Vanishing Discount Method in HJ Systems

• The vanishing discount method is a rigorous framework for studying the asymptotic behavior of discounted Hamilton–Jacobi equations using viscosity solution theory and convex duality.
• It leverages generalized Mather and Green–Poisson measures to represent solutions and ensure convergence under coercivity, convexity, and quasi-monotonicity conditions.
• The method underpins selection principles in ergodic theory and optimal control, guiding the uniqueness and stability of limiting viscosity solutions in nonlinear and weakly-coupled systems.

The vanishing discount method is a rigorous analytical framework for studying the asymptotic behavior of solutions to discounted Hamilton–Jacobi equations and systems as the discount parameter tends to zero. This method is central to modern ergodic theory, weak KAM analysis, viscosity solution theory, optimal control, and Mather theory, particularly in describing the limiting selection mechanisms for nonlinear equations, monotone systems, and degenerate boundary-value problems (Ishii et al., 2019, Ishii et al., 2019, Wang et al., 22 Sep 2025, Ishii et al., 2016, Terai, 2020, Ishii et al., 2016).

1. Formal Statement of the Vanishing Discount Problem

Given a compact manifold or domain $T^n$ and a finite index set $I = \{1, \ldots, m\}$, the vanishing discount problem for nonlinear, monotone weakly-coupled systems takes the following general form: For each $\delta > 0$, seek a vector-valued function $u^\delta = (u_i^\delta)_{i\in I} \in C(T^n)^m$ that solves, in the viscosity sense,

$$(P^\delta)\qquad \delta u_i^\delta(x) + H_i(x, Du^\delta(x), u^\delta(x)) = 0, \quad x \in T^n,\quad i \in I,$$

where $$H = (H_i)_{i\in I}: T^n \times \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}$$ is continuous and satisfies:

• Coercivity in the gradient: $$\lim_{|p| \to \infty} \inf_{x \in T^n, |u| \leq R} H_i(x, p, u) = +\infty$$ for each $R > 0$;
• Joint convexity in $(p,u)$: $(p,u) \mapsto H_i(x,p,u)$ is convex for each $(x,i)$;
• Quasi-monotonicity: for $u, v \in \mathbb{R}^m$ with $(u-v)_k = \max_{j}(u_j-v_j) \geq 0$, $H_k(x,p,u) \geq H_k(x,p,v)$ for all $(x,p)$;
• Existence of limit (ergodic) solution: The system $(P^0)$ with $\delta = 0$ admits a continuous solution (Ishii et al., 2019).

This formalism applies to scalar equations, actively coupled nonlinear systems, and can be extended to infinite-dimensional settings and noncompact domains (Terai, 2020, Ishii et al., 2019).

2. Convex Duality, Mather Measures, and Green–Poisson Measures

A cornerstone of the vanishing discount method is the representation of solutions via convex duality, utilizing generalized Mather measures for (possibly nonlinear, weakly coupled) systems. For each $i \in I$, the associated Lagrangian is defined by Legendre duality: $$L_i(x, \xi, \eta) = \sup_{(p, u) \in \mathbb{R}^n \times \mathbb{R}^m} \left\{ \xi \cdot p + \eta \cdot u - H_i(x, p, u) \right\}.$$ The set of Mather measures $\mathcal{P}^0$ is constructed as families of nonnegative Borel measures $\mu = (\mu_i)_{i\in I}$ on $T^n \times \mathbb{R}^n \times \mathbb{R}^m$, satisfying normalization and "closedness" constraints: $$\sum_{i\in I} \int S^0(\xi, \eta) \, \mu_i(dx \, d\xi \, d\eta) \leq 1,\qquad \int \left[\xi \cdot D\phi_i(x) + \eta \phi_i(x)\right] \mu_i(dx\,d\xi\,d\eta) = 0$$ for all test functions $\phi = (\phi_i)_{i \in I} \in C^1(T^n)^m$ (Ishii et al., 2019).

For each $(z, k, \delta)$, Green–Poisson measures $\mathcal{P}(z, k, \delta) \subset \mathcal{P}^0$ encode constraints connecting $\delta$-discounted solutions to the test point: $$\int \left[ \xi \cdot D\psi_i + \eta \psi_i + \delta \psi_i \right] \mu_i = \psi_k(z),\qquad \forall \psi \in C^1(T^n)^m.$$ The value $u_k^\delta(z)$ admits the dual variational representation: $$u_k^\delta(z) = \min_{\mu \in \mathcal{P}(z,k,\delta)} \sum_{i \in I} \int L_i(x, \xi, \eta) \, \mu_i(dx\,d\xi\,d\eta).$$ The minimum is attained for a Green–Poisson measure, linking analytic and measure-theoretic perspectives.

