Proof of Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here.

When a level $N\ge N_{0}$ is fixed we abbreviate $\overline{w}^{0}=\overline{w}^{0}_{N,\tau}$, $\underline{w}^{0}=\underline{w}^{0}_{N,\tau}$, $\overline{w}=\overline{w}_{N,\tau}$, $\underline{w}=\underline{w}_{N,\tau}$, $p_{0}=p_{0,N}$ and $\mu_{x}=\mu^{N}_{x}$. Two facts are used throughout. First, $0\le\omega_{g}(s)\le2b_{g}$ for $s\ge0$: the set defining $\omega_{g}(s)$ contains $|g(\mu)-g(\mu)|=0$, and each of its elements is at most $|g(\mu)|+|g(\nu)|\le2b_{g}$ by The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §data. Second, every $\nu\in\mathcal{D}$ is atomless: $\mathcal{D}$ is contained in $\mathcal{D}_{\log}$ by The Confined Logarithmic-Energy Pair on the Wasserstein Space of the Real Line §pair, and every element of $\mathcal{D}_{\log}$ gives measure $0$ to each one-point set by The Logarithmic Energy of a Probability Measure on the Real Line §energy, which is atomlessness. Since every point of $W_{N}$ is ordered (as recorded in Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function), Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §optimal, applied with $\hat{\mu}=\nu$, gives for all $N\ge N_{0}$, $x,y\in W_{N}$ and $\nu\in\mathcal{D}$

Step 1 (The hypotheses of the convolution lemma). Fix $N\ge N_{0}$. We check the hypotheses of Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty for the data listed in the statement. The set $W_{N}$ is nonempty and open by The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §open. It is convex: for $x,y\in W_{N}$, $0\le t\le1$ and $i<j$, $(tx_{i}+(1-t)y_{i})-(tx_{j}+(1-t)y_{j})=t(x_{i}-x_{j})+(1-t)(y_{i}-y_{j})$ is positive, because both differences are positive by The Weyl Chamber of Ordered Points in Euclidean Space and $t,1-t$ are nonnegative with sum $1$. By The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §penalty, $P_{N}$ is a penalty on $W_{N}$ with monotone gradient and $\kappa_{N}$ is positive; $\lambda$ is positive and $\theta=1$ is nonnegative. The dissipation inequality with $\varepsilon=\varepsilon_{0}$ and $C=C_{0}N$ is The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §dissipation, where $0<\varepsilon_{0}<1$. Write $G(x)=N\,g(\mu_{x})$. The function $\rho(s)=N\,\omega_{g}(s/\sqrt{N})$ maps $[0,\infty)$ into $[0,\infty)$ and is nondecreasing because $\omega_{g}$ is. For $x,y\in W_{N}$, Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §lipschitz (in force with dimension $1$ by N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §particles) gives $N\,W_{2}(\mu_{x},\mu_{y})^{2}\le\lVert x-y\rVert^{2}$, hence $W_{2}(\mu_{x},\mu_{y})\le\lVert x-y\rVert/\sqrt{N}$ since both sides are nonnegative, and therefore

Further $M=\lambda^{-1}Nb_{g}\ge0$, $\eta=\frac{1}{N}>0$, $\tau>0$, and $\theta\eta=\frac{1}{N}\le\varepsilon_{0}$ because $N\varepsilon_{0}\ge N_{0}\varepsilon_{0}\ge1$. The operator $F$ of that lemma (control cost $1$, running cost $G$, potential $P_{N}$, discount $\lambda$, noise intensity $\kappa_{N}$) is the operator $F_{N}$ of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §particle-equation: by The N-Particle Dyson Hamilton-Jacobi Equation with a Mean-Field Running Cost §operator and The Dyson Hamilton-Jacobi Equation for N Controlled Particles in the Weyl Chamber §operator, $F_{N}$ is the penalty-drift operator on $W_{N}$ with these coefficients and with potential $H_{b_{N}}+\sum_{k}V(x_{k})$, which is $P_{N}$ by The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §hypotheses. By The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §particle-equation and Well-Posedness of the Dyson Hamilton-Jacobi Equation in the Weyl Chamber below the Collision Threshold §well-posed, $v_{N}$ is both a viscosity subsolution and a viscosity supersolution of $F$ on $W_{N}$, with $|v_{N}(x)|\le M$. So Parts A and B apply with $u=v=v_{N}$.

