Hellinger-Kantorovich (aka WFR) spaces and gradient flows Alexander Mielke Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin Institut für Mathematik, Humboldt-Universität zu Berlin www.wias-berlin.de/people/mielke/ Workshop on Optimal Transport: from Theory to Applications Interfacing Dynamical Systems, Optimization, and Machine Learning Humboldt-Universität zu Berlin, 11.–15. März 2024 Research partially supported by Berlin Mathematics Research Center

Joint work with Vaios Laschos Matthias Liero Giuseppe Savaré Jia-Jie Zhu • M-Zhu: Approximation, Kernelization, and Entropy-Dissipation of Gradient Flows: from Wasserstein to Fisher-Rao. In prep. (2024). https://jj-zhu.github.io/ • Laschos-M: EVI on the HK and the SHK spaces. arXiv:2207.09815v3 (2023). • Liero-M-Savaré: Fine prop. of geodesics and geodesic λ-convexity for the HK distance. Arch Rat Mech Analysis (2023). • Laschos-M: Geometric properties of cones with applications on the HK space, new distance prob. J Funct Analysis (2019) • Liero-M-Savaré:. Optimal entropy-transport problems and a new HK distance. Inventiones mathematicae (2018) A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 2 (27)

Overview 1. Introduction: unbalanced transport 2. “Hellinger” versus “Fisher-Rao” 3. Geodesics for the Hellinger-Kantorovich distance 4. Geodesic Λ-convexity on (M(X), H Kα,β) 5. Reaction-diffusion equations as gradient flows 6. EVIλ flows in (M(X), H Kα,β) 7. EVIλ flows in (P(X), SH Kα,β) A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024

1. Introduction: unbalanced transport Sequence of images or data sets: distinguish between motion/transport growhth/shape change birth/death of new objects A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 3 (27)

1. Introduction: unbalanced transport Sequence of images or data sets: distinguish between motion/transport growhth/shape change birth/death of new objects Cloud motion shows all of this: A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 3 (27)

1. Introduction: unbalanced transport Application closer to my research: reaction-diffusion system Hydrogen production and distribution transport = diffusion (in a pipeline) growth = chemical reactions 2 H2 0 2 H2 + 1 O2 . H2 O water from Ocean to Sahara ∂t ρH2O = div σ(∇ρH2O + ρH2O ∇V ) use solar power to produce hydrogen H2   ˙ ρH2O ˙ ρH2 ˙ ρO2   = κf (x)ρ2 H2O − κb(x)ρ2 H2 ρO2   −2 2 1   transport hydrogen to big cities ∂t ρH2 = div σ(∇ρH2 + ρH2 ∇V ) A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 4 (27)

1. Introduction: unbalanced transport Liero-M’13: certain reaction-diffusion systems have a gradient structure vector of densities ρ = (ρ1, ..., ρn) ∈ [0, ∞[n Boltzmann entropy E(ρ) = X n i=1 λB ρi πi(x) πi(x) dx λB (r) = r log r − r + 1 ∂tρ = D∆ρ + R(ρ) = −KRDS(ρ)DE(ρ) Onsager operator (= inv. Riem. tensor) KRDS(ρ) = Kdiff(ρ) + Kreact(ρ) diffusion via Otto transport KOtto(ρ)ξ = − div(ρ∇ξ) growth (creation or annihilation) via reaction A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 5 (27)

1. Introduction: unbalanced transport Liero-M’13: certain reaction-diffusion systems have a gradient structure vector of densities ρ = (ρ1, ..., ρn) ∈ [0, ∞[n Boltzmann entropy E(ρ) = X n i=1 λB ρi πi(x) πi(x) dx λB (r) = r log r − r + 1 ∂tρ = D∆ρ + R(ρ) = −KRDS(ρ)DE(ρ) Onsager operator (= inv. Riem. tensor) KRDS(ρ) = Kdiff(ρ) + Kreact(ρ) diffusion via Otto transport KOtto(ρ)ξ = − div(ρ∇ξ) growth (creation or annihilation) via reaction Today, only the scalar case n = 1: Kα,β(ρ)ξ = α KOtto(ρ)ξ for diffusion + β Kreact(ρ)ξ for growth = −α div(ρ∇ξ) + βρξ A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 5 (27)