3. Convergence and Selection Principle

Under the structural hypotheses (coercivity, convexity, monotonicity), the family $\{u^\delta\}_{\delta > 0}$ is equi-Lipschitz and uniformly bounded. As $\delta \to 0^+$,

$$u^\delta \rightarrow u^0 \quad \text{in} \;\; C(T^n)^m,$$

where $u^0$ is the unique viscosity solution to the ergodic system $(P^0)$,

$$H_i(x, Du^0(x), u^0(x)) = 0, \quad x \in T^n,\quad i \in I.$$

The limiting solution $u^0$ is selected by the Mather measures: for any weak-* limit $\mu^0$ of the Green–Poisson measures $\mu^\delta$,

$$\sum_{i \in I} \int u^0_i(x) \, \mu^0_i(dx\,d\xi\,d\eta) \leq 0.$$

Uniqueness in $C(T^n)^m$ is enforced by a comparison argument exploiting monotonicity and the duality structure (Ishii et al., 2019, Ishii et al., 2016).

4. Analytical Framework and Proof Strategies

The proofs are built on:

• Duality and convex analysis: Solutions of the discounted problem admit dual representations as minimizing measures over suitable cones defined by subsolution constraints.
• Tightness and compactness: Coercivity ensures measures $\mu^\delta$ are supported on a fixed compact set. Weak-* compactness allows passage to limiting Mather measures as $\delta \to 0$.
• Selection principles: Comparison of sub- and supersolutions via dual inequalities forces uniqueness of accumulation points and thus uniform convergence for the full family.
• Holonomy and closedness: The constraints on measures generalize classical holonomy conditions from Aubry–Mather theory, ensuring dynamical invariance and the correct selection of limiting solutions (Ishii et al., 2019, Ishii et al., 2016).

5. Illustrative Examples and Special Cases

• Linear coupling: For $H_i(x, p, u) = G_i(x, p) + \sum_j b_{ij}(x) u_j$, monotonicity reduces to specific matrix conditions (row sums, off-diagonal nonpositivity). The measure-theoretic method recovers classical linear system convergence results.
• Nonlinear monotone coupling: The formalism encompasses $H_i(x,p,u) = G_i(x,p) + F_i(u_i, u_{i+1})$ with $F_i$ convex and monotone. The same convex–duality approach yields convergence (Ishii et al., 2019).
• Necessity of convexity: Convexity in $u$ is crucial; counterexamples in the scalar, nonconvex case demonstrate failure of convergence (Ishii, 2022).
• Infinite systems: The method extends to infinite-dimensional systems with compact index sets and integral couplings, provided coercivity and monotonicity are maintained (Terai, 2020).
• Control-theoretic and weak KAM link: Discounted HJB equations on $T^n \times I$ describe optimal switching Markov-ODE systems; Mather and Green–Poisson measures generalize minimizing actions to the coupled system (Ishii et al., 2019).

6. Extensions and Limitations

While the vanishing discount method enforces convergence and selection under coercivity, convexity, monotonicity, and solvability, the framework's limitations are sharply illustrated by counterexamples:

• Systems with insufficient coupling structure: Even with convex Hamiltonians and monotone couplings, full convergence may fail for weakly-coupled systems not satisfying strong irreducibility or diagonal dominance (Ishii, 2022).
• Nonconvexities and degenerate cases: Failure of convexity in either $p$ or $u$ may obstruct selection and yield non-uniqueness or oscillatory accumulation points, emphasizing the importance of the duality and measure-theoretic criteria.

7. Summary Table: Key Elements in the Vanishing Discount Method

Component	Description	Reference
Discounted system	$\delta u + H(x,Du,u) = 0$ on $T^n$, monotone nonlinear $H$	(Ishii et al., 2019)
Mather measures	Minimizing Borel measures, generalize classical Mather sets	(Ishii et al., 2019, Ishii et al., 2016)
Green–Poisson measures	Discounted analogues, dual representations for $u^\delta$	(Ishii et al., 2019)
Convergence theorem	Equi-Lipschitz, uniform boundedness, selection via measures	(Ishii et al., 2019, Ishii et al., 2016)
Selection principle	Limiting $u^0$ maximal subsolution respecting Mather measures	(Ishii et al., 2019)

The vanishing discount method thus provides a comprehensive framework for the selection and convergence of solutions of discounted Hamilton–Jacobi systems, grounded in convex duality, measure representations, and monotonicity analysis. It is the preferred tool for rigorously extracting ergodic limits in both scalar and system settings under well-defined structural hypotheses.