The derived quantities of that lemma are those of the statement: $p_{0}$ is the least value of $P_{N}$ on $W_{N}$; the lemma's $r(x)$ has square $2\tau\bigl(2\lambda^{-1}Nb_{g}+\frac{1}{N}(P_{N}(x)-p_{0})\bigr)=r_{N}(x)^{2}$, so $r=r_{N}$; $\theta_{1}=1-\frac{1}{2N\varepsilon_{0}}=\theta_{1,N}$ and $\theta_{2}=1+\frac{1}{N\varepsilon_{0}}=\theta_{2,N}$; and $g_{1}(x)=G(x)+\frac{1}{N}C_{0}N+N\,\omega_{g}(r_{N}(x)/\sqrt{N})=g^{+}_{N}(x)$, likewise $g_{2}=g^{-}_{N}$. Thus $F_{1}$ and $F_{2}$ of the lemma are the two operators of clause (v). Since $p_{0}=P_{N}(x_{*})$ for some $x_{*}\in W_{N}$, The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §bounds gives $p_{0}\ge e_{*}N$. Dividing $r_{N}(y)^{2}$ by $N$,

Step 2 (Clause 1). Fix $N\ge N_{0}$ and $x,y\in W_{N}$.

(i) By Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty §sub-bounds and $p_{0}\ge e_{*}N$, $v_{N}(x)-\frac{1}{N}P_{N}(x)\le\overline{w}^{0}(x)\le\lambda^{-1}Nb_{g}-\frac{p_{0}}{N}\le\lambda^{-1}Nb_{g}+|e_{*}|$. As $-L_{N}<0\le\lambda^{-1}Nb_{g}+|e_{*}|$ and $N\ge1$,

By Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty §inf-bounds, $-\lambda^{-1}Nb_{g}-|e_{*}|\le-M+\frac{p_{0}}{N}\le\underline{w}^{0}(x)\le v_{N}(x)+\frac{1}{N}P_{N}(x)$, and $L_{N}>0$, so

With $c=\lambda^{-1}b_{g}+|e_{*}|+1$ this gives $-c(N+|P_{N}(x)|)\le-cN\le-L_{N}\le\overline{w}(x)\le cN$ and $-cN\le\underline{w}(x)\le L_{N}\le cN\le c(N+|P_{N}(x)|)$, the growth conditions of Upper and Lower Half-Relaxed Limits along Empirical Measures of Functions on the Weyl Chambers §upper and Upper and Lower Half-Relaxed Limits along Empirical Measures of Functions on the Weyl Chambers §lower; so $\bar{u}_{\tau}$ and $\underline{u}_{\tau}$ are defined.

(ii) By Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty §sub-semiconvex and Semiconvex Function on a Convex Subset of $\mathbb{R}^n$, $h(z)=\overline{w}^{0}(z)+\frac{1}{2\tau}\lVert z\rVert^{2}$ is convex on $W_{N}$. So is $k(z)=-L_{N}+\frac{1}{2\tau}\lVert z\rVert^{2}$, because $t\lVert a\rVert^{2}+(1-t)\lVert b\rVert^{2}-\lVert ta+(1-t)b\rVert^{2}=t(1-t)\lVert a-b\rVert^{2}\ge0$ for $0\le t\le1$. For $a,b\in W_{N}$, $0\le t\le1$ and $z=ta+(1-t)b$,

so $\max(h,k)=\overline{w}+\frac{1}{2\tau}\lVert\cdot\rVert^{2}$ is convex, that is, $\overline{w}$ is semiconvex on $W_{N}$ with constant $\tau^{-1}$. By The Maximum of Two Upper Semicontinuous Functions §duality, $-\underline{w}=\max(-\underline{w}^{0},-L_{N})$ pointwise, and $-\underline{w}^{0}$ is semiconvex with constant $\tau^{-1}$ by Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty §inf-semiconcave; the same argument applies.

