Proof of Classical Solutions of the N-Particle Hamilton-Jacobi Equation Lift to Classical Solutions of the Lifted Equation, and a Bounded One is the Lifted Viscosity Solution

Each result cited is universally quantified over the data in its own statement, and is applied to the data named at the point of use. The rules of Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic in an Ordered Field for rearranging sums and products of real numbers and for adding and scaling inequalities (among them $s-t=s+(-1)t$) are used without further mention. Every application below of a result or notion on the Wasserstein space in dimension $dN$ is made at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level), and this is not repeated at each use.

Step 1 (the data and the two operators). By The N-Particle Potential of a Confining Potential is a Confining Potential on the Configuration Space §regularity and The N-Particle Potential of a Confining Potential is a Confining Potential on the Configuration Space §confining, $V_{N}$ is of class $C^{2}$ on $\mathbb{R}^{dN}$, with gradient $DV_{N}(x)$, and is a confining potential on $\mathbb{R}^{dN}$. By The N-Particle Hamilton-Jacobi Equation with Individual and Common Noise on the Configuration Space §common-noise, $\Gamma_{N}\in\mathcal{M}_{p\times dN}(\mathbb{R})$. Since $c$ is continuous for the Euclidean distance and the absolute-value metric, it is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps; being bounded, it is integrable with respect to every probability measure on $\mathbb{R}^{dN}$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, so the data of The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space are in place. Let $F$ be the $N$-particle Hamilton-Jacobi operator of The N-Particle Hamilton-Jacobi Equation with Individual and Common Noise on the Configuration Space §operator and $F_{N}$ the lifted $N$-particle Hamilton-Jacobi operator of The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space §operator, both with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, control cost $\theta$, common-noise matrix $\Gamma$ and running cost $c$. By The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space §operator and The Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics in a Confining Potential on the Wasserstein Space §operator, $F_{N}$ is the Hamilton-Jacobi operator with common noise and penalty drift of the pair $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$ with discount $\lambda_{0}$, common-noise matrix $\Gamma_{N}$, control cost $\theta$ and running cost $G(P)=\int_{\mathbb{R}^{dN}}c\,dP$, a second-order equation operator over $\mathcal{D}_{N,\Sigma}$. By The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §pair, applied at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level) with the confining potential $V_{N}$ and $\sigma$, this quadruple is a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{dN})$, so $\mathcal{D}_{N,\Sigma}\subseteq\mathcal{D}_{N}\subseteq\mathcal{P}_{2}(\mathbb{R}^{dN})$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair.

By The N-Particle Hamilton-Jacobi Equation with Individual and Common Noise on the Configuration Space §equation, $v$ is a classical subsolution (supersolution) of the $N$-particle equation exactly when $F(x,v(x),Dv(x),D^{2}v(x))\le0$ (respectively $\ge0$) for every $x\in\mathbb{R}^{dN}$, in the sense of Classical Subsolution and Supersolution of a Second-Order Equation with $n=dN$ and $U=\mathbb{R}^{dN}$. By The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space §equation, The Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics in a Confining Potential on the Wasserstein Space §equation and The Discounted Hamilton-Jacobi Equation with Common Noise and a Penalty Drift on the Wasserstein Space §equation, an intrinsic test function $u$ on $\mathcal{D}_{N,\Sigma}$ is a classical subsolution, supersolution or solution of the lifted $N$-particle equation exactly when it is a classical subsolution, supersolution or solution of $F_{N}$ on $\mathcal{D}_{N,\Sigma}$ in the sense of Classical Sub- and Supersolutions of a Second-Order Equation on the Wasserstein Space §subsolution, Classical Sub- and Supersolutions of a Second-Order Equation on the Wasserstein Space §supersolution and Classical Sub- and Supersolutions of a Second-Order Equation on the Wasserstein Space §solution; and a function $\mathcal{D}_{N}\to\mathbb{R}$ is a viscosity solution of the lifted equation exactly when it is a viscosity solution of $F_{N}$ relative to the penalty pair.

