We develop a global differential calculus on the $L^2$-Wasserstein space over a closed Riemannian manifold, based on derivations of cylinder functions rather than on the standard pointwise approach. Within this framework we define some fundamental geometric tools. In particular, we show that the Levi-Civita connection on the base manifold lifts to the unique torsion-free connection compatible with the'extended'Otto metric. The corresponding Riemann tensor is exactly the lift of the base Riemann tensor, which shows that - in this framework - the correction terms of the classical gradient formalism are not intrinsic curvature terms, but arise from the projection onto the measure-dependent gradient distribution. This allows us to revisit with a global and purely differential approach the smooth computations by J. Lott, Comm. Math. Phys., 277(2):423-437, 2007, reaching partially different conclusions.
In this paper, we introduce a hybrid metric geometry on the space of absolutely continuous probability densities that combines optimal transport (Wasserstein geometry) and log-ratio composition (Aitchison geometry). The hybrid distance $D_\alpha$ is defined through a Benamou--Brenier-type dynamical formulation that couples spatial transport with a centered reaction term preserving total mass.We prove that $D_\alpha$ is a genuine metric and establish comparison estimates with the Wasserstein and Aitchison distances. In particular, we show that the topology induced by $D_\alpha$ is stronger than the narrow topology and weaker than the supremum topology generated by the Wasserstein and Aitchison metrics. We further prove that the metric space is geodesic. Within this framework, we develop the foundations of a gradient flow theory in the sense of Ambrosio--Gigli--Savar\'e, including the characterization of absolutely continuous curves, metric derivatives, metric slopes, and formal Jordan--Kinderlehrer--Otto schemes. We also investigate hybrid barycenters and their connections with Wasserstein barycenters and Aitchison barycenters. Finally, we discuss several partial differential equations, including logistic diffusion, Allen--Cahn equations with log-ratio constraints, and chemotaxis models with logarithmic growth, as formal gradient flows associated with the hybrid geometry, and compare the proposed framework with the Wasserstein--Fisher--Rao metric.
A. Soglo, K. W. Houédanou, Jamal Adetola et al.· 0 citations
We develop an intrinsic calculus for smooth functions of paths of arbitrarily low regularity on smooth manifolds. The regularity of paths is defined in terms of a $p$-th variation tensor along a sequence of partitions, for an arbitrary integer $p$; this tensor is constructed as a local symmetric tensor measure along the path. We define pathwise integrals of closed one-forms along paths with finite $p$-th variation and derive a change of variable formula for smooth functions of such paths. For $p=2$, our results give a manifold version of H. F\"ollmer's pathwise It\^o calculus. Our construction only requires an affine connection on the manifold and may be viewed as a higher-order analogue of L. Schwartz's second-order differential geometry. The connection provides a splitting of higher-order tangent vectors into symmetric tensor components and leads to a geometric transfer principle: the change-of-variable formula defines an intrinsic, connection-independent functional of the reduced $p$-jet of the test function, whose canonical highest-order component is determined by the $p$-th variation tensor. Although our results are purely geometric, they apply to manifold-valued stochastic processes with highly irregular paths and yield a higher-order It\^o-type calculus for such processes. We illustrate this calculus for exponential lifts of fractional Brownian motions to Riemannian manifolds and Lie groups.
We present a systematic study on the role of invariant connections in geometric mechanics. We first develop a comprehensive theory of reduction under left- and right-invariant connections on Lie groups, showing that the Euler--Poincar\'e and Lie--Poisson equations are independent of the connection. We then introduce a novel connection-dependent variational principle, where the Lagrangian depends on the velocity parallel-transported back to the initial point of the curve. For Cartan--Schouten connections, this leads to an integro-differential Euler--Poincar\'e equation that exhibits two distinct sources of nonlocality in time: a path-dependent term encoded in the parallel transport and a future-dependent term arising from a curvature integral. We reformulate this equation as a two-point boundary value problem and present a two-level numerical scheme. The general theory is illustrated on the Heisenberg group, where the equations simplify due to nilpotency, and on the rotation group, where the full integro-differential structure is retained.
Several inequivalent notions of tangent space exist for diffeological spaces, which has hindered the development of Riemannian geometry in this setting. On Blohmann's elastic diffeological spaces, the tangent functor carries a genuine tangent structure in the sense of Cockett and Cruttwell, and we use this to develop Riemannian geometry. We introduce connections adapted to this setting and show that they induce covariant derivatives on vector fields satisfying the usual axioms, together with the associated curvature tensor, whose basic properties we record following Cockett--Cruttwell. Given a Riemannian metric, we establish a Koszul formula and prove that the Levi-Civita covariant derivative is unique if it exists, while its existence remains open in general. We then develop the theory of geodesics and of the energy functional: geodesics are always critical paths of the energy functional, and on elastic spaces for which the real line is a curve object, completeness of the geodesic spray yields existence and uniqueness of geodesics, as well as the converse statement that critical paths are geodesics. As a fundamental class of examples, we show that for a closed manifold $M$ and an elastic Riemannian diffeological space $N$ these structures on the mapping space $C^\infty(M,N)$ are obtained pointwise from those on $N$, under a natural tangent-commuting hypothesis that holds automatically when $N$ is a manifold; in particular, the geodesic spray of $C^\infty(M,N)$ is complete whenever that of $N$ is.
In this article, we consider discontinuous martingales on tangent bundles over submanifolds of Euclidean space. First, we introduce a connection rule on tangent bundles and establish the It\^o calculus for discontinuous semimartingales on tangent bundles. Then we focus on harmonic maps with respect to non-local Dirichlet forms and show that the derivative of harmonic maps along infinitesimal symmetries induces discontinuous martingales on tangent bundles. This process may be viewed as a stochastic Jacobi field along the image martingale. We also introduce the stochastic parallel transport of tangent vectors along c\`adl\`ag semimartingales on Riemannian submanifolds with projected jumps. Using the parallel transport, we obtain the mean-value property for the differential of harmonic maps involving a jump part expressed through the second fundamental form. We also obtain a derivative formula for harmonic maps for isotropic L\'evy processes with a Brownian component on compact Riemannian manifolds.
We formulate the inverse problem in information geometry within a two-point tensorial framework and solve it for general metric-affine manifolds, without imposing any curvature or torsion constraints. The construction is explicit and starts directly from the given geometric data: the metric tensor is paired to the affine structure, realized through a local parallelism obtained from parallel transport. This yields a contrast bi-form inducing the original metric-affine manifold. The inverse problems for statistical manifolds admitting torsion and for statistical manifolds are then recovered by homotopical reduction. In this way, suitable pre-contrast and contrast functions are obtained, including several established constructions for statistical manifolds and SMATs. We apply the general theory to reductive homogeneous pseudo-Riemannian manifolds endowed with invariant affine connections. Particular attention is devoted to semisimple Lie groups with Cartan-Schouten connections and to odd-dimensional spheres equipped with Berger metrics.
Florio M. Ciaglia, Giuseppe Marmo, M. Pacelli et al.· 0 citations
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