(iii) Let $\overline{w}$ be differentiable at $x$. By claim 1 of A Derivative Matrix is the Jacobian Matrix, and is Unique all partial derivatives of $\overline{w}$ exist at $x$, and $D\overline{w}(x)$ is the vector of them (Gradient of a Real-Valued Function on a Euclidean Open Set). If $\overline{w}^{0}(x)>-L_{N}$: $\overline{w}^{0}$ is semiconvex on the open convex set $W_{N}$ by Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty §sub-semiconvex, hence continuous by Local Lipschitz Bound and Continuity for a Semiconvex Function on an Open Convex Set §continuity (with $S=W_{N}$); so there is $\delta>0$ such that every $z$ with $\lVert z-x\rVert<\delta$ lies in $W_{N}$ and satisfies $\overline{w}^{0}(z)>-L_{N}$, hence $\overline{w}(z)=\overline{w}^{0}(z)$. For $0<|s|<\delta$ the difference quotients of $\overline{w}$ and $\overline{w}^{0}$ at $x$ in each coordinate direction coincide, so the partial derivatives of $\overline{w}^{0}$ exist at $x$ and equal those of $\overline{w}$, and Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty §sub-gradient gives $\lVert D\overline{w}(x)\rVert=\lVert D\overline{w}^{0}(x)\rVert\le\tau^{-1}r_{N}(x)$. If $\overline{w}^{0}(x)\le-L_{N}$: then $\overline{w}(x)=-L_{N}\le\overline{w}(z)$ for all $z\in W_{N}$ by (1a), so for each $i$ the quotients $(\overline{w}(x+se_{i})-\overline{w}(x))/s$ are nonnegative for small $s>0$ and nonpositive for small $s<0$; their limit $\partial_{i}\overline{w}(x)$ is $0$, and $\lVert D\overline{w}(x)\rVert=0\le\tau^{-1}r_{N}(x)$. For $\underline{w}$ the argument is the same with Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty §inf-gradient, the continuity of $\underline{w}^{0}$ following from that of the semiconvex function $-\underline{w}^{0}$, and in the case $\underline{w}^{0}(x)\ge L_{N}$ the point $x$ being a maximum point of $\underline{w}$ by (1b).

(iv) Let $\Delta=\frac{1}{2\tau}\lVert x-y\rVert(3\lVert x-y\rVert+2r_{N}(y))\ge0$. By Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty §sub-transfer, $\overline{w}(x)=\max(\overline{w}^{0}(x),-L_{N})\le\max(\overline{w}^{0}(y)+\Delta,-L_{N}+\Delta)=\overline{w}(y)+\Delta$. By Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty §inf-transfer, $\underline{w}(x)\ge\min(\underline{w}^{0}(y)-\Delta,L_{N}-\Delta)=\underline{w}(y)-\Delta$.

(v) The function $\overline{w}^{0}$ is upper semicontinuous on $W_{N}$, being a viscosity subsolution of $F_{1}$ by Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty §subsolution (see Viscosity Subsolution and Supersolution of a Second-Order Equation); the constant $-L_{N}$ is upper semicontinuous; so $\overline{w}$ is upper semicontinuous by The Maximum of Two Upper Semicontinuous Functions §max-usc. Let $\varphi:W_{N}\to\mathbb{R}$ be of class $C^{2}$ and let $\overline{w}-\varphi$ have a local maximum at $x$ relative to $W_{N}$: there is $\delta>0$ with $\overline{w}(z)-\varphi(z)\le\overline{w}(x)-\varphi(x)$ for $z\in W_{N}$, $\lVert z-x\rVert<\delta$. If $\overline{w}^{0}(x)\ge-L_{N}$, then $\overline{w}(x)=\overline{w}^{0}(x)$ and for such $z$, $\overline{w}^{0}(z)-\varphi(z)\le\overline{w}(z)-\varphi(z)\le\overline{w}^{0}(x)-\varphi(x)$; so $\overline{w}^{0}-\varphi$ has a local maximum at $x$ and Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty §subsolution gives $F_{1}(x,\overline{w}(x),D\varphi(x),D^{2}\varphi(x))\le0$. If $\overline{w}^{0}(x)<-L_{N}$, then $\overline{w}(x)=-L_{N}$ and for such $z$, $-L_{N}-\varphi(z)\le\overline{w}(z)-\varphi(z)\le-L_{N}-\varphi(x)$; so $\varphi$ has a local minimum at $x$ relative to the open set $W_{N}$, and claim 2 of First- and Second-Order Conditions at a Local Extremum of a Function of Class $C^2$ gives $D\varphi(x)=0$ and $0\preceq D^{2}\varphi(x)$. Then $D^{2}\varphi(x)$ is positive semidefinite by The Positive Semidefinite Ordering Compared by Differences, so its diagonal entries $e_{i}\cdot D^{2}\varphi(x)e_{i}$, and hence its trace, are nonnegative. Since $g^{+}_{N}(x)\ge-Nb_{g}$ (as $|g|\le b_{g}$, $C_{0}\ge0$, $\omega_{g}\ge0$) and $\kappa_{N}\ge0$, the formula of The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space §operator gives

