EVI Gradient Flow in Metrics & Beyond

• EVI Gradient Flow is a framework defined by variational inequalities that captures steepest descent dynamics in both metric and non-metric spaces.
• It provides structural guarantees, including contractivity and energy dissipation, to ensure well-posedness and robust convergence analysis.
• The theory underpins advanced numerical schemes and applications in optimal control, crowd motion, and Hamilton–Jacobi equations, accommodating complex cost functions.

An Evolution Variational Inequality (EVI) Gradient Flow is a dynamical system framework defined by the steepest descent evolution of an energy functional, characterized not by differential equations directly, but by a powerful and intrinsic variational inequality. EVI theory unifies and generalizes a vast class of gradient flows in metric and non-metric settings—including the Wasserstein space of probability measures, non-Euclidean geometries, spaces equipped with Bregman or entropic transport costs, and even settings relevant to information geometry and optimal control. EVI gradient flows yield structure theorems for contractivity, regularity, and well-posedness, provide the backbone for the convergence analysis of time-incremental variational schemes, and underlie the rigorous paper of Hamilton–Jacobi equations and controlled evolution in metric measure spaces.

1. Mathematical Formulation and Definition

An EVI gradient flow on a metric space $(X,d)$ with a lower semicontinuous energy functional $\varphi:X\to(-\infty,+\infty]$ is a curve $u:(0,\infty)\to X$ that, for any $v\in\mathrm{Dom}(\varphi)$, satisfies the evolution variational inequality: $$\frac{1}{2} \frac{d}{dt} d^2(u(t),v) + \frac{\lambda}{2} d^2(u(t),v) \leq \varphi(v) - \varphi(u(t))$$ almost everywhere in $t>0$, for some $\lambda\in\mathbb{R}$ (Muratori et al., 2018). The parameter $\lambda$ quantifies the (geodesic) convexity of $\varphi$ along curves. When the underlying geometry is not metric, the generalization replaces $d^2$ by an appropriate cost $c(x,y)$ (Aubin-Frankowski et al., 1 May 2025): $d^+(\varphi \circ x)(t) + A_c(x,x(t)) \leq 0$ with $A_c$ an appropriate remainder term. For flows in Wasserstein space $\mathcal{P}_2(\Omega)$, the canonical EVI is written with the squared Wasserstein distance: $$\frac{1}{2} \frac{d}{dt} W_2^2(\rho(t),\nu) + \frac{\lambda}{2} W_2^2(\rho(t),\nu) \leq E(\nu) - E(\rho(t))$$ for absolutely continuous curves $\rho:[0,T]\to\mathcal{P}_2(\Omega)$ and energy $E$ (Gallouët et al., 2022).

In the most general setting (Aubin-Frankowski et al., 1 May 2025), the cost $c$ need only be nonnegative, vanishing only on the diagonal, and may lack symmetry or the triangle inequality (e.g., Bregman or KL divergences).

2. Structural Properties and Well-Posedness

EVI gradient flows exhibit a suite of structural results, providing strong control over their evolution:

• Uniqueness and Contractivity: The EVI implies $\lambda$-contractivity of the associated semigroup: for two EVI solutions $u^1,u^2$ with initial data $u_0^1,u_0^2$,

$$d(u^1(t),u^2(t)) \leq e^{-\lambda (t-s)} d(u^1(s),u^2(s))\quad \forall~0\leq s < t$$

ensuring uniqueness and stability (Muratori et al., 2018).

• Energy Dissipation and Regularity: The solution path $t\mapsto\varphi(u(t))$ is nonincreasing and, for $\lambda<0$, even (locally semi-)convex. The metric derivative equals the minimal local slope:

$$\varphi'(u(t)) = -\lvert \dot{u}(t)\rvert^2 = -\lvert \partial\varphi \rvert^2(u(t))$$

for almost every $t$ (Muratori et al., 2018).