Step 2 (gradient and translation Hessian of $\varphi_{v}$). By Integrals of Functions with Bounded First and Second Derivatives are Intrinsic Test Functions on the Wasserstein Space §integrable and Integrals of Functions with Bounded First and Second Derivatives are Intrinsic Test Functions on the Wasserstein Space §test, applied at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level) with $f=v$ and the constant $M$, the function $v$ is Borel and integrable with respect to every $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, each entry of $D^{2}v$ is Borel and bounded, and $\varphi_{v}$ is an intrinsic test function on $\mathcal{P}_{2}(\mathbb{R}^{dN})$ with

for every $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$. By Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §restriction, applied at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level) with $Q=\mathcal{P}_{2}(\mathbb{R}^{dN})$ and $Q'$ equal to $\mathcal{D}_{N}$ or to $\mathcal{D}_{N,\Sigma}$ (subsets of $Q$ by Step 1), $\varphi_{v}$ is an intrinsic test function on $\mathcal{D}_{N}$ and on $\mathcal{D}_{N,\Sigma}$, with the same gradients along couplings and translation Hessians.

Step 3 (the terms of $F_{N}$ at $\varphi_{v}$). Fix $P\in\mathcal{D}_{N,\Sigma}$. By The Langevin Free-Energy Pair of a Confining Potential on the Wasserstein Space §pair, read with the potential $V_{N}$, $P$ has finite Fisher information, with score $\xi_{P}$, and $\int_{\mathbb{R}^{dN}}\lVert DV_{N}\rVert^{2}\,dP<\infty$, so the gradient map $\nabla V_{N}:x\mapsto DV_{N}(x)$ has a class $\nabla V_{N}\in L^{2}(P;\mathbb{R}^{dN})$. By Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields §tangent, applied at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level) with $\mu=P$ and $f=v$, the gradient map of $v$ is Borel with $\int_{\mathbb{R}^{dN}}\lVert Dv\rVert^{2}\,dP<\infty$. Recall from Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields that $L^{2}(P;\mathbb{R}^{dN})$ is the space of The Space of Square-Integrable Random Vectors on the probability space $(\mathbb{R}^{dN},\mathcal{B}(\mathbb{R}^{dN}),P)$, with $\langle\eta,\eta'\rangle_{P}=\int_{\mathbb{R}^{dN}}\eta\cdot\eta'\,dP$, a real inner product space by The Space of Square-Integrable Random Vectors §inner-product.

(a) The drift term. By Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §operations, applied on that probability space to the representatives $\nabla V_{N}$ and $Dv$, the function $x\mapsto DV_{N}(x)\cdot Dv(x)$ is integrable with respect to $P$, and $\langle\nabla V_{N},Dv\rangle_{P}=\int_{\mathbb{R}^{dN}}DV_{N}\cdot Dv\,dP$.

(b) The score term. By Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields §score-identity, applied at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level) with $\mu=P$ and $f=v$ (whose first and second partial derivatives are bounded by $M$), $\Delta v$ is bounded and Borel, hence integrable with respect to $P$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, and $\langle\xi_{P},Dv\rangle_{P}=-\int_{\mathbb{R}^{dN}}\Delta v\,dP$. Moreover, for every $x\in\mathbb{R}^{dN}$, $\Delta v(x)=\sum_{i=1}^{dN}\partial_{i}\partial_{i}v(x)$ by The Laplacian of a Twice Continuously Differentiable Function §laplacian, $\partial_{i}\partial_{i}v(x)$ is the $i$-th diagonal entry of $D^{2}v(x)$ by Hessian Matrix of a C^2 Function, and so $\Delta v(x)=\mathrm{tr}\bigl(D^{2}v(x)\bigr)$ by Trace of a Real Square Matrix.

(c) The combined drift. By conditions (b) and (c) of Real Inner Product Space §inner-product in $L^{2}(P;\mathbb{R}^{dN})$, together with (a) and (b),

(d) The control term. By Real Inner Product Space §norm and Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §operations (the identity $\mathbb{E}[X\cdot X]=\mathbb{E}[\lVert X\rVert^{2}]$ on that probability space), $\lVert Dv\rVert_{P}^{2}=\langle Dv,Dv\rangle_{P}=\int_{\mathbb{R}^{dN}}\lVert Dv\rVert^{2}\,dP$; the Borel function $x\mapsto\lVert Dv(x)\rVert^{2}$ (a composition of Borel maps by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps) is nonnegative with finite integral, hence integrable with respect to $P$.