Hence $\overline{w}$ is a viscosity subsolution of $F_{1}$ on $W_{N}$. Symmetrically, $\underline{w}^{0}$ is lower semicontinuous by Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty §supersolution, so $\underline{w}$ is lower semicontinuous by The Maximum of Two Upper Semicontinuous Functions §min-lsc; if $\underline{w}-\varphi$ has a local minimum at $x$ relative to $W_{N}$, then either $\underline{w}^{0}(x)\le L_{N}$, $\underline{w}^{0}-\varphi$ has a local minimum at $x$ and Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty §supersolution gives $F_{2}(x,\underline{w}(x),D\varphi(x),D^{2}\varphi(x))\ge0$, or $\underline{w}^{0}(x)>L_{N}$, $\varphi$ has a local maximum at $x$, claim 1 of First- and Second-Order Conditions at a Local Extremum of a Function of Class $C^2$ gives $D\varphi(x)=0$ and $D^{2}\varphi(x)\preceq0$, so $\operatorname{tr}(D^{2}\varphi(x))\le0$, and with $g^{-}_{N}(x)\le Nb_{g}$

The functions $g^{\pm}_{N}$ are bounded: $|g^{\pm}_{N}(x)|\le Nb_{g}+C_{0}+2Nb_{g}$. They are Borel, that is, measurable with respect to the Borel $\sigma$-algebra $\mathcal{B}(W_{N})$ of the metric subspace $W_{N}$ of $\mathbb{R}^{N}$ and $\mathcal{B}(\mathbb{R})$. Indeed, $G_{1}(x)=G(x)\pm C_{0}$ is continuous on $W_{N}$ by Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §empirical, and $P_{N}$ is continuous on $W_{N}$ (as recorded in Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty); both are therefore measurable by claims 3 and 2 of Borel Measurability and Bounded Integration on a Metric Space. Let $T(p)=N\,\omega_{g}\bigl((2\tau(2\lambda^{-1}Nb_{g}+(p-p_{0})/N))^{1/2}/\sqrt{N}\bigr)$ for $p\ge p_{0}$ and $T(p)=0$ for $p<p_{0}$; then $T(P_{N}(x))=N\,\omega_{g}(r_{N}(x)/\sqrt{N})$. On the closed set $E=[p_{0},\infty)$, which lies in $\mathcal{B}(\mathbb{R})$ by claims 4 and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, $T$ is nondecreasing (the radicand is nondecreasing in $p$, square roots of nonnegative numbers preserve order, and $\omega_{g}$ is nondecreasing), so $T$ is measurable by A Function Nondecreasing on a Borel Subset of the Real Line and Vanishing Outside It is Borel §borel, and $G_{2}=T\circ P_{N}$ is measurable since $G_{2}^{-1}(B)=P_{N}^{-1}(T^{-1}(B))$. By claims 2 and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, $x\mapsto(G_{1}(x),\pm G_{2}(x))$ is measurable into $\mathcal{B}(\mathbb{R}^{2})$ (the map $s\mapsto-s$ being continuous, $-G_{2}$ is measurable in the same way), and the addition map $\mathbb{R}^{2}\to\mathbb{R}$, being continuous, is measurable by claim 3(a) of that lemma; composing as before, $g^{\pm}_{N}=G_{1}\pm G_{2}$ is measurable.

