Proof of Viscosity Sub- and Supersolutions on the Torus Wasserstein Space are Entropy-Penalised Viscosity Sub- and Supersolutions

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

Claim 1 is proved in Steps 0 to 7; Step 8 lists precisely what changes for claim 2. The choices are made in this order: the constants $\delta_{0}$, $C_{R}$ and the function $c$ (Step 0); then $\varphi$, $\delta$ and $\mu$ (Step 2); then the times $s_{n}$, the functions $\psi_{n}$ and the maximisers $\mu_{n}$ (Step 2); then the integer $K$ and the index $n_{0}$ (Step 5); then the constants $B$, $M$ and $C$ (Step 6).

Step 0 (Constants). Let $\delta_{0}>0$ and, for every real $R>0$, $C_{R}\ge0$ be as in the score-perturbation bound. For a real $r\ge0$ the set of integers $k$ with $1\le k\le r+1$ is finite and contains $1$; let $c(r)$ be the largest of the numbers $C_{k}$ with $k$ in this set. Then $c(r)\ge0$, and $c$ is nondecreasing on $[0,\infty)$, because for $r\le r'$ the set belonging to $r$ is contained in the set belonging to $r'$. We show that $\delta_{0}$ and $c$ have the property required in Entropy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space §sub (and, in Step 8, in Entropy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space §super).

Step 1 (Three auxiliary facts). (a) By The Torus Wasserstein Space is a Sequentially Compact Metric Space in which Optimal Couplings Exist §metric and The Torus Wasserstein Space is a Sequentially Compact Metric Space in which Optimal Couplings Exist §compact, $\mathcal{P}(\mathbb{T}^{d})$ is sequentially compact in the metric space $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$, hence compact by A Sequentially Compact Subset of a Metric Space is Compact; it is nonempty (the Dirac measure at the origin belongs to it, by Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures §measure and $0\in Q$). Hence every upper semicontinuous function on $\mathcal{P}(\mathbb{T}^{d})$ attains a maximum, by claim 1 of Semicontinuous Functions Attain Their Extrema on a Compact Set; every lower semicontinuous one attains a minimum, by claim 2 of that theorem; and every continuous one attains a minimum and a maximum, by Extreme Value Theorem on a Compact Subset of a Metric Space.

(b) Let $E$ be real, and let $(\lambda_{j})_{j\in\mathbb{N}}$ be a sequence in $\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ with $\mathrm{Ent}(\lambda_{j})\le E$ for every $j$ that converges to some $\bar\lambda$. Then $\bar\lambda\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ by The Entropy of Measures on the Torus: Nonnegativity, Absolute Continuity and Closed Sublevel Sets §closed, and for every real $\varepsilon>0$ there is $J$ with $\mathrm{Ent}(\lambda_{j})>\mathrm{Ent}(\bar\lambda)-\varepsilon$ for every $j\ge J$. Indeed, otherwise the set of indices $j$ with $\mathrm{Ent}(\lambda_{j})\le\mathrm{Ent}(\bar\lambda)-\varepsilon$ is infinite; listing it increasingly gives a subsequence of $(\lambda_{j})$, which converges to $\bar\lambda$ by A Subsequence of a Convergent Sequence Has the Same Limit, and The Entropy of Measures on the Torus: Nonnegativity, Absolute Continuity and Closed Sublevel Sets §closed applied to it with the bound $\mathrm{Ent}(\bar\lambda)-\varepsilon$ gives $\mathrm{Ent}(\bar\lambda)\le\mathrm{Ent}(\bar\lambda)-\varepsilon$, which is absurd.

