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Optimal transport in competition with reaction: the Hellinger-Kantorovich distance and geodesic curves
Matthias Liero, Alexander Mielke, Giuseppe Savaré
TL;DR
The paper asks how to develop a metric theory for reaction–diffusion systems with both transport and mass change. It defines and characterizes the Hellinger–Kantorovich distance through an Onsager formulation and cone-space optimal transport, then constructs geodesics and establishes their main properties. The resulting geometry switches from transport to creation or annihilation beyond a distance threshold, with uniqueness and semiconcavity depending on the setting.
Problem
The paper asks whether abstract metric theory can be extended to reaction–diffusion systems whose dynamics combine diffusion with mass generation and absorption.
Method
It defines a distance from an Onsager operator combining Wasserstein diffusion and creation–annihilation, and equivalently characterizes it by Wasserstein transport between lifts to a cone space.
Results
For Dirac masses, creation and annihilation are shorter than transport when separation exceeds π/2; geodesics are unique with an absolutely continuous endpoint, but HK is not semiconcave in dimensions at least two.
Takeaways & Limitations
The Hellinger–Kantorovich geometry provides a framework for studying finite nonnegative measures and subsequent gradient-system formulations of scalar reaction–diffusion equations.
Takeaways & Limitations
The paper leaves the characterization of all geodesics and the nonbranching question for future work.
Abstract
from arXiv · showhide
We discuss a new notion of distance on the space of finite and nonnegative measures which can be seen as a generalization of the well-known Kantorovich-Wasserstein distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and its properties. In fact the distance can be equivalently described by an optimal transport problem on the cone space over the underlying metric space. We give a construction of geodesic curves and discuss their properties.
1 Introduction
The paper develops the Hellinger–Kantorovich distance as a metric combining Wasserstein transport with mass creation and annihilation. It characterizes this distance dynamically and geometrically, constructs geodesics through cone-space lifts, and analyzes their properties and limitations.
- Motivation and construction: The Onsager operator combines a Wasserstein diffusion part with a creation–annihilation part, generating a distance on finite nonnegative measures rather than only probability measures.The resulting distance is formulated through a generalized Benamou–Brenier minimization over curves connecting measures.
- Motivation and construction: The Hellinger–Kantorovich distance balances transport against mass creation or annihilation, with transport never occurring beyond distance π√(α/β).The paper specializes much of the introduction to α = 1 and β = 4.
- Geometric characterization: The distance is equivalently characterized by lifting measures to a cone space and minimizing a Wasserstein distance between optimal lifts.This cone formulation directly yields geodesic structure, and the paper proves its equality with the dynamical distance D1,4.
- Geodesic examples: For Dirac masses, transport is optimal up to separation π/2, whereas beyond π/2 creation and annihilation produce a shorter curve with squared length a0 + a1.One-mass transport minimizers exist only below separation π; at larger distances the relevant value can be only an infimum.
- Geodesic examples: The construction yields explicit geodesics whose total mass is simultaneously 2-convex and 2-concave, while critical Dirac configurations can admit infinitely many geodesics.The paper also describes radial transport with partial annihilation for nearby mass and pure Hellinger annihilation for farther mass.
- Geodesic properties: The paper establishes geodesic Λ-convexity criteria for linear functionals by transferring convexity to the cone geometry.It also identifies Hellinger geodesics and prepares a metric theory for scalar reaction–diffusion equations.
- Geodesic properties: Geodesics are unique when one endpoint is absolutely continuous, but the distance is not semiconcave in dimensions two or higher.These properties distinguish the Hellinger–Kantorovich geometry from the Wasserstein setting.
2 Gradient structures for reaction-diffusion equations
This section formulates reaction-diffusion equations as gradient systems with Onsager operators and derives their induced dissipation distances. It specializes the framework to the Hellinger–Kantorovich distance, defined for finite nonnegative measures and connecting transport with reaction.
- Gradient systems: Reaction-diffusion systems are represented by a state space, driving functional, and Onsager operator, with the operator mapping thermodynamic forces to rates.The Onsager operator is symmetric and positive semidefinite; its inverse defines a metric tensor and an associated geodesic distance.
