selection there exists a measurable map with . For and Borel with we set , and for we set . In this case is the unique optimal dynamical plan between and . Again, the family of -spaces is a class that satisfies this property [GRS].
Definition 2.1 (Generalized smooth metric measure spaces).
We say is a generalized smooth metric measure space if there exists an open smooth manifold , and a Riemannian metric on with induced distance function such that the metric completion of the metric space is isometric to , and for any optimal dynamical plan such that is a geodesic and we have that
In particular, , and if we choose the constant geodesic with for all where is any measurable set of finite -measure, one gets that is of -measure . We call the set of regular points in .
The condition (1) yields that is essentially non-branching and that for -almost every pair there exists a unique geodesic between and . Moreover, for each pair there is a unique dynamcial optimal coupling such that where is the space of geodesic in , , and is induced by a map. To see this note that and transport geodesics are contained in . Then, since one can choose an exhaustion of by compact sets, we can assume that and are compactly supported in . Then, the claim follows from statements in [CEMS01] and since geodesics are unique.
Examples of generalized smooth metric measure spaces in the sense of the previous definition are Riemannian manifolds with boundary that are geodesically convex, cones, suspensions [BS14] and warped products [Ket13]. Moreover, in an upcoming paper of the author with Ilaria Mondello, it will be shown that stratified spaces are generalized smooth provided certain assumptions on tangent cones at singular points. This result will also show that orbifolds are generalized smooth.
-forms and vector fields
Assume is a generalized smooth metric measure space.
A -form is a measurable map with .
We say for if
where and compact.
Similar, we can consider measurable and -integrable vector fields on . Note that a vector field on yields a -form via . In the context of generalized smooth metric measures space this is the natural isomorphism between vectorfields and -forms, and we will often switch between these viewpoints. If is an optimal dynamical coupling with bounded compression, the line integral
exists -almost surely, and it does not depend on the parametrization of up to changes of orientation. Moreover, for any -Wasserstein geodesic with bounded compression, we set .
The case of arbirtrary metric measure spaces
Let be any function. The Hopf-Lax semigroup is defined by is the -transform of . We say a function is -concave if there exist such that . If is compact, by Kantorovich duality for any pair with bounded densities there exist a Lipschitz function such that
for any geodesic with bounded compression. For instance, see [GH].
If we follow the approach of Gigli in [Giga], there is also a well-defined notion of -integrable -form for general metric measure spaces , and one can define the dual coupling as measurable function on where is a Sobolev function. Note that in this context does not necessarily exist. The notion of line integral along a geodesic is more subtle, but if we consider a -Wasserstein geodesic that has bounded compression, then we can define where is a Kantorovich potential for , and is the density of . Since and are bounded, is well-defined if is -integrable. Note that in a smooth context , and therefore in smooth context we have
In the following we just write for . Also note, that in general there is no identification between -forms and vectorfields.
For we define the Boltzmann-Shanon entropy by
and otherwise. Given a number , we define the -Rény entropy functional with respect to by
where denotes the density of the absolutely continuous part in the Lebesgue decomposition of . In the case the -Rény entropy is . If is finite, then
Moreover, if is finite and , then is lower semi-continuous. If is -finite one has to assume an exponential growth condition [AGMR15] to guarantee lower semi continuity.
If there is a -form , we also define by
and otherwise. If , then .
For two numbers and we define
is the solution of the initial value problem
The modified distortion coefficients for number and are given by
3. Curvature-dimension condition for nonsymmetric diffusions
Let be a generalized smooth metric measure space, and let be an -integrable -form. We say satisfies the curvature-dimension condition for and if and only if for each pair there exists a dynamical optimal plan with
where . We call any such -form admissible.
If we replace by in the previous definition we say satisfies the reduced curvature-dimension condition .
Let be a metric measure space, and let be an -integrable -form in the sense of [Giga].
We say satisfies the curvature-dimension condition if and only if for each pair with bounded densities there exists a geodesic with bounded compression and a potential as in (2) such that
where and .
Equivalently, the map is -convex.
satisfies the entropic curvature-dimension condition if and only if for each pair with bounded densities there exists a geodesic with bounded compression and a potential as in (2) such that
where and . That is the map is -convex in the sense of [EKS15].
Definition 3.2 makes sense for any possibly non-smooth metric measure space.
But for simplicity, for the rest of the article we always assume that is a generalized smooth metric measure space.
Some of the statements that we prove for generalized smooth metric measure spaces extend to arbitrary metric measure spaces but in general not without additional assumptions.
Let and be generalized smooth metric measure spaces. A map is a smooth metric measure space isomorphism if is a metric measure space isomorphism, and if is a diffeomophism between the subsets of regular points and .
Let be a generalized smooth metric measure space, and let be an -intergrable -form. Assume satisfies the condition . Then the following properties hold.
For define the generalized smooth metric measures space . Then satisfies .
For a convex subset define the generalized smooth metric measure space . Then satisfies .
Let be a generalized smooth metric measure space, and let be a smooth metric measure space isomorphism. Then satisfies the condition .
We check (iii). We define on . If is a geodesic in , then is a geodesic in . The line integral of along is
Then, the statement follows like similar results for metric measure spaces that satisfy a curvature-dimension condition (for instance see [Stu06b]). ∎
Let be a generalized smooth metric measure space, an -integrable -form, and and . Then the following statements are equivalent:
For each pair there exists an optimal dynamical plan with such that
for and -a.e. . is the density of w.r.t. .
Moreover, the condition is equivalent with (ii) if the coefficients are replaced by the coefficients .
First, we observe that in the context of generalized smooth metric measure spaces up to a set measure zero optimal couplings between -absolutly continuous measures are unique
(also compare with the remark after Definition 2.1).
“(i)(ii)”: Let be with bounded support, and let be the optimal coupling between and . Let be an -stable generator of the Borel -field of . For each we define a disjoint covering of of sets by where and .
We define and set if . Then we consider the marginal measures and that are -absolutely continuous, and is the unique optimal coupling. Since geodesics are -almost surely unique, the dynamical optimal plan is the unique optimal dynamical coupling between its endpoints where . Therefore, satisfies the -inequality for every and . In particular, is -absolutely continuous. Then, we define a dynamical coupling between and by is optimal since
is an optimal coupling. Therefore, we can apply Lemma 3.11 in [EKS15]:
Since the measures for are mutually singular,
are mutually singular as well.
Now, for we consider the measure . Since it decomposes into mutually singular, absolutely continuous measures with densities , is absolutely continuous as well, and by mutual singularity of the measure its density is . Again, since geodesics are -almost surely unique we have that , and for every . From the -inequality for we have
This holds for every and . Since and are mutually disjoint, by summing up the previous inequality holds for and as well. Since the family is a generator for the -field, we have for -almost every