(c) Let $(\lambda_{n})_{n\in\mathbb{N}}$ converge to $\lambda$ in $\mathcal{P}(\mathbb{T}^{d})$ and let $(t_{n})_{n\in\mathbb{N}}$ be real numbers with $0<t_{n}\le\tfrac12$ converging to $0$. By the triangle inequality for the metric $W_{\mathbb{T}}$ (The Torus Wasserstein Space is a Sequentially Compact Metric Space in which Optimal Couplings Exist §metric) and Heat Smoothing on the Torus: Distance to the Identity, Contraction, a Smooth Positive Periodic Density, Finite Entropy, and Finite Fisher Information §distance,

$$W_{\mathbb{T}}(S_{t_{n}}\lambda_{n},\lambda)\le W_{\mathbb{T}}(S_{t_{n}}\lambda_{n},\lambda_{n})+W_{\mathbb{T}}(\lambda_{n},\lambda)\le\sqrt{d\,t_{n}}+W_{\mathbb{T}}(\lambda_{n},\lambda),$$

so $(S_{t_{n}}\lambda_{n})_{n\in\mathbb{N}}$ converges to $\lambda$.

Step 2 (Setup). Let $u$ be an upper semicontinuous viscosity subsolution of $(\mathrm{E})$. Fix an intrinsic test function $\varphi$ on $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$, a real $\delta$ with $0<\delta\le\delta_{0}$, and $\mu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ with $G(\nu)<G(\mu)$ for every $\nu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ with $\nu\ne\mu$, where

$$G(\nu):=u(\nu)-\varphi(\nu)-\delta\,\mathrm{Ent}(\nu)\qquad(\nu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})).$$

The measure $\mu$ is absolutely continuous by The Entropy of Measures on the Torus: Nonnegativity, Absolute Continuity and Closed Sublevel Sets §nonnegative, so $\nabla\varphi(\mu)\in T_{\mu}$ by Intrinsic Test Functions on the Torus Wasserstein Space §differentiability; put $r:=\lVert\nabla\varphi(\mu)\rVert_{\mu}$. The function $\varphi$ is continuous on $\mathcal{P}(\mathbb{T}^{d})$ by Intrinsic Test Functions on the Torus Wasserstein Space §continuity.

For $n\in\mathbb{N}$ put $s_{n}:=1/(2(n+1))$, so that $0<s_{n}\le\tfrac12$ and $(s_{n})$ converges to $0$, and

$$\psi_{n}:=\varphi\circ S_{s_{n}}+\delta\,\mathrm{Ent}\circ S_{s_{n}}.$$

By Heat Regularisation of an Intrinsic Test Function: a Laplacian Test Function whose Laplacian is Minus the Pairing of the Gradient with the Score §test and Heat Regularisation of the Entropy: a Laplacian Test Function whose Laplacian is Minus the Fisher Information §test, $\varphi\circ S_{s_{n}}$ and $\mathrm{Ent}\circ S_{s_{n}}$ are Laplacian test functions, with gradient fields $g^{n}(\nu,x):=A^{\nu}_{s_{n}}\nabla\varphi(S_{s_{n}}\nu)(x)$ and $h^{n}(\nu,x):=A^{\nu}_{s_{n}}\xi_{S_{s_{n}}\nu}(x)$ respectively, and $\nabla(\varphi\circ S_{s_{n}})(\nu)=A^{\nu}_{s_{n}}\nabla\varphi(S_{s_{n}}\nu)$ and $\nabla(\mathrm{Ent}\circ S_{s_{n}})(\nu)=A^{\nu}_{s_{n}}\xi_{S_{s_{n}}\nu}$ for every $\nu\in\mathcal{P}(\mathbb{T}^{d})$. Both are intrinsic test functions on $\mathcal{P}(\mathbb{T}^{d})$ by Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space §test. By Laplacian Test Functions on the Torus: Uniqueness of the Gradient Field, a Criterion, and Linearity of the Test Classes §linear, $\psi_{n}$ is a Laplacian test function with gradient field $g^{n}+\delta h^{n}$, and

$$\nabla\psi_{n}(\nu)=A^{\nu}_{s_{n}}\nabla\varphi(S_{s_{n}}\nu)+\delta\,A^{\nu}_{s_{n}}\xi_{S_{s_{n}}\nu}\qquad(\nu\in\mathcal{P}(\mathbb{T}^{d})).$$

