How is the optimal control problem for the Dyson equation solved?

The Dyson equation controls the mean-field spectrum

The Dyson equation describes the mean-field evolution of the spectrum of large random matrices. It arises as the limit, as the matrix size N goes to infinity, of the empirical measure of the eigenvalues of the Dyson Brownian motion, a stochastic particle system with repulsive logarithmic Coulomb interaction. In the paper's notation, the empirical measure is m_{N,t} = (1/N) sum of Dirac masses at the eigenvalues, and its N to infinity limit solves the Dyson equation with a Hilbert transform term H[m].

The control problem studied here adds a measurable velocity alpha to that evolution. Given a control alpha mapping time and space to the real line, the controlled measure evolves according to a continuity equation in which the flux is m times (H[m] + alpha). The payoff to be minimized combines a quadratic running cost, one half of alpha squared integrated against the measure, plus a running cost F[m_s] and a terminal cost U_T(m_T). The optimization is over pairs (alpha, m) that solve the controlled equation, keep the measure path continuous in the 2-Wasserstein space, and have finite energy.

Regular and singular terminal data are treated separately

The paper distinguishes two terminal conditions. In the first, the terminal cost U_T is a continuous function on the space of probability measures with finite second moment. The authors link this case to soft large deviations. In the second, U_T is the indicator of a single target measure nu, equal to zero at nu and infinity elsewhere; this case is tied to large deviation asymptotics of transition probabilities.

That distinction matters because the target problem, where the terminal measure is fixed, requires the terminal measure to have finite free entropy. The authors note that assuming finite entropy of the terminal measure turns out to be necessary for a solution to exist. So the two regimes are not interchangeable: one allows a spread of terminal costs, the other pins the endpoint to a specific measure and imposes an extra admissibility condition on that measure.

Admissible controls are constructed and optimal ones exist

A central contribution is a set of new results on the existence of solutions to the controlled Dyson equation for general controls alpha. The authors state that well-posedness of the controlled equation for general controls was mostly open, since only cases with smooth vector fields had been investigated. They construct admissible controls and prove that optimal ones exist.

The compactness result behind this is Proposition 3.1: for a pair (alpha, m) in the admissible set A(mu, t), there is a constant C depending only on the energy, the time horizon, and the second moment of the initial measure, such that the second moment of m_s stays bounded and the free entropy E(m_s) is bounded by C times (1 + (s - t)^(-1/2)). The same proposition gives an energy identity bounding the integral of m cubed over a time interval by twice the control energy plus terms involving the entropy drop. In other words, bounded control energy buys compactness and regularity of the measure path.

The Hilbert transform term makes the HJB equation singular

Without the Hilbert transform and with F identically zero, the minimization is the problem studied by Benamou and Brenier in mass transport, and the associated HJB equation on the Wasserstein space has been analyzed in earlier work. The authors state that when H is present the problem is more complicated and singular, and that showing well-posedness of the HJB equation and that the value function solves it required new ideas on Hamilton-Jacobi equations on Wasserstein spaces as well as new results on the Dyson equation with a drift.

The HJB equation the paper derives is backward in time on the product of the Wasserstein space and the time interval. It involves the horizontal, or Wasserstein, derivative D_mu U of the value function, its squared norm integrated against the measure, and a term pairing D_mu U with the Hilbert transform H[mu]. The terminal condition is U(., T) = U_T. The value function U(mu, t), defined as the infimum of the payoff over admissible controls, is shown to be the unique viscosity solution of this equation.

Viscosity solutions are defined in the space of measures

Because the HJB equation lives on an infinite-dimensional space of measures, the paper works with the theory of viscosity solutions in the space of measures. Proposing an appropriate notion of viscosity solution and establishing its well-posedness, especially in the singular terminal case where U_T is the indicator of a target measure, are described by the authors as two of the major contributions of the paper.

The analysis uses the 2-Wasserstein space of probability measures with finite second moment, the horizontal derivative of functions on that space, and a C^{1,1}-type regularity class. A key technical tool is an approximation of the squared Wasserstein distance and of the optimal transport cost by regularized maps, which lets the authors handle the fact that the squared Wasserstein distance need not be differentiable everywhere. The free entropy E(m) = -(1/2) double integral of log|x - y| dm(x) dm(y) also plays a central role; it is lower semicontinuous with respect to weak convergence of measures, and along the Dyson flow it is decreasing.

The value function is the unique viscosity solution

For locally uniformly continuous F and U_T, Theorem 6.2 states that U is the unique continuous viscosity solution of the HJB equation satisfying U(mu, T) = U_T(mu) for all mu. For the target problem with a fixed terminal measure nu, Theorem 6.3 gives a characterization under additional conditions: the candidate V must be locally uniformly continuous, bounded below by C^{-1} W_2^2(., nu) - C, satisfy V(mu, nu, t) at least -C(T - t), tend to zero as t goes to T along the diagonal V(nu, nu, t), and tend to infinity locally uniformly as t goes to T away from nu. Under those conditions, V is the value function of the target problem.

The proof of the target case uses the comparison principle and a dynamic programming relation. The authors also construct, by a penalization argument, a family U_epsilon whose terminal datum is (1/(2 epsilon)) W_2^2(., nu); Proposition 6.6 shows that as epsilon goes to zero this family converges to the value function of the target problem. This is how the singular terminal condition is reached as a limit of regular ones.

The result covers quadratic costs and finite second moments

The paper restricts attention to initial data in the space of probability measures with finite second moment, and it shows that for controls of finite cost the dynamics stay in that space. The authors state that their results on the control of the Dyson equation and the resulting HJB equation extend to more general than quadratic energies and separated running costs, namely to costs L(alpha, m) that are uniformly convex in alpha, leaving the details to the interested reader. That extension is stated, not proved in the excerpts available here.

The motivation is the study of large deviations of the Dyson Brownian motion. The authors say that in a subsequent paper they provide a complete and direct proof of large deviations for the spectrum of large random matrices and study the asymptotic behavior of transition probabilities of the Dyson Brownian motion. So the present paper is the control-theoretic and HJB foundation; the large deviation conclusions themselves are deferred to that companion work and are not established here.

The singular terminal case is the sharpest test

The most decision-relevant distinction in this paper is between the regular terminal case and the singular one. In the regular case, uniqueness of the continuous viscosity solution follows once F and U_T are locally uniformly continuous, and the proof is described as a consequence of earlier results. In the singular case, where U_T is the indicator of a target measure nu, the proof is more involved because it relies on the comparison principle, and the terminal measure must have finite free entropy.

For a reader judging what this paper actually delivers, the priority is therefore the singular terminal problem: it is where the new notion of viscosity solution and its well-posedness are most needed, where the finite-entropy condition on nu becomes a real admissibility threshold, and where the penalized family U_epsilon with terminal datum (1/(2 epsilon)) W_2^2(., nu) provides the bridge from regular to singular data. The regular case is the baseline; the singular case is the one that changes how the HJB equation must be posed and solved.

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