Step 3 (Clause 2). Two general remarks. For $\nu\in\mathcal{D}$, positive $\rho\le\rho'$ and $N_{1}\ge N_{1}'\ge N_{0}$, the sets of Upper and Lower Half-Relaxed Limits along Empirical Measures of Functions on the Weyl Chambers satisfy $S(\nu,\rho,N_{1})\subseteq S(\nu,\rho',N_{1}')$, so

For $\nu\in\mathcal{D}$ and positive $\rho$ let $N'(\nu,\rho)$ be the natural number $N_{1}$ given by The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §recovery for $\nu$ and $\varepsilon=\min(\rho,1)$: for every $N\ge N'(\nu,\rho)$ there is $y\in W_{N}$ with

Let $\mu\in\mathcal{D}$, $\rho>0$ and $N_{1}\ge N_{0}$. Every element $\overline{w}_{N,\tau}(x)/N$ of the set $S(\mu,\rho,N_{1})$ formed with $f_{N}=\overline{w}_{N,\tau}$ satisfies, by (1a), $\overline{w}_{N,\tau}(x)/N\le\lambda^{-1}b_{g}+|e_{*}|/N\le\lambda^{-1}b_{g}+|e_{*}|/N_{1}$; so $\bar{u}_{\tau}(\mu)\le s^{+}(\mu,\rho,N_{1})\le\lambda^{-1}b_{g}+|e_{*}|/N_{1}$ for every $N_{1}\ge N_{0}$, whence $\bar{u}_{\tau}(\mu)\le\lambda^{-1}b_{g}$. Next, for $N\ge\max(N_{1},N'(\mu,\rho))$ take $y$ as in (Rec) for $\nu=\mu$; then $\overline{w}_{N,\tau}(y)/N\in S(\mu,\rho,N_{1})$ and, by (1a) and $|v_{N}|\le\lambda^{-1}Nb_{g}$ (The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §particle-equation),

As $N$ is arbitrarily large, $s^{+}(\mu,\rho,N_{1})\ge-\lambda^{-1}b_{g}$, and taking the greatest lower bound, $\bar{u}_{\tau}(\mu)\ge-\lambda^{-1}b_{g}$. For $f_{N}=\underline{w}_{N,\tau}$, (1b) gives $s^{-}(\mu,\rho,N_{1})\ge-\lambda^{-1}b_{g}-|e_{*}|/N_{1}$, hence $\underline{u}_{\tau}(\mu)\ge-\lambda^{-1}b_{g}$ as $N_{1}$ is arbitrary; and with $y$ as above, $\underline{w}_{N,\tau}(y)/N\le v_{N}(y)/N+P_{N}(y)/N^{2}\le\lambda^{-1}b_{g}+(|\mathcal{E}(\mu)|+1)/N$ for all $N\ge\max(N_{1},N'(\mu,\rho))$, so $s^{-}(\mu,\rho,N_{1})\le\lambda^{-1}b_{g}$ and $\underline{u}_{\tau}(\mu)\le\lambda^{-1}b_{g}$.

Step 4 (Transfer at a level). Let $N\ge N_{0}$ and $x,y\in W_{N}$, and put $d=\lVert x-y\rVert/\sqrt{N}$ and $\Phi(d,R)=\frac{1}{2\tau}d(3d+2R)$, which is nondecreasing in each of $d\ge0$ and $R\ge0$. Dividing clause (iv) by $N$,

If $P_{N}(y)\le c'N$ for a real $c'$, then $(P_{N}(y)-p_{0})/N^{2}\le(|c'|+|e_{*}|)/N$ since $p_{0}\ge e_{*}N$, so by (R) and monotonicity of square roots