By Laplacian Test Functions on the Torus: Uniqueness of the Gradient Field, a Criterion, and Linearity of the Test Classes §unique, $g^{n}+\delta h^{n}$, $g^{n}$ and $h^{n}$ are the gradient fields of $\psi_{n}$, $\varphi\circ S_{s_{n}}$ and $\mathrm{Ent}\circ S_{s_{n}}$ used in The Laplacian of a Laplacian Test Function on the Torus Wasserstein Space §laplacian. Since each partial derivative of a component of $g^{n}(\nu,\cdot)+\delta h^{n}(\nu,\cdot)$ is the corresponding combination of partial derivatives, and the integral is linear (claim 2 of Linearity and Monotonicity of the Lebesgue Integral), that definition gives $\mathcal{L}\psi_{n}=\mathcal{L}(\varphi\circ S_{s_{n}})+\delta\,\mathcal{L}(\mathrm{Ent}\circ S_{s_{n}})$; so Heat Regularisation of an Intrinsic Test Function: a Laplacian Test Function whose Laplacian is Minus the Pairing of the Gradient with the Score §laplacian and Heat Regularisation of the Entropy: a Laplacian Test Function whose Laplacian is Minus the Fisher Information §laplacian yield, for every $\nu\in\mathcal{P}(\mathbb{T}^{d})$,

$$\mathcal{L}\psi_{n}(\nu)=-\bigl\langle\nabla\varphi(S_{s_{n}}\nu),\xi_{S_{s_{n}}\nu}\bigr\rangle_{S_{s_{n}}\nu}-\delta\,\mathcal{I}(S_{s_{n}}\nu).\qquad(2.1)$$

The function $\psi_{n}$ is continuous by Intrinsic Test Functions on the Torus Wasserstein Space §continuity, so $u-\psi_{n}$ is upper semicontinuous (at a point, given $\varepsilon>0$, take the smaller of the radii provided by the upper semicontinuity of $u$ and the continuity of $\psi_{n}$, each for $\varepsilon/2$). By Step 1(a) we may choose $\mu_{n}\in\mathcal{P}(\mathbb{T}^{d})$ with $u(\nu)-\psi_{n}(\nu)\le u(\mu_{n})-\psi_{n}(\mu_{n})$ for every $\nu\in\mathcal{P}(\mathbb{T}^{d})$.

Notation: $\nu_{n}:=S_{s_{n}}\mu_{n}$, which lies in $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})\cap\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})\cap\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ by Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation §measures; $F_{n}:=\nabla\varphi(\nu_{n})\in T_{\nu_{n}}$; $\xi_{n}:=\xi_{\nu_{n}}$; $I_{n}:=\mathcal{I}(\nu_{n})=\lVert\xi_{n}\rVert_{\nu_{n}}^{2}$ by Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §information; $p_{n}:=A^{\mu_{n}}_{s_{n}}F_{n}$ and $z_{n}:=A^{\mu_{n}}_{s_{n}}\xi_{n}$. Thus $p_{n}=\nabla(\varphi\circ S_{s_{n}})(\mu_{n})$ and $z_{n}=\nabla(\mathrm{Ent}\circ S_{s_{n}})(\mu_{n})$, both in $T_{\mu_{n}}$ by Intrinsic Test Functions on the Torus Wasserstein Space §differentiability, and $\nabla\psi_{n}(\mu_{n})=p_{n}+\delta z_{n}$.