Overview 1. Introduction: unbalanced transport 2. “Hellinger” versus “Fisher-Rao” 3. Geodesics for the Hellinger-Kantorovich distance 4. Geodesic Λ-convexity on (M(X), H Kα,β) 5. Reaction-diffusion equations as gradient flows 6. EVIλ flows in (M(X), H Kα,β) 7. EVIλ flows in (P(X), SH Kα,β) A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024

2. “Hellinger” versus “Fisher-Rao” The Hellinger-Kantorovich distance was introduced independently and named Wasserstein-Fisher-Rao distance WFR(µ0, µ1) French group: Chizat, Peyré, Schmitzer, Vialard: An interpolating distance be- tween optimal transport and Fisher–Rao metrics. Found Comput Math (2015) Chizat, Peyré, Schmitzer, Vialard: Unbalanced optimal transport: geometry and Kantorovich formulation. J Funct Analysis (2018) Portugese group: Kondratyev, Monsaingeon, Vorotnikov: A new optimal transport dis- tance on the space of finite Radon measures. Adv Diff Eqns (2016). Gallouët, Monsaingeon: A JKO splitting scheme for Kantovorich-Fisher-Rao gradient flows. SIAM Math Analysis (2017) A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 6 (27)

2. “Hellinger” versus “Fisher-Rao” Hellinger: dissertation [Hell1907] and habilitation [Hell1909] introduce integrals of the type b a u(t)df1df2 dg for functions u, f1, f2, g, g ∈ C0([a, b]) where additionally g and h are increasing and (dfj)2 ≤ dg dg. Kakutani in [Kak’48] uses the Radon-Nikodým derivative, to define the so-called Hellinger integral h(ν0, ν1) = Ω ν0(dω)ν1(dω) and defines what is nowadays called the Hellinger distance He(ν0, ν1) = 2−2h(ν0, ν1) 1/2 = Ω dν0 dg − dν1 dg 2 dg for µ, ν ∈ P(Ω). With this, (M(Ω), He) is a geodesic metric space, but not (P(Ω), He). • 1955→ “Hellinger distance” is consistently used (MathSciNet 630 hits) • [Rao’63] explicitly introduces the Hellinger integral and distance A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 7 (27)

2. “Hellinger” versus “Fisher-Rao” Fisher-Rao distance originates from [Rao’45] using the “Fisher information metric” [Fis1921] (finite-dimensional) parameter manifold Θ and function p : Θ → P(Ω) defines parametrized family µ = p(·, θ) ∈ P(Ω) θ ⊂ Θ Fisher’s information matrix = Riemannian metric tensor on TΘ: GFi(θ)v1, v2 = Ω Dθ(log p(ω, θ) [v1] Dθ(log p(ω, θ) [v2] p(dω, θ), for v1, v2 ∈ TθΘ. [Rao’45] defines the so-called Fisher-Rao distance as the geodesic distance dFR : Θ×Θ → [0, ∞[ induced by the Riemannian tensor GFi . A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 8 (27)

2. “Hellinger” versus “Fisher-Rao” Taking Θ = P(Ω) itself with µ = p(µ) one obtains indeed, as a special case of Fisher’s information metric, the infinitesimal metric GHe(µ)v1, v2 = Ω dv1 dµ dv2 dµ dµ = Ω dv1dv2 dµ (as in [Hell1905]) The induced geodesic distance on M(Ω) is given by 1 2 He(µ0, µ1) The induced geodesic distance on P(Ω) is Bhattacharya’s distance [Bha35] Bh(ν0, ν1) = 2 arcsin 1 2 He(ν0, ν1) ν∈P(Ω) ν √ ν √ ν∈Sn √ ν0 √ ν1 He(ν0, ν1) Bh(ν0, ν1) A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 9 (27)