Step 5 (Clause 3). Let $\mu,\mu'\in\mathcal{D}$, $W=W_{2}(\mu,\mu')$ and $K'=|\mathcal{E}(\mu')|+1+|e_{*}|$. The choices are made in the order $\theta'$, $\rho$, $N_{1}$. Let $\theta'>0$. As $\bar{u}_{\tau}(\mu')$ is a greatest lower bound, there are $\rho_{0}>0$ and $N_{1}^{0}\ge N_{0}$ with $s^{+}(\mu',\rho_{0},N_{1}^{0})\le\bar{u}_{\tau}(\mu')+\theta'$. Let $0<\rho\le\min(\rho_{0},1)$, and let $N_{1}\ge\max(N_{1}^{0},N'(\mu',\rho))$. Since $\bar{u}_{\tau}(\mu)\le s^{+}(\mu,\rho,N_{1})$, which is a least upper bound, there are $N\ge N_{1}$ and $x\in W_{N}$ with $W_{2}(\mu_{x},\mu)<\rho$ and $\overline{w}_{N,\tau}(x)/N>\bar{u}_{\tau}(\mu)-\rho$. By (Rec) for $\nu=\mu'$ at this level $N\ge N'(\mu',\rho)$ there is $y\in W_{N}$ with $W_{2}(\mu_{y},\mu')<\rho$ and $P_{N}(y)\le N(|\mathcal{E}(\mu')|+1)$. Then $\overline{w}_{N,\tau}(y)/N\in S(\mu',\rho,N_{1})$, so by (M) $\overline{w}_{N,\tau}(y)/N\le s^{+}(\mu',\rho_{0},N_{1}^{0})\le\bar{u}_{\tau}(\mu')+\theta'$. By (Q) with $\nu=\mu$, The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §triangle and The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §symmetry, $d\le W_{2}(\mu_{x},\mu)+W_{2}(\mu_{y},\mu')+W<W+2\rho$; by (B) with $c'=|\mathcal{E}(\mu')|+1$, $r_{N}(y)/\sqrt{N}\le(R_{\tau}^{2}+2\tau K'/N_{1})^{1/2}$. So (T) gives

The right side depends continuously on $1/N_{1}\ge0$; letting $N_{1}\to\infty$, then $\rho\to0$, then $\theta'\to0$ yields $\bar{u}_{\tau}(\mu)\le\bar{u}_{\tau}(\mu')+\Phi(W,R_{\tau})$, the first inequality of clause 3. For the second: given $\theta'>0$ choose $\rho_{0},N_{1}^{0}$ with $s^{-}(\mu',\rho_{0},N_{1}^{0})\ge\underline{u}_{\tau}(\mu')-\theta'$; for $\rho,N_{1}$ as above, since $s^{-}(\mu,\rho,N_{1})\le\underline{u}_{\tau}(\mu)$ is a greatest lower bound there are $N\ge N_{1}$ and $x\in W_{N}$ with $W_{2}(\mu_{x},\mu)<\rho$ and $\underline{w}_{N,\tau}(x)/N<\underline{u}_{\tau}(\mu)+\rho$; with $y$ from (Rec) for $\mu'$, (M) gives $\underline{w}_{N,\tau}(y)/N\ge\underline{u}_{\tau}(\mu')-\theta'$, and the second inequality of (T) gives $\underline{u}_{\tau}(\mu)+\rho>\underline{u}_{\tau}(\mu')-\theta'-\Phi(W+2\rho,(R_{\tau}^{2}+2\tau K'/N_{1})^{1/2})$; the same limits give $\underline{u}_{\tau}(\mu)\ge\underline{u}_{\tau}(\mu')-\Phi(W,R_{\tau})$.

Step 6 (Clause 4). Since $(N_{k})$ is a strictly increasing sequence of natural numbers, $N_{k}\ge N_{1}+k-1\ge k$. Let $\varepsilon>0$. Choose $\rho>0$ and $\hat{N}\ge N_{0}$ with $s^{+}(\mu,\rho,\hat{N})<\bar{u}_{\tau}(\mu)+\varepsilon/2$ (greatest lower bound), and then $k_{1}$ with $N_{k}\ge\hat{N}$ and $W_{2}(\mu^{N_{k}}_{x^{k}},\mu)<\rho$ for $k\ge k_{1}$. For such $k$, $\overline{w}_{N_{k},\tau}(x^{k})/N_{k}\in S(\mu,\rho,\hat{N})$, so $\overline{w}_{N_{k},\tau}(x^{k})/N_{k}<\bar{u}_{\tau}(\mu)+\varepsilon/2$. Symmetrically, with $\rho',\hat{N}'$ such that $s^{-}(\mu,\rho',\hat{N}')>\underline{u}_{\tau}(\mu)-\varepsilon/2$, there is $k_{2}$ with $\underline{w}_{N_{k},\tau}(x^{k})/N_{k}>\underline{u}_{\tau}(\mu)-\varepsilon/2$ for $k\ge k_{2}$. This proves the first display with $k_{0}=\max(k_{1},k_{2})$. If moreover $P_{N_{k}}(x^{k})\le c'N_{k}$ for all $k$, choose $k_{3}$ with $|c'|/k<\varepsilon/2$ for $k\ge k_{3}$; then by (1a) and (1b), for $k\ge\max(k_{1},k_{2},k_{3})$,