- Reaction-diffusion equations: For reaction-diffusion systems, the Onsager structure combines mobility-driven diffusion with a reaction matrix, producing diffusion and reaction terms in the gradient-flow equation.The resulting equation uses D(c)=M(c)D^2F(c) for diffusion and R(c)=H(c)DF(c) for reaction.
- Dissipation distances: The induced dissipation distance is characterized by minimizing an action over sufficiently smooth curves connecting the endpoint measures.A dual formulation uses the variable ξ(s)=K(c(s)) ċ(s), avoiding direct inversion of the Onsager operator.
- Hellinger–Kantorovich specialization: The Hellinger–Kantorovich distance arises from the scalar operator Kα,β(c)ξ=−div(αc∇ξ)+βcξ and is defined on all finite nonnegative measures.For β>0, geodesics between distinct probability measures can have mass below one at every interior parameter.
- Limiting cases: The limiting cases recover scaled Hellinger distance when α=0 and the transport-only case when β=0.In the Hellinger case, geodesics interpolate square roots of densities.
3 The Hellinger–Kantorovich distance
This section characterizes the Hellinger–Kantorovich distance through its dynamic action formulation and scaling properties. The analysis uses the continuity equation with transport and reaction fields and reduces general parameters by rescaling.
- Dynamic formulation: The distance Dα,β is equivalently expressed through a continuity equation coupling a vector transport field Ξ with a scalar reaction field ξ.The vector field represents transport, while the scalar field accounts for mass generation or annihilation.
- Parameter scaling: The parameters satisfy the scaling relation Dα,β(µ0,µ1)=Dα/β,1(µ0,µ1)/√β, allowing analysis to focus on β=1 or the convenient choice α=1, β=4.The paper primarily studies α=1 and β=4, while recovering other positive parameters by scaling.
- Transport limit: For β=0 and α=1, the coupled formulation reduces to the classical Benamou–Brenier transport case.A full justification of the coupled geodesic system is cited for the reaction-diffusion setting.
3.1 The optimal curves for Dα,β with one or two mass-points
This section analyzes geodesics represented by moving or splitting mass points and identifies when such paths are optimal. It shows that transport and reaction trade off according to distance, with a threshold at π in the one-point problem.
- One-mass-point curves: Restricting competitors to curves γx,a(s)=a(s)δx(s) always gives an upper bound, and becomes exact up to a distance threshold where the path is geodesic.The paper states that this restricted minimizer is a geodesic up to a threshold in the Euclidean endpoint distance.
- One-mass-point curves: The mass-point path is parameterized by x(s)=(1−ρ(s))x0+ρ(s)x1, so its speed is |ẋ(s)|=ρ̇(s)L while a(s) controls mass variation.The action separates transport through ρ̇ and reaction through ȧ.
- One-mass-point curves: For the one-mass-point problem, a minimizer exists for L<π, whereas for L≥π minimizing sequences concentrate transport at an intermediate parameter.For L≥π, the amplitude approaches a quadratic profile with zero mass at the concentration point and ρ̇ converges to a Dirac mass.
- Two-mass-point curves: For two separated mass points, splitting mass can outperform a single moving mass point when endpoint distance lies between π/2 and π.The paper reports that the one-mass-point estimate is sharp if and only if the distance is at least π/2 in the stated comparison.
- Geometric interpretation: The optimal curves balance transport against creation or annihilation, and transport never occurs over distances longer than π.This trade-off is the defining geometric feature of the Hellinger–Kantorovich distance.
3.2 Optimal transport on the cone
The cone construction embeds mass-location pairs into a geodesic cone, where optimal transport can be analyzed through a Wasserstein distance with a reservoir at the cone tip.
- The cone CΩ identifies all zero-radius points with a tip o and represents positive-radius points as [x, r].
- For interval domains, the cone is visualized as a Euclidean sector, with geodesics switching to rays through the tip when angular separation reaches π.
- The cone distance is geodesic, and its explicit interpolator supplies constant-speed geodesics between cone points.
- Lifting measures to the cone and minimizing the associated Wasserstein distance yields the Hellinger–Kantorovich transport formulation.