Overview 1. Introduction: unbalanced transport 2. “Hellinger” versus “Fisher-Rao” 3. Geodesics for the Hellinger-Kantorovich distance 4. Geodesic Λ-convexity on (M(X), H Kα,β) 5. Reaction-diffusion equations as gradient flows 6. EVIλ flows in (M(X), H Kα,β) 7. EVIλ flows in (P(X), SH Kα,β) A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024

3. Geodesics for the Hellinger-Kantorovich distance α ≥ 0 and β ≥ 0 interpolate between pure Hellinger distance He = H K0,4 (i.e. (α, β) = (0, 4)) pure Kantorovich-Wasserstein distance W = H K1,0 (i.e. (α, β) = (1, 0)) More general, we have H Kα,0 = 1 √ α W and H K0,β = 2 √ β He. A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 10 (27)

3. Geodesics for the Hellinger-Kantorovich distance α ≥ 0 and β ≥ 0 interpolate between pure Hellinger distance He = H K0,4 (i.e. (α, β) = (0, 4)) pure Kantorovich-Wasserstein distance W = H K1,0 (i.e. (α, β) = (1, 0)) More general, we have H Kα,0 = 1 √ α W and H K0,β = 2 √ β He. In general, H Kα,β is defined via the dynamic Benamou-Brenier formulation (for RDS in Liero-M’13) H Kα,β(µ0, µ1)2 := inf 1 0 X α|∇ξs|2+βξ2 s µs(dx) ds µ0 = µ0, µ1 = µ1, s → (µs, ξs) solves (gCE) α,β , where (gCE) α,β ∂sµs + div α∇ξsµs = βξsµs in the sense of distributions. A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 10 (27)

3. Geodesics for the Hellinger-Kantorovich distance Any minimizer s → µs is called a (constant-speed) geodesic. Equations for geodesics (justified in [LiMiSa18OETP,LiMiSa23FPGG]) ∂sµs + div α∇ξsµs = βξsµs, ∂sξs + α 2 |∇ξs|2 + β 2 ξ2 s = 0. A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 11 (27)

3. Geodesics for the Hellinger-Kantorovich distance Any minimizer s → µs is called a (constant-speed) geodesic. Equations for geodesics (justified in [LiMiSa18OETP,LiMiSa23FPGG]) ∂sµs + div α∇ξsµs = βξsµs, ∂sξs + α 2 |∇ξs|2 + β 2 ξ2 s = 0. The second equation gives ξt = Pα,β t−s ξs) for 0 < s < t < 1, where Pα,β τ η (x) = inf 2 βτ 1 − cos β/(4α)|x−y| 1 + τ β/2 η(y) |x−y| ≤ π α β . One can check that the formal limits α → 0 and β → 0 recover the classical cases corresponding to H Kα,0 = 1 √ α W and H K0,β = 2 √ β He. A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 11 (27)

3. Geodesics for the Hellinger-Kantorovich distance Remarkable fact: transport only occurs over distances ≤ ∗ = π α/β In general, geodesic curves between µ0, µ1 ∈ M(X) consists of three pieces: • A0 : mass of µ0 located further from sppt(µ1) than ∗ is annihilated, • A1 : mass of µ1 located further from sppt(µ0) than ∗ is created, • Atra : mass of µ0 located less than ∗ from sppt(µ1) is transported but again with changing mass along the way. Transport over a distance of exactly ∗ is more difficult, in particular non-uniqueness of geodesics may happen. sppt µ1 A1 sppt µ0 A0 ∗ ∗ Atra A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 12 (27)