• Asymptotic Behavior: For $\lambda>0$, solutions converge exponentially to the unique minimizer. Quantitative results such as $d(u(t),\bar{u})\leq d(u_0,\bar{u})e^{-\lambda t}$ hold.
• Extension to General Costs: With general $c$, contractivity and energy dissipation are restated in terms of $c(x(t),\tilde{x}(t))$ and the appropriate “$c$-cost derivative” (Aubin-Frankowski et al., 1 May 2025).
• Relation to Geodesic Convexity: If $\varphi$ is (approximately) $\lambda$-convex along geodesics, then EVI flows inherit additional stability and regularity properties (Muratori et al., 2018, Gallouët et al., 2022). The EVI can in fact serve as a characterization of geodesic (displacement) convexity.

3. Connection with Minimizing Movement Schemes

The EVI gradient flow structure provides a blueprint for time-discrete variational schemes and their convergence analysis:

• JKO (Minimizing Movement) Scheme: In metric spaces and particularly in Wasserstein space, the discrete sequence $(U^n)$ is defined by:

$$U^n \in \operatorname{argmin}_{U\in X} \left\{ \varphi(U) + \frac{1}{2\tau} d^2(U,U^{n-1}) \right\}$$

(Muratori et al., 2018). Under lower semicontinuity and weak (geodesic) convexity, interpolated curves converge to the EVI gradient flow.

• Higher-Order Time Discretization: The EVBDF2 scheme (Gallouët et al., 2022), based on a backward differentiation formula of order two, introduces extrapolation in Wasserstein space and yields a discrete EVI at each time-step. Under $E$ being $\lambda$-convex along geodesics, convergence to the continuous EVI gradient flow is established.
• General Cost Splitting and Explicit/Implicit Schemes: For general $c$, alternating minimization ("splitting") and backward Euler schemes converge to EVI flows provided new cross-convexity/c-concavity notions hold (Aubin-Frankowski et al., 1 May 2025).
• Error Analysis: Uniform error estimates of order $1/2$ in the timestep, independent of geometric curvature or compactness, are available (Muratori et al., 2018).
• Approximate Minimization: Use of Ekeland’s variational principle yields relaxed minimization allowing for lack of compactness (Muratori et al., 2018).

4. EVI Flows in Applications and Special Settings

Crowd motion with hard congestion: The evolution of a density $\rho$ under hard constraints $0\leq\rho(x)\leq 1$ is formulated as an EVI gradient flow in Wasserstein space. The functional

$$\Phi(\rho) = \int_\Omega D(x)\rho(x)dx + \mathbf{I}_K(\rho)$$

with the indicator $\mathbf{I}_K$ of the admissible densities $K$, is not everywhere finite and not geodesically convex; careful analysis of the discrete JKO scheme is required, yet convergence to an EVI gradient flow holds, with the velocity field characterized by a projection onto admissible motions (Maury et al., 2010).

Information geometry and Weyl structure: In dually flat information geometry, gradient flows can be recast as pre-geodesics in a Weyl integrable geometry, with the non-metricity determined by the local “dissipation rate.” The Weyl covector $\omega_k = -\partial_k \ln \eta^2(\theta)$ encodes how the metric changes along the flow, providing geometric insight into the energy dissipation encoded in the EVI formalism (Wada, 2022).

Hamilton–Jacobi and Controlled Gradient Flows: The EVI structure underpins viscosity solutions of HJ equations in metric measure spaces. In controlled Wasserstein gradient flows, a “controlled EVI” allows derivation of regularity, energy, and metric bounds for optimizing sequences. Viscosity solution theory leverages cylindrical test functions (depending only on squared distances) to circumvent the lack of smooth differentiability, with upper and lower Hamiltonian bounds constructed directly from the EVI (Conforti et al., 2023, Conforti et al., 4 Jan 2024).

Generalized Costs: Extension to non-metric settings (e.g., Bregman, KL, Sinkhorn divergences) is nontrivial: EVI flows and contractivity results still hold under suitable cross-convexity, provided the dissipation cost $c$ satisfies the (Diss) nonnegativity structure (Aubin-Frankowski et al., 1 May 2025).

5. Challenges and Limitations

Several intrinsic challenges are associated with EVI gradient flow theory.

