Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy

In the setting of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation, let $R$ be the wall radius of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §parameters, let $\mathcal{E}:\mathcal{D}\to\mathbb{R}$ be as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy, and assume $\mathcal{D}\ne\varnothing$. Semicontinuity on $\mathcal{D}$ refers to the metric space $(\Sigma_{d,R},W_{2})$ of The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points, which contains $\mathcal{D}$ by The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds. Fix a real $e$ with $e\le\mathcal{E}(\lambda)$ for every $\lambda\in\mathcal{D}$, as provided by The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds. Let $u,v:\Sigma^{2}_{d}\to\mathbb{R}$, let $b\ge0$ be real with $|u(\lambda)|\le b$ and $|v(\lambda)|\le b$ for every $\lambda\in\Sigma^{2}_{d}$. For reals $\delta\ge0$ and $\alpha>0$ let $\Psi_{\delta,\alpha}:\mathcal{D}\times\mathcal{D}\to\mathbb{R}$ be

$$\Psi_{\delta,\alpha}(\mu,\nu)=u\bigl(\kappa_{d}(\mu)\bigr)-v\bigl(\kappa_{d}(\nu)\bigr)-\tfrac{\alpha}{2}W_{2}(\mu,\nu)^{2}-\delta\,\mathcal{E}(\mu)-\delta\,\mathcal{E}(\nu).$$

1. (Bounds)¶ For all reals $\delta\ge0$ and $\alpha>0$, every value of $\Psi_{\delta,\alpha}$ is at most $2b-2\delta e$, and the supremum $S(\delta,\alpha)$ of the values of $\Psi_{\delta,\alpha}$ is a real number; moreover $S(\delta,\alpha)\le S(0,\alpha)-2\delta e$.

2. (Diagonal)¶ For every real $\alpha>0$, $S(0,\alpha)\ge u(\kappa_{d}(\lambda))-v(\kappa_{d}(\lambda))$ for every $\lambda\in\mathcal{D}$.

3. (Monotone in the strength)¶ $S(0,\alpha')\ge S(0,\alpha)$ for all reals $\alpha,\alpha'$ with $0<\alpha'\le\alpha$.

4. (Doubling strength)¶ Let $\delta\ge0$, $\theta\ge0$ and $\alpha,\alpha'$ be reals with $0<\alpha'<\alpha$, and let $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ satisfy $\Psi_{\delta,\alpha}(\mu,\nu)\ge S(\delta,\alpha)-\theta$. Then

$$S(\delta,\alpha)-\theta+\tfrac{\alpha-\alpha'}{2}\,W_{2}(\mu,\nu)^{2}\le S(\delta,\alpha').$$

5. (Vanishing weight)¶ For all reals $\alpha>0$ and $\varepsilon>0$ there is a real $\delta_{1}>0$ with $S(\delta,\alpha)\ge S(0,\alpha)-\varepsilon$ for every real $\delta$ with $0<\delta\le\delta_{1}$.

6. (Perturbed maximiser)¶ Let $\delta>0$ be real, and assume that the function $\mu\mapsto u(\kappa_{d}(\mu))-\delta\,\mathcal{E}(\mu)$ on $\mathcal{D}$ is upper semicontinuous on $\mathcal{D}$ and the function $\nu\mapsto v(\kappa_{d}(\nu))+\delta\,\mathcal{E}(\nu)$ on $\mathcal{D}$ is lower semicontinuous on $\mathcal{D}$. Then there is a real $c$ with the following property. For all reals $\alpha>0$ and $\theta$ with $0<\theta<1$ there are $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}$, sequences $(\mu_{k})_{k\in\mathbb{N}}$ and $(\nu_{k})_{k\in\mathbb{N}}$ in $\mathcal{D}$ and positive reals $\varepsilon_{k}$ $(k\in\mathbb{N})$ such that the series $\sum_{k=1}^{\infty}\varepsilon_{k}$ converges with $\sum_{k=1}^{\infty}\varepsilon_{k}\le\theta$; $\mathcal{E}(\hat{\mu})\le c$, $\mathcal{E}(\hat{\nu})\le c$, and $\mathcal{E}(\mu_{k})\le c$ and $\mathcal{E}(\nu_{k})\le c$ for every $k\in\mathbb{N}$; $\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})\ge S(\delta,\alpha)-\theta$; for every $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ the series $\sum_{k=1}^{\infty}\varepsilon_{k}\bigl(W_{2}(\mu,\mu_{k})^{2}+W_{2}(\nu,\nu_{k})^{2}\bigr)$ converges; and, with $\Phi(\mu,\nu)$ denoting $\Psi_{\delta,\alpha}(\mu,\nu)$ minus the sum of that series,

$$\Phi(\mu,\nu)<\Phi(\hat{\mu},\hat{\nu})\qquad\text{for every }(\mu,\nu)\in\mathcal{D}\times\mathcal{D}\text{ with }(\mu,\nu)\ne(\hat{\mu},\hat{\nu}).$$