3. Geodesics for the Hellinger-Kantorovich distance sppt µ1 A1 sppt µ0 A0 ∗ ∗ Atra A0 = x ∈ sppt µ0 dist(x, sppt µ1) > ∗ A1 = x ∈ sppt µ1 dist(x, sppt µ0) > ∗ Atra = x ∈ sppt µ0 dist(x, sppt µ1) < ∗ geodesic: µs = (1−s)2µ0|A0 + Ts # q2 s µ0|Atra + s2µ1|A1 with growth factor q2 s (x) = 1+s β 4 ξ0(x) 2 + αβ 4 s2 |∇ξ0(x)|2 and transport map Ts(x) = x + 4α β arctan s √ αβ 2+sβξ0(x) ∇ξ0(x) . Observe that • β = 0 gives qs ≡ 1 and Ts(x) = x + αs∇ξ0(x) and • α = 0 gives qs = (1+sβξ0/4)2 and Ts(x) = x A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 13 (27)

Overview 1. Introduction: unbalanced transport 2. “Hellinger” versus “Fisher-Rao” 3. Geodesics for the Hellinger-Kantorovich distance 4. Geodesic Λ-convexity on (M(X), H Kα,β) 5. Reaction-diffusion equations as gradient flows 6. EVIλ flows in (M(X), H Kα,β) 7. EVIλ flows in (P(X), SH Kα,β) A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024

4. Geodesic Λ-convexity on (M(X), H Kα,β ) E : M(X) → R∞ is geod. Λ-convex, where Λ ∈ R, iff along cs-geodesics E(µs) ≤ (1−s)E(µ0) + s E(µ1) − Λ s−s2 2 H Kα,β(µ0, µ1)2 for s ∈ [0, 1]. The total mass functional is exact quadratic µs(X) = (1−s)µ0(X) + s µ1(X) − β 2 s−s2 2 H Kα,β(µ0, µ1)2, i.e. µ → aµ(X) is (aβ/2)-convex for all a ∈ R. Linear functional µ → X V (x)µ(dx) are geod. Λ-cvx iff (cf. [LiMSa16]) αD2V (x)+β 2 V (x) αβ/4 ∇V (x) αβ/4 ∇V (x) β 2 V (x) ≥ Λ Id+1 in R(d+1)×(d+1). A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 14 (27)

4. Geodesic Λ-convexity on (M(X), H Kα,β ) Internal energies E(µ) = X E(ρ) dx + E∞ µ⊥ with µ = ρL + µ⊥, L ⊥ µ⊥ Theorem (Necessary & sufficient conditions for αβ > 0) [LiMSa23] E defined by E : [0, ∞[ → R∞ is geodesic Λ-convex on M(X), H Kα,β) iff the auxiliary function NE(γ, δ) = (δ/γ)dE γ2+d/δd satisfies (a) (γ, δ) → NE(γ, δ) − 2Λ β γ2 is convex, (b) δ → (d−1) NE(γ, δ) is non-decreasing. A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 15 (27)

4. Geodesic Λ-convexity on (M(X), H Kα,β ) Internal energies E(µ) = X E(ρ) dx + E∞ µ⊥ with µ = ρL + µ⊥, L ⊥ µ⊥ Theorem (Necessary & sufficient conditions for αβ > 0) [LiMSa23] E defined by E : [0, ∞[ → R∞ is geodesic Λ-convex on M(X), H Kα,β) iff the auxiliary function NE(γ, δ) = (δ/γ)dE γ2+d/δd satisfies (a) (γ, δ) → NE(γ, δ) − 2Λ β γ2 is convex, (b) δ → (d−1) NE(γ, δ) is non-decreasing. • Reducing the conditions to γ ≡ 1, one obtains McCann’s condition for geodesic convexity in (M(X), W). • Reducing to δ ≡ 1 gives the conditions for geodesic Hellinger convexity. • However, the joint condition is strictly stronger, see below. A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 15 (27)