(e) The common-noise term. For $j\in[p]$ write $g_{j}=\gamma_{j}^{\oplus}\in\mathbb{R}^{dN}$, with coordinates $g_{j,i}$ for $i\in[dN]$. For $X\in\mathcal{S}(dN)$, The N-Particle Hamilton-Jacobi Equation with Individual and Common Noise on the Configuration Space §common-noise and claim 4 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum (with $n=dN$, $M=X$ and $w=z=g_{j}$ for each $j$) give

Apply (T) with $X=D^{2}v(x)$, which lies in $\mathcal{S}(dN)$ by Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, and put $T(x)=\mathrm{tr}(\Gamma_{N}^{\top}\Gamma_{N}D^{2}v(x))$. Each entry $x\mapsto D^{2}v(x)_{il}$ is bounded and Borel (Step 2), hence integrable with respect to $P$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, and its integral is the entry $(\bar{X}_{P})_{il}$ by Integrals of Functions with Bounded First and Second Derivatives are Intrinsic Test Functions on the Wasserstein Space §integrable. Applying Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §integrable and Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear first to the innermost sum in (T) (with $m=dN$, the functions $D^{2}v(\cdot)_{il}$ and the coefficients $g_{j,i}g_{j,l}$), then to the middle and outer sums (with coefficients $1$), we obtain that $T$ is integrable with respect to $P$ and

the last equality being (T) with $X=\bar{X}_{P}\in\mathcal{S}(dN)$ (Step 2).

Step 4 (the integral identity). For $x\in\mathbb{R}^{dN}$ put $h(x)=F\bigl(x,v(x),Dv(x),D^{2}v(x)\bigr)$. By The N-Particle Hamilton-Jacobi Equation with Individual and Common Noise on the Configuration Space §operator and (b),

a linear combination, with the real coefficients $\lambda_{0},\tfrac{\theta}{2},1,-\tfrac{\sigma^{2}}{2},-\tfrac{1}{2},-1$, of the six functions $v$, $\lVert Dv\rVert^{2}$, $DV_{N}\cdot Dv$, $\mathrm{tr}(D^{2}v)=\Delta v$, $T$ and $c$, each integrable with respect to $P$ by Step 2, (d), (a), (b), (e) and Step 1. By Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §integrable and Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear (with $m=6$), $h$ is integrable with respect to $P$ and

all integrals over $\mathbb{R}^{dN}$. On the other hand, by Step 2 and The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space §operator, with $\varphi_{v}(P)=\int v\,dP$,

Substituting (e), (d) and (c) into the second, third and fourth terms, the two right-hand sides agree, so

Step 5 (clauses 1 and 2). The zero function is integrable with respect to $P$ with integral $0$, by claim 2 of Linearity and Monotonicity of the Lebesgue Integral applied to $0\cdot h+0\cdot h$. If $v$ is a classical subsolution of the $N$-particle equation, then $h(x)\le0$ for every $x\in\mathbb{R}^{dN}$ (Step 1), so $\int h\,dP\le0$ by the monotonicity in claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and by (I) $F_{N}(P,\varphi_{v}(P),\nabla\varphi_{v}(P),H_{\varphi_{v}}(P))\le0$ for every $P\in\mathcal{D}_{N,\Sigma}$. As $\varphi_{v}$ is an intrinsic test function on $\mathcal{D}_{N,\Sigma}$ (Step 2), it is a classical subsolution of $F_{N}$ on $\mathcal{D}_{N,\Sigma}$ by Classical Sub- and Supersolutions of a Second-Order Equation on the Wasserstein Space §subsolution, that is (Step 1) a classical subsolution of the lifted $N$-particle equation: clause 1. If $v$ is a classical supersolution, then $0\le h(x)$ for every $x$, so $0\le\int h\,dP$ by the same monotonicity, and Classical Sub- and Supersolutions of a Second-Order Equation on the Wasserstein Space §supersolution gives clause 2 in the same way.

Step 6 (clause 3: $\varphi_{v}$ on $\mathcal{D}_{N}$ is a viscosity solution). Assume the hypotheses of clause 3. By clauses 1 and 2, $\varphi_{v}$ is a classical subsolution and a classical supersolution of $F_{N}$ on $\mathcal{D}_{N,\Sigma}$, hence a classical solution of $F_{N}$ on $\mathcal{D}_{N,\Sigma}$ by Classical Sub- and Supersolutions of a Second-Order Equation on the Wasserstein Space §solution. Write $w:\mathcal{D}_{N}\to\mathbb{R}$ for the restriction of $\varphi_{v}$ to $\mathcal{D}_{N}$. We record the hypotheses of Classical Sub- and Supersolutions of a Degenerate Elliptic Equation on the Wasserstein Space are Viscosity Sub- and Supersolutions for the penalty pair $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$ (Step 1) and the operator $F_{N}$.

