Proof of Uniqueness of a Continuous Viscosity Solution of the Mean-Field Control Equation on the Torus

Each result cited below is universally quantified over the data in its own statement.

Step 1 (The control Hamiltonian as a Hamiltonian of the setting). By Control Data and the Control Hamiltonian on the Torus Wasserstein Space §data the datum $(b,f)$ has a constant $L\ge0$, and by Control Data and the Control Hamiltonian on the Torus Wasserstein Space §hamiltonian the control Hamiltonian $H_{b,f}$ assigns a real number $H_{b,f}(\mu,p)$ to every $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and every $p\in T_{\mu}$. It is therefore an admissible choice of the Hamiltonian $H$ of The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation §hamiltonian, and we take $H=H_{b,f}$ throughout. By The Discounted Hamilton-Jacobi-Bellman Equation of Mean-Field Control with Idiosyncratic Noise on the Torus §equation, the equation $(\mathrm{E}_{b,f})$ is the equation $(\mathrm{E})$ of The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space §laplacian-form for this Hamiltonian, so the viscosity sub- and supersolutions and solutions of $(\mathrm{E}_{b,f})$ in the sense of Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space are exactly those of $(\mathrm{E})$ with $H=H_{b,f}$.

Step 2 (The three hypotheses). By The Control Hamiltonian is Continuous Along Couplings and Satisfies the Score-Perturbation Bound and the Structure Condition §continuous, $H_{b,f}$ is continuous along couplings; by The Control Hamiltonian is Continuous Along Couplings and Satisfies the Score-Perturbation Bound and the Structure Condition §score-perturbation, it satisfies the score-perturbation bound (with $\delta_{0}=\sigma^{2}/4$ and $C_{R}=2(R+L)^{2}/\sigma^{2}$); and by The Control Hamiltonian is Continuous Along Couplings and Satisfies the Score-Perturbation Bound and the Structure Condition §structure, it satisfies the structure condition. These are the three assumptions on the Hamiltonian in Comparison and Uniqueness for Continuous Viscosity Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space, and the notion of continuity of functions $\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ is the same in both statements.

Step 3 (Conclusion). Let $u,v:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ be continuous, with $u$ a viscosity subsolution and $v$ a viscosity supersolution of $(\mathrm{E}_{b,f})$. By Step 1 they are a continuous viscosity subsolution and supersolution of $(\mathrm{E})$ with $H=H_{b,f}$, so Comparison and Uniqueness for Continuous Viscosity Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space §comparison, whose hypotheses hold by Step 2, gives $u(\mu)\le v(\mu)$ for every $\mu\in\mathcal{P}(\mathbb{T}^{d})$. This proves Uniqueness of a Continuous Viscosity Solution of the Mean-Field Control Equation on the Torus §comparison. Likewise, a continuous viscosity solution of $(\mathrm{E}_{b,f})$ is a continuous viscosity solution of $(\mathrm{E})$ with $H=H_{b,f}$, and by Comparison and Uniqueness for Continuous Viscosity Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space §uniqueness there is at most one of these. This proves Uniqueness of a Continuous Viscosity Solution of the Mean-Field Control Equation on the Torus §uniqueness.