Step 7 (Clause 5). Let $\mu\in\mathcal{D}$. Define $N_{k}$ and $z^{k}\in W_{N_{k}}$ recursively: put $\hat{N}_{1}=N_{0}$; given $\hat{N}_{k}\ge N_{0}$, since $\bar{u}_{\tau}(\mu)\le s^{+}(\mu,1/k,\hat{N}_{k})$, which is a least upper bound, choose $N_{k}\ge\hat{N}_{k}$ and $z^{k}\in W_{N_{k}}$ with $W_{2}(\mu^{N_{k}}_{z^{k}},\mu)<1/k$ and $\overline{w}_{N_{k},\tau}(z^{k})/N_{k}>\bar{u}_{\tau}(\mu)-1/k$, and put $\hat{N}_{k+1}=N_{k}+1$. Then $(N_{k})$ is strictly increasing with $N_{1}\ge N_{0}$. Let $x^{k}\in W_{N_{k}}$ with $\alpha_{k}=W_{2}(\mu^{N_{k}}_{x^{k}},\mu)\to0$ and $P_{N_{k}}(x^{k})\le c'N_{k}$. Apply (T) at level $N_{k}$ with $(x,y)=(z^{k},x^{k})$: by (Q) with $\nu=\mu$, $d_{k}=\lVert z^{k}-x^{k}\rVert/\sqrt{N_{k}}\le1/k+\alpha_{k}$, and by (B) with $N_{k}\ge1$, $r_{N_{k}}(x^{k})/\sqrt{N_{k}}\le B:=(R_{\tau}^{2}+2\tau(|c'|+|e_{*}|))^{1/2}$. Hence

and $\frac{1}{k}+\Phi(\frac{1}{k}+\alpha_{k},B)\to0$. Given $\varepsilon>0$, this bound and clause 4 (Step 6) give $k_{0}$ with $|\overline{w}_{N_{k},\tau}(x^{k})/N_{k}-\bar{u}_{\tau}(\mu)|<\varepsilon$ for $k\ge k_{0}$; so the sequence converges to $\bar{u}_{\tau}(\mu)$. For the lower limit the construction is repeated independently. Put $\hat{N}'_{1}=N_{0}$; given $\hat{N}'_{k}\ge N_{0}$, since $s^{-}(\mu,1/k,\hat{N}'_{k})\le\underline{u}_{\tau}(\mu)$ and $s^{-}(\mu,1/k,\hat{N}'_{k})$ is a greatest lower bound, choose $N'_{k}\ge\hat{N}'_{k}$ and $z'^{k}\in W_{N'_{k}}$ with $W_{2}(\mu^{N'_{k}}_{z'^{k}},\mu)<1/k$ and $\underline{w}_{N'_{k},\tau}(z'^{k})/N'_{k}<\underline{u}_{\tau}(\mu)+1/k$, and put $\hat{N}'_{k+1}=N'_{k}+1$; then $(N'_{k})$ is strictly increasing with $N'_{1}\ge N_{0}$. Now let $x'^{k}\in W_{N'_{k}}$ be arbitrary with $\alpha'_{k}=W_{2}(\mu^{N'_{k}}_{x'^{k}},\mu)\to0$ and $P_{N'_{k}}(x'^{k})\le c'N'_{k}$ for every $k$. By (Q) with $\nu=\mu$, $d'_{k}=\lVert z'^{k}-x'^{k}\rVert/\sqrt{N'_{k}}\le1/k+\alpha'_{k}$, and by (B), $r_{N'_{k}}(x'^{k})/\sqrt{N'_{k}}\le B$ with the same $B$. The second inequality of (T) at level $N'_{k}$ with $(x,y)=(z'^{k},x'^{k})$ gives