Step 3 (Comparison with $\mu$ and an entropy bound). Taking $\nu=\mu$ in the maximality of $\mu_{n}$, and using $\mathrm{Ent}(S_{s_{n}}\mu)\le\mathrm{Ent}(\mu)$ from Heat Smoothing on the Torus: Distance to the Identity, Contraction, a Smooth Positive Periodic Density, Finite Entropy, and Finite Fisher Information §entropy together with $\delta>0$,

$$u(\mu_{n})-\varphi(\nu_{n})-\delta\,\mathrm{Ent}(\nu_{n})\ge u(\mu)-\varphi(S_{s_{n}}\mu)-\delta\,\mathrm{Ent}(S_{s_{n}}\mu)\ge u(\mu)-\varphi(S_{s_{n}}\mu)-\delta\,\mathrm{Ent}(\mu).\qquad(3.1)$$

By Step 1(a) there are reals $U$, $m$, $m'$ with $u\le U$ and $m\le\varphi\le m'$ on $\mathcal{P}(\mathbb{T}^{d})$. Then (3.1) gives $\mathrm{Ent}(\nu_{n})\le E:=\delta^{-1}\bigl(U-m-u(\mu)+m'\bigr)+\mathrm{Ent}(\mu)$ for every $n$.

Step 4 ($\mu_{n}\to\mu$ and $u(\mu_{n})\to u(\mu)$). First, every subsequence $(\mu_{n_{k}})_{k\in\mathbb{N}}$ has a further subsequence converging to $\mu$. By Step 1(a) there are a strictly increasing $(k_{j})_{j\in\mathbb{N}}$ and $\bar\mu$ with $(\mu_{n_{k_{j}}})_{j}$ converging to $\bar\mu$. Put $m_{j}:=n_{k_{j}}$; this is strictly increasing by claim 2 of A Subsequence of a Subsequence is a Subsequence, so $m_{j}\ge j$ and $(s_{m_{j}})_{j}$ converges to $0$. By Step 1(c), $(\nu_{m_{j}})_{j}$ converges to $\bar\mu$ and $(S_{s_{m_{j}}}\mu)_{j}$ converges to $\mu$. By Step 1(b) with the bound $E$ of Step 3, $\bar\mu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$. Let $\varepsilon>0$. By the upper semicontinuity of $u$ at $\bar\mu$, the continuity of $\varphi$ at $\bar\mu$ and at $\mu$, and Step 1(b), there is $J$ such that $u(\mu_{m_{J}})<u(\bar\mu)+\varepsilon$, $\varphi(\nu_{m_{J}})>\varphi(\bar\mu)-\varepsilon$, $\varphi(S_{s_{m_{J}}}\mu)<\varphi(\mu)+\varepsilon$ and $\mathrm{Ent}(\nu_{m_{J}})>\mathrm{Ent}(\bar\mu)-\varepsilon$. Inserting these into (3.1) with $n=m_{J}$,

$$G(\bar\mu)+(2+\delta)\varepsilon>u(\mu_{m_{J}})-\varphi(\nu_{m_{J}})-\delta\,\mathrm{Ent}(\nu_{m_{J}})\ge u(\mu)-\varphi(S_{s_{m_{J}}}\mu)-\delta\,\mathrm{Ent}(\mu)>G(\mu)-\varepsilon.$$

As $\varepsilon>0$ is arbitrary, $G(\bar\mu)\ge G(\mu)$, so $\bar\mu=\mu$ by the strict maximality of $\mu$.

Now suppose that $(\mu_{n})$ did not converge to $\mu$. Then there are a real $\varepsilon_{0}>0$ and infinitely many $n$ with $W_{\mathbb{T}}(\mu_{n},\mu)\ge\varepsilon_{0}$; listing them increasingly gives a subsequence no further subsequence of which converges to $\mu$, contradicting the previous paragraph. Hence $(\mu_{n})$ converges to $\mu$, and by Step 1(c) so do $(\nu_{n})$ and $(S_{s_{n}}\mu)$.