• Non-finite-valued and Non-geodesically Convex Energies: Many natural energies (e.g., those including hard constraints) lead to functionals that take the value $+\infty$ outside an admissible set and fail to be geodesically convex (e.g., crowd motion with mass constraint). The lack of global convexity prohibits direct application of standard metric gradient-flow theory, necessitating ad hoc compactness and tightness arguments in the analysis of convergence for variational schemes (Maury et al., 2010).
• Non-metric Cost Functions: For Bregman, Sinkhorn, or entropic divergences, geometric techniques must be refined, as the lack of symmetry and triangle inequality complicates compactness and contractivity. General cost-based cross-convexity must be exploited to retain the necessary energy and distance controls in splitting and approximation algorithms (Aubin-Frankowski et al., 1 May 2025).
• Computation of Projections and Controls: In cases with constraints or projections (macroscopic crowd motion), characterization of the projected velocity and pressure variables, even at the discrete level, is analytically and numerically involved (Maury et al., 2010).
• Lack of Smooth Structure: Metric and measure-theoretic settings often lack a differentiable structure, requiring EVI and viscosity solution techniques that are derivative-free and rely solely on variational and metric properties (Conforti et al., 2023, Conforti et al., 4 Jan 2024).

6. Broader Implications and Impact

EVI gradient flows provide a rigorous, unifying language for gradient flows in a wide array of geometric, analytic, and control-theoretic contexts.

• Analysis of Dissipative Dynamics: EVI formulations rigorously encode the interplay of geometry and dissipation, supporting well-posedness theory for nonlinear evolution equations in spaces lacking smooth geometry (probability metrics, entropic transport, etc.).
• Numerical Schemes for Evolution Equations: Convergence analysis of minimizing movement, higher-order time stepping (e.g., EVBDF2), and splitting schemes is grounded on the EVI structure, enabling provably accurate and stable algorithms for diverse PDEs, including those with nonsmooth constraints or nonstandard costs (Muratori et al., 2018, Gallouët et al., 2022, Aubin-Frankowski et al., 1 May 2025).
• Controlled and Mean-Field Evolution: Derivative-free viscosity and dynamic programming methods for Hamilton–Jacobi equations in measure spaces rely critically on EVI arguments, opening the door for infinite-dimensional optimal control and mean field game analysis (Conforti et al., 2023, Conforti et al., 4 Jan 2024).
• Bridging to Information Geometry and Non-Riemannian Settings: The geometric reinterpretation of gradient flows via Weyl geometry consolidates the understanding of dissipation and dynamics in statistical and information geometry (Wada, 2022).
• Extensibility Beyond Metric Spaces: The generalization to Bregman, entropic, and other costs is particularly relevant for modern applications in data science, machine learning (mirror descent, KL dynamics), and statistical physics, extending theoretical guarantees to non-metric evolutionary regimes (Aubin-Frankowski et al., 1 May 2025).

7. Summary Table: Key EVI Formulations and Properties

Formulation	Setting	Energy/Cost Structure
Classical EVI w/ $d^2$	$(X,d)$ metric space, $\varphi$ $\lambda$-convex	$$\frac{1}{2}\frac{d}{dt} d^2(u(t),v) + \frac{\lambda}{2} d^2 \leq \varphi(v) - \varphi(u(t))$$
Wasserstein EVI	$\mathcal{P}_2(\Omega)$, $E$ convex along geodesics	$$\frac{1}{2}\frac{d}{dt} W_2^2(\rho(t),\nu) + \frac{\lambda}{2} W_2^2 \leq E(\nu) - E(\rho(t))$$
General cost $c$	Arbitrary (quasi-)metric, cost $c$, cross-convex energy	$d^+(\varphi \circ x)(t) + A_c(x, x(t)) \leq 0$
Controlled EVI (Wasserstein)	$\mathcal{P}_2(\mathbb{R}^d)$, controlled SDEs/McKean–Vlasov	Generalized EVI with control term in continuity equation and test function increments

EVI gradient flow theory thus constitutes a foundational tool for modern analysis and applied mathematics, yielding insight, algorithms, and structure results across metric, non-metric, constrained, stochastic, and geometric evolution problems.