4. Geodesic Λ-convexity on (M(X), H Kα,β ) Internal energies E(µ) = X E(ρ) dx + E∞ µ⊥ with µ = ρL + µ⊥, L ⊥ µ⊥ Theorem (Necessary & sufficient conditions for αβ > 0) [LiMSa23] E defined by E : [0, ∞[ → R∞ is geodesic Λ-convex on M(X), H Kα,β) iff the auxiliary function NE(γ, δ) = (δ/γ)dE γ2+d/δd satisfies (a) (γ, δ) → NE(γ, δ) − 2Λ β γ2 is convex, (b) δ → (d−1) NE(γ, δ) is non-decreasing. • Multiples of the mass functional Ma(ρ) = aρ give NMa (γ, δ) = aγ2. Hence, adding Ma shifts Λ-convexity exactly by aβ/2. A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 15 (27)

4. Geodesic Λ-convexity on (M(X), H Kα,β ) Internal energies E(µ) = X E(ρ) dx + E∞ µ⊥ with µ = ρL + µ⊥, L ⊥ µ⊥ Theorem (Necessary & sufficient conditions for αβ > 0) [LiMSa23] E defined by E : [0, ∞[ → R∞ is geodesic Λ-convex on M(X), H Kα,β) iff the auxiliary function NE(γ, δ) = (δ/γ)dE γ2+d/δd satisfies (a) (γ, δ) → NE(γ, δ) − 2Λ β γ2 is convex, (b) δ → (d−1) NE(γ, δ) is non-decreasing. Examples for α > 0 and β > 0: E(ρ) = ρm with m > 1 giving Λ = 0 and E(ρ) = aρ gives Λ = aβ/2 Boltzmann entropy E(ρ) = ρ log ρ not geod. Λ convex for any Λ ∈ R. E(ρ) = −ρθ is geod. convex (with Λ = 0) iff θ ∈ d/(d+2), 1/2 . restriction to d = 1 or 2. McCann’s condition needs θ ≥ (d−1)/d, Hellinger convexity needs θ ≤ 1/2. A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 15 (27)

4. Geodesic Λ-convexity on (M(X), H Kα,β ) Main idea of proof: establish enough regularity of the geodesic curves reduce to the pure transport part reduce to the case with Lebesgue densities rewrite transport part of the geodesics in the form µs = ρs(x)dx with ρs(Ts(x)) = ρ0(x)qs(x)2/ det DTs(x) Then, set γs = ρ0(x)1/2qs and δs = ρ0(x)1/2qs(det DTs(x))1/d to obtain E(µs )= Ts(Atra) E(ρs (y)) dy = Atra E ρ0 (x)qs (x)2 det DTs (x) det DTs (x) dx = Atra NE γs (x), δs (x) dx. Using prop. (a) and (b) of NE and explicit represent. of qs and Ts , the result follows via some lengthy and nontrivial computations. A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 16 (27)

Overview 1. Introduction: unbalanced transport 2. “Hellinger” versus “Fisher-Rao” 3. Geodesics for the Hellinger-Kantorovich distance 4. Geodesic Λ-convexity on (M(X), H Kα,β) 5. Reaction-diffusion equations as gradient flows 6. EVIλ flows in (M(X), H Kα,β) 7. EVIλ flows in (P(X), SH Kα,β) A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024

5. Reaction-diffusion equations as gradient flows Onsager operators for H Kα,β on M(X) and SH Kα,β on P(X) are given by K α,β H K (µ)ξ = −α div(µ∇ξ) + βµξ K α,β SH K (ν)ξ = −α div(ν∇ξ) + βν ξ− X ξ dν [keeps mass constraint X dν = 1 ] For functionals E(µ) = X E(ρ)−ρV dx the induced reaction-diffusion equations take the form ∂tρ = α div ρ∇(E (ρ)−V ) − βρ (E (ρ)−V ) + δSH Kβρ X ρ (E (ρ)−V ) dx, with δS H K = 0 for H K and δS H K = 1 for SH K. Existence of weak solutions: [GalMon17JKOS,GaLaMo19UOTS, DimChi20TGMH, Flei21MMAC, ...] A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 17 (27)