Let $\varepsilon>0$. By the upper semicontinuity of $u$ at $\mu$, the continuity of $\varphi$ at $\mu$, and Step 1(b) applied to $(\nu_{n})$ with the bound $E$, there is $N$ such that for every $n\ge N$: $u(\mu_{n})<u(\mu)+\varepsilon$, $|\varphi(\nu_{n})-\varphi(\mu)|<\varepsilon$, $|\varphi(S_{s_{n}}\mu)-\varphi(\mu)|<\varepsilon$ and $\mathrm{Ent}(\nu_{n})>\mathrm{Ent}(\mu)-\varepsilon$. Then (3.1) gives

$$u(\mu_{n})\ge u(\mu)-\varphi(S_{s_{n}}\mu)+\varphi(\nu_{n})-\delta\,\mathrm{Ent}(\mu)+\delta\,\mathrm{Ent}(\nu_{n})>u(\mu)-(2+\delta)\varepsilon,$$

so $|u(\mu_{n})-u(\mu)|<(2+\delta)\varepsilon$ for $n\ge N$. Hence $(u(\mu_{n}))$ converges to $u(\mu)$.

Step 5 (The viscosity inequality at $\mu_{n}$ and the score perturbation). Since $\psi_{n}$ is a Laplacian test function and $\mu_{n}$ maximises $u-\psi_{n}$ over $\mathcal{P}(\mathbb{T}^{d})$, Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space §sub applies to $\psi_{n}$ and $\mu_{n}$; with $\nabla\psi_{n}(\mu_{n})=p_{n}+\delta z_{n}$ and (2.1) at $\nu=\mu_{n}$ it gives

$$\rho\,u(\mu_{n})+H(\mu_{n},p_{n}+\delta z_{n})+\frac{\sigma^{2}}{2}\langle F_{n},\xi_{n}\rangle_{\nu_{n}}+\frac{\sigma^{2}\delta}{2}\,I_{n}\le0.\qquad(5.1)$$

Since $\mu\in\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$, $(\mu_{n})$ converges to $\mu$ and $(s_{n})$ converges to $0$, Heat Regularisation of an Intrinsic Test Function: a Laplacian Test Function whose Laplacian is Minus the Pairing of the Gradient with the Score §convergence shows that $((\nu_{n},F_{n}))_{n}$ and $((\mu_{n},p_{n}))_{n}$ both converge along couplings to $(\mu,\nabla\varphi(\mu))$. Let $\gamma_{n}\in\Pi(\mu_{n},\mu)$ be couplings as in that definition for the second sequence. On the probability space $(\mathbb{R}^{d+d},\gamma_{n})$ the maps $X_{n}:=p_{n}\circ\mathrm{pr}_{1}$ and $Y:=\nabla\varphi(\mu)\circ\mathrm{pr}_{2}$ (Borel representatives) are square-integrable with norms $\lVert p_{n}\rVert_{\mu_{n}}$ and $r$, by the change-of-variables formula, the marginals of $\gamma_{n}$ being $\mu_{n}$ and $\mu$; and $\lVert X_{n}-Y\rVert^{2}=D_{\gamma_{n}}(p_{n},\nabla\varphi(\mu))$. The triangle inequality for the norm of the real Hilbert space $L^{2}(\mathbb{R}^{d+d};\mathbb{R}^{d})$ over $\gamma_{n}$ (The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations §hilbert) gives $\bigl|\lVert p_{n}\rVert_{\mu_{n}}-r\bigr|\le D_{\gamma_{n}}(p_{n},\nabla\varphi(\mu))^{1/2}$, which converges to $0$. Hence $(\lVert p_{n}\rVert_{\mu_{n}})$ converges to $r$, and in the same way $(\lVert F_{n}\rVert_{\nu_{n}})$ converges to $r$.

Let $K$ be the least positive integer with $K>r$. Then $K\le r+1$ (if $K\ge2$, then $K-1$ is a positive integer that is not larger than $r$), so $C_{K}\le c(r)$ by Step 0. Since $r<K$, there is $n_{0}$ with $\lVert p_{n}\rVert_{\mu_{n}}\le K$ for every $n\ge n_{0}$.