- With a sufficiently large reservoir, optimal plans avoid true transport beyond π/2, while the optimal cost stabilizes once the reservoir exceeds κ∗.
3.3 The Hellinger–Kantorovich distance
The Hellinger–Kantorovich distance is defined by lifting measures to the cone and optimizing cone transport, equivalently capturing transport and mass creation or annihilation.
- HK is defined by lifting measures to cone measures and minimizing the Wasserstein distance induced by the cone metric.
- HK has optimal lifts, satisfies HK(µ0, µ1)^2 ≤ µ0(Ω) + µ1(Ω), and induces the weak topology on finite nonnegative measures.
- For Dirac masses separated by more than π/2, the optimal plan uses pure Hellinger reaction; at π/2, transport and reaction combine through a convex family of optimal plans.
- The logarithmic-entropy transport functional provides an equivalent minimization characterization with an attained calibration measure.
- The distance interpolates between Wasserstein transport as β→0 and the Hellinger distance when α=0.
3.4 Geodesic curves induced by optimal transport plans
HK geodesics are obtained by projecting cone geodesics associated with optimal lifts and transport plans, but they need not be unique.
- An optimal cone plan generates an HK geodesic by pushing it through the cone geodesic interpolator and projecting the resulting measures.
- Nonuniqueness of optimal transport plans can produce different HK geodesics, including infinitely many in symmetric examples.
- At separation π/2, both a moving single-mass curve and a split endpoint-mass curve are geodesics, with further geodesic multiplicity possible.
- The resulting curve is a constant-speed geodesic for the Hellinger–Kantorovich distance.
4 Equivalence to the dynamical formulation
The paper proves that the cone-transport and dynamical Onsager formulations of HK are equivalent, and characterizes absolutely continuous curves through dynamic plans and continuity equations.
- The central equivalence is HK(µ0, µ1) = D1,4(µ0, µ1), identifying cone transport with the Onsager dynamical distance.
- The dynamical formulation uses a modified continuity equation with vector and scalar fields whose L2(µ)-norms control the HK metric derivative.
- Characteristic-based representations require regularity; without it, uniqueness of characteristics and the explicit formula are not guaranteed.
- Every HK-absolutely-continuous curve admits a dynamic plan whose projected cone marginals represent the curve and attain the metric derivative.
- All HK geodesics arise by projecting geodesics in the cone Wasserstein space associated with optimal lifts and plans.
5 Geodesic curves for HK
The section characterizes Hellinger–Kantorovich geodesics through cone-space lifts and examines their convexity, uniqueness, multiplicity, and curvature properties. It shows that geodesic behavior depends sharply on transport distance, absolute continuity, and the dimension of the domain.
- 5.1 Geodesic convexity: The total mass along a geodesic is a convex quadratic function of the interpolation parameter, with m′′(s) = 2HK(µ0, µ1)^2 ≥ 0.This follows from the cone-space representation and the quadratic structure used to compute the mass.
- 5.1 Geodesic convexity: Linear functionals are Λ-convex along HK geodesics exactly when their cone-lifted integrands [x, r] 7→r2Φ(x) are Λ-convex on the cone.The characterization uses optimal cone-space plans and applies the convexity of the lifted functional.
- 5.2 Geodesic connections for two Dirac measures: At the critical distance |y0−y1| = π/2, an infinite-dimensional convex set of geodesics connects two Dirac measures.This multiplicity arises from many optimal lifts to the cone space.
- 5.5 Towards a characterization of all geodesic connecting two measures: Every optimal cone-space plan generates a geodesic, while distinct plans may project to the same measure-valued geodesic.The set of optimal plans is convex, but projection can remove distinctions between cone-space geodesics.
- 5.5 Towards a characterization of all geodesic connecting two measures: If one endpoint is absolutely continuous with respect to Lebesgue measure, the connecting Hellinger–Kantorovich geodesic and its normalized optimal plan are unique.The operator N eliminates redundancies in the optimal representation.
- 5.6 Failure of positive curvature: HK is not K-semiconcave for any finite K in dimensions two or higher, contrasting with the one-dimensional PC-space result.The paper also states that HK is not a positively curved Alexandrov space when the domain is at least two-dimensional.