5. Reaction-diffusion equations as gradient flows Choose V ≡ 0 and E(ρ) = 2 3 − ρ1/2 + 1 3 ρ3/2 and initial datum µ(0) = 0. We obtain the PDE ∂tρ = ∆ 1 2 ρ1/2 + 1 6 ρ3/2 + 2 ρ1/2 − ρ3/2 in X, ∇ρ · n = 0 on ∂X. A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 18 (27)

5. Reaction-diffusion equations as gradient flows Choose V ≡ 0 and E(ρ) = 2 3 − ρ1/2 + 1 3 ρ3/2 and initial datum µ(0) = 0. We obtain the PDE ∂tρ = ∆ 1 2 ρ1/2 + 1 6 ρ3/2 + 2 ρ1/2 − ρ3/2 in X, ∇ρ · n = 0 on ∂X. It is easily seen that there are multiple solutions with µ(0) = 0, namely µζ(t) = 0 for t ∈ [0, T] and µζ(t) = tanh(t−ζ) 2 for t ≥ ζ. Thus, for weak solutions we indeed have non-uniqueness. A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 18 (27)

5. Reaction-diffusion equations as gradient flows Energy-dissipation estimates (ongoing joint work with JJ Zhu) • E0 = infρ∈M(X) E(ρ) = 0, using E(ρ) = ρ1/2+2 3 (ρ1/2−1)2 ≥ 0, E(1) = 0. • π ≡ 1 is the unique minimizer of E. − d dt E(ρ) = − X E (ρ)∂tρ dx = − E (ρ), Kα,β(ρ)E (ρ) = α X ρ ∇E (ρ)|2 ≥0 dx + β X ρ|E (ρ)|2 ≥3 8 E(ρ) dx ≥ 3β 8 E(ρ) A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 19 (27)

5. Reaction-diffusion equations as gradient flows Energy-dissipation estimates (ongoing joint work with JJ Zhu) • E0 = infρ∈M(X) E(ρ) = 0, using E(ρ) = ρ1/2+2 3 (ρ1/2−1)2 ≥ 0, E(1) = 0. • π ≡ 1 is the unique minimizer of E. − d dt E(ρ) = − X E (ρ)∂tρ dx = − E (ρ), Kα,β(ρ)E (ρ) = α X ρ ∇E (ρ)|2 ≥0 dx + β X ρ|E (ρ)|2 ≥3 8 E(ρ) dx ≥ 3β 8 E(ρ) We conclude E(ρ(t)) ≤ e−3βt/8E(ρ(0)) for all(???) ρ(0) ∈ M(X) This cannot hold for all weak solutions!! A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 19 (27)

5. Reaction-diffusion equations as gradient flows Up(r) = rp−pr+p−1 p(p−1) on M(X): p-divergences Dp(ρ µ) = X Up dρ dµ dµ U1(r) = r log r − r+1 U0(r) = r−1− log r cp = inf rUp (r)2 Up(r) r > 0 = 1/(1−p) for p ≤ 1/2, 0 for p > 1/2. Theorem [M-Zhu’24?] Assume p ≤ 1/2 and Dp(ρ0 π) < ∞. Then, all curves of maximal slope t → ρ(t) for teh grad.-flow equation ∂tρ = −Kα,β(ρ)DρDp(ρ π), ρ(0) = ρ0 satisfy the decay estimate Dp(ρ(t) π) ≤ e−cpβtDp(ρ0 π). see https://jj-zhu.github.io/ Open problem: Interaction of Log-Sobolev inequal. (α>0) and Hellinger estimates (β>0) A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 20 (27)