(P1) The pair has regular penalised maxima by The Langevin Free-Energy Pair is Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima §regular, and is Wasserstein-coercive by The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §coercive, both applied at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level) with the confining potential $V_{N}$ on $\mathbb{R}^{dN}$ and $\sigma$.

(P2) $\mathcal{E}_{N}$ is lower semicontinuous on $\mathcal{D}_{N}$ relative to $\mathcal{D}_{N}$, by The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §growth, applied at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level) with the same data.

(P3) $F_{N}$ is degenerate elliptic, by The Hamilton-Jacobi Operator with Common Noise and Penalty Drift is Degenerate Elliptic, applied at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level) to the penalty pair of Step 1, with the positive numbers $\lambda_{0}$ and $\theta$, the natural number $p$, $\Gamma_{N}\in\mathcal{M}_{p\times dN}(\mathbb{R})$ and the function $G:\mathcal{P}_{2}(\mathbb{R}^{dN})\to\mathbb{R}$; by Step 1, $F_{N}$ is the operator named there for these data.

(P4) $\varphi_{v}$ is an intrinsic test function on $\mathcal{D}_{N}$ (Step 2).

(P5) $w$ has penalty-subordinate growth from above and from below. Indeed, $v$ is bounded, so there is $B\in\mathbb{R}$ with $0\le B$ and $|v(x)|\le B$, that is $-B\le v(x)\le B$ (claim 6 of Properties of the Absolute Value in an Ordered Field), for every $x\in\mathbb{R}^{dN}$ (Bounded Real-Valued Function on a Set). Let $P\in\mathcal{D}_{N}$. The constant function $1=\mathbf{1}_{\mathbb{R}^{dN}}$ is bounded and Borel, hence integrable with respect to $P$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures), with $\int1\,dP=P(\mathbb{R}^{dN})=1$ by The Integral of an Indicator Function is the Measure of the Set and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures; by claim 2 of Linearity and Monotonicity of the Lebesgue Integral the constants $B$ and $-B$ (the multiples $B\cdot1$ and $(-B)\cdot1$) are integrable with integrals $B$ and $-B$, and, $v$ being integrable (Step 2), the monotonicity there gives $-B\le\int v\,dP\le B$, that is $-B\le w(P)\le B$. By The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth, applied at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level) to the Wasserstein-coercive penalty pair (Step 1 and (P1)) with the bounds $B$ and $-B$, $w$ has penalty-subordinate growth from above and from below.

By (P1)-(P5), Classical Sub- and Supersolutions of a Degenerate Elliptic Equation on the Wasserstein Space are Viscosity Sub- and Supersolutions §solution, applied at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level) to the pair, the degenerate elliptic operator $F_{N}$ over $\mathcal{D}_{N,\Sigma}$ and $u=\varphi_{v}$, a classical solution of $F_{N}$ on $\mathcal{D}_{N,\Sigma}$, shows that $w$ is a viscosity solution of $F_{N}$ relative to the pair, hence (Step 1) a viscosity solution of the lifted $N$-particle equation. By (P5), $|w(P)|\le B$ for every $P\in\mathcal{D}_{N}$ (claim 6 of Properties of the Absolute Value in an Ordered Field), so $w$ is bounded.

Step 7 (clause 3: uniqueness). Since $c$ is bounded there is $b\in\mathbb{R}$ with $|c(x)|\le b$ for every $x\in\mathbb{R}^{dN}$ (Bounded Real-Valued Function on a Set); $c$ is uniformly continuous by hypothesis, and $0<\theta\le1$. Hence Well-Posedness of the Lifted N-Particle Hamilton-Jacobi Equation: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness applies with the data $V$, $\lambda_{0}$, $\sigma$, $\theta$, $p$, $\Gamma$, $c$ and $b$, whose viscosity solutions are those of the lifted $N$-particle equation of the statement, functions on the same domain $\mathcal{D}_{N}$: if $u:\mathcal{D}_{N}\to\mathbb{R}$ is any bounded viscosity solution of the lifted equation, then $u$ and $w$ are two bounded viscosity solutions, so $u=w$. Thus $w$ is the only bounded viscosity solution of the lifted $N$-particle equation, which completes clause 3.