5. Reaction-diffusion equations as gradient flows Main observation: ρ(t) = 0 is a curve of maximal slope if and only if ∂Dp(0 π) α,β = 0 A calculation gives ∂Dp(0 π) α,β := lim sup H K(0,ρ)→0 Dp(0 π)−Dp(ρ π) H Kα,β(0, µ) =        0 for p > 1/2, 2 β|X| for p = 1/2, ∞ for p < 1/2. For π = 1 dx, one can use the relations • H Kα,β(0, ρ) = 2 √ β He(0, ρ) = 2 √ β ρ(X)1/2. • Dp(0 1) − Dp(ρ 1) = X ρp−pρ p(1−p) dx, especially D1/2 (ρ π) = 2He(ρ, π)2 A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 21 (27)

Overview 1. Introduction: unbalanced transport 2. “Hellinger” versus “Fisher-Rao” 3. Geodesics for the Hellinger-Kantorovich distance 4. Geodesic Λ-convexity on (M(X), H Kα,β) 5. Reaction-diffusion equations as gradient flows 6. EVIλ flows in (M(X), H Kα,β) 7. EVIλ flows in (P(X), SH Kα,β) A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024

6. EVIλ flows in (M(X), H Kα,β ) Uniqueness of solutions can be obtained via Evolutionary Variational Inequalities (EVI) Definition of EVI solutions [AmGiSa05, MurSav20, MurSav24?] • Metric gradient systems (M, E, D) • (M, D) is a geodesic space and E is geodesically Λ-convex A curve u : [0, ∞[ → M is called EVIΛ -solution if it satisfies 1 2 d dt D(u(t), w)2+ Λ 2 D(u(t), w)2 ≤ E(w)−E(u(t)) for all w ∈ dom(E). Important consequence: Uniqueness and Lipschitz continuity D(u(t), u(t)) ≤ e−Λ(t−s)D(u(s), u(s)) for 0 ≤ s < t. However, existence theory for EVI solutions highly nontrivial, cf. [MurSav24?] A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 22 (27)

6. EVIλ flows in (M(X), H Kα,β ) Combining results from [LaschosM19] and from [MurSav24?] one obtains: Theorem [LaschosM arXiv23] If E(µ) = X E(dµ/dx) dx is geodesically Λ-convex on (M(X), H Kα,β), then there is a global EVIΛ flow on dom(E)H K . A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 23 (27)

6. EVIλ flows in (M(X), H Kα,β ) Combining results from [LaschosM19] and from [MurSav24?] one obtains: Theorem [LaschosM arXiv23] If E(µ) = X E(dµ/dx) dx is geodesically Λ-convex on (M(X), H Kα,β), then there is a global EVIΛ flow on dom(E)H K . Corollary. ρ(t) = tanh(t)2 is the unique EVI solution (and curve of maximal slope) for the gradient system (M(X), E, H K1,4) for E(ρ) = 2 3 − ρ1/2 + 1 3 ρ3/2 and initial datum ρ0 = 0. In contrast, the PDE ∂tρ = ∆ 1 2 ρ1/2 + 1 6 ρ3/2 + 2 ρ1/2 − ρ3/2 in X, ∇ρ · n = 0 on ∂X. has many weak solutions. A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 23 (27)

6. EVIλ flows in (M(X), H Kα,β ) Ideas of the proof: [MurSav24?] new theory relies on the so-called local-angle condition and K-concavity of the squared norm norm u → 1 2 D(w, u)2. This approach was extended in [LasMie22?EVIH] based on previous results in [LasMie19GPCA] on K-concavity on suitable subsets of M(X), where measures have densities with upper and lower bounds. A general existence result for EVIλ flows in (M(X), H Kα,β) was obtained, by constructing solutions via the minimizing movement scheme, such that lower and upper bounds for the densities propagate and K-concavity can be exploited at least on finite time horizons. By density and the Λ-contraction property of EVI flows, these results lead to global EVI flows. A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 24 (27)

Overview 1. Introduction: unbalanced transport 2. “Hellinger” versus “Fisher-Rao” 3. Geodesics for the Hellinger-Kantorovich distance 4. Geodesic Λ-convexity on (M(X), H Kα,β) 5. Reaction-diffusion equations as gradient flows 6. EVIλ flows in (M(X), H Kα,β) 7. EVIλ flows in (P(X), SH Kα,β) A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024

7. EVIλ flows in (P(X), SH Kα,β ) Restriction of H Kα,β to the probability measures P(X) gives a metric space, but not a geodesic space. M(X) P(X) M(X) P(X) H K geodesic SH K geodesic A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 25 (27)

7. EVIλ flows in (P(X), SH Kα,β ) Restriction of H Kα,β to the probability measures P(X) gives a metric space, but not a geodesic space. M(X) P(X) M(X) P(X) H K geodesic SH K geodesic (P(X), SH Kα,β) is geodesic if the spherical H K distance is defined via SH Kα,β(ν0, ν1) := 2 arcsin √ β 4 H Kα,β(ν0, ν1) ∈ [0, π/2]. The Onsager operator (inverse Riemannian tensor) for ν ∈ P(X) reads KSH Kα,β (ν)ξ = −α div ν∇ξ + β ν ξ − X νξ dx A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 25 (27)

7. EVIλ flows in (P(X), SH Kα,β ) Fundamental is the following scaling relation: H Kα,β(r2 0 µ0, r2 1 µ1)2 = = r0r1H Kα,β(r0µ0, r1µ1)2 + 4 β (r2 0 − r0r1)µ0(X) + (r2 1 − r0r1)µ1(X) , • (M(X), H Kα,β) is a metric cone over (P(X), SH Kα,β) • explicit geodesics s → νSH K(s) = n(s)µH K(σ(s)), where n(s) and σ(s) are explicitly given in [LaschosM19]. µH K(t) νS H K(s) EVI-flow theory on (P(X), E, SH Kα,β) works as in the H K case caveat: geodesic Λ-convexity on (P(X), SH Kα,β) largely unknown. only positive result µ → X (−ρ)θ dx with θ ∈ d/(d+2), 1/2 A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 26 (27)

Conclusion H Kα,β allows for gradient flows with unbalanced transport Geodesics and distance for H K and SH K can be characterized explicity Geodesic Λ-convexity for H K is understood, but not for SH K. EVIΛ flows for can be constructed in “good” cases, but many important cases are not Λ-convex Functional inequalities need more research Working on M(X) includes µ = 0 which may be degenerate A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 27 (27)

Conclusion H Kα,β allows for gradient flows with unbalanced transport Geodesics and distance for H K and SH K can be characterized explicity Geodesic Λ-convexity for H K is understood, but not for SH K. EVIΛ flows for can be constructed in “good” cases, but many important cases are not Λ-convex Functional inequalities need more research Working on M(X) includes µ = 0 which may be degenerate Thank you for your attention • M-Zhu: Approximation, Kernelization, and Entropy-Dissipation of Gradient Flows: from Wasserstein to Fisher-Rao. In prep. (2024). https://jj-zhu.github.io/ • M: An Introduction to the Analysis of Gradient Systems. Lecture Course (2022/23) arXiv2306.05026. • Laschos-M: EVI on the HK and the SHK spaces. arXiv:2207.09815v3 (2023). • Liero-M-Savaré: Fine prop. of geodesics and geodesic λ-convexity for the HK distance. Arch Rat Mech Ana (2023). • Laschos-M: Geometric properties of cones with applications on the HK space, new distance prob. J Funct Ana (2019) • Liero-M-Savaré:. Optimal entropy-transport problems and a new HK distance. Inventiones mathematicae (2018) A. Mielke, HK spaces and gradient flows, Berlin, 11 März 2024 27 (27)