The score $\xi_{n}$ has a Borel representative $\mathbb{R}^{d}\to\mathbb{R}^{d}$ that is square-integrable with respect to $\nu_{n}=S_{s_{n}}\mu_{n}$, so Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings §average gives $\lVert z_{n}\rVert_{\mu_{n}}^{2}\le\lVert\xi_{n}\rVert_{\nu_{n}}^{2}=I_{n}$. For $n\ge n_{0}$ the score-perturbation bound, applied with $\mu_{n}$, $p=p_{n}$, $\zeta=z_{n}$, $R=K$ and $\delta$, gives

$$H(\mu_{n},p_{n}+\delta z_{n})\ge H(\mu_{n},p_{n})-\frac{\sigma^{2}\delta}{4}\lVert z_{n}\rVert_{\mu_{n}}^{2}-C_{K}\delta\ge H(\mu_{n},p_{n})-\frac{\sigma^{2}\delta}{4}I_{n}-C_{K}\delta,$$

and with (5.1), for every $n\ge n_{0}$,

$$\rho\,u(\mu_{n})+H(\mu_{n},p_{n})+\frac{\sigma^{2}}{2}\langle F_{n},\xi_{n}\rangle_{\nu_{n}}+\frac{\sigma^{2}\delta}{4}\,I_{n}\le C_{K}\delta.\qquad(5.2)$$

Step 6 (A uniform bound on the Fisher information). Since $p_{n}\in T_{\mu_{n}}$, $\nabla\varphi(\mu)\in T_{\mu}$ and $((\mu_{n},p_{n}))_{n}$ converges along couplings to $(\mu,\nabla\varphi(\mu))$, the continuity of $H$ along couplings shows that $(H(\mu_{n},p_{n}))_{n}$ converges to $H(\mu,\nabla\varphi(\mu))$. The real sequences $(u(\mu_{n}))$, $(H(\mu_{n},p_{n}))$ and $(\lVert F_{n}\rVert_{\nu_{n}})$ converge (Step 4, the previous sentence and Step 5), hence are bounded (claim 2 of Uniqueness of Limits and Boundedness of Convergent Real Sequences); choose reals $B\ge0$ and $M\ge0$ with $\lVert F_{n}\rVert_{\nu_{n}}\le B$ and $C_{K}\delta-\rho\,u(\mu_{n})-H(\mu_{n},p_{n})\le M$ for every $n$. By the Cauchy-Schwarz inequality in the real Hilbert space $L^{2}(\nu_{n};\mathbb{R}^{d})$ (Optimal Transport on the Flat Torus: Standing Notation §fields and The Cauchy-Schwarz Inequality in a Real Inner Product Space), $|\langle F_{n},\xi_{n}\rangle_{\nu_{n}}|\le B\sqrt{I_{n}}$. Put $a:=\sigma^{2}\delta/4>0$, $b:=\sigma^{2}B/2$ and $t_{n}:=\sqrt{I_{n}}$. Then (5.2) gives $a\,t_{n}^{2}\le M+b\,t_{n}$ for $n\ge n_{0}$. If $t_{n}>2b/a$, then $b\,t_{n}<a\,t_{n}^{2}/2$, whence $a\,t_{n}^{2}/2<M$; so in all cases $I_{n}=t_{n}^{2}\le\max\{4b^{2}/a^{2},\,2M/a\}$ for $n\ge n_{0}$. Let $C$ be the largest of $\max\{4b^{2}/a^{2},2M/a\}$ and $I_{0},\dots,I_{n_{0}-1}$; then $I_{n}\le C$ for every $n$.

Step 7 (Passage to the limit). Apply Lower Semicontinuity of the Torus Fisher Information and Closedness of the Score Along Couplings to the sequence $(\nu_{n})$ in $\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$, which converges to $\mu$ (Step 4), with the bound $C$ of Step 6. By Lower Semicontinuity of the Torus Fisher Information and Closedness of the Score Along Couplings §lsc, $\mu\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$. By Lower Semicontinuity of the Torus Fisher Information and Closedness of the Score Along Couplings §pairing, applied with $\eta=\nabla\varphi(\mu)\in T_{\mu}$ and $\eta_{n}=F_{n}\in L^{2}(\nu_{n};\mathbb{R}^{d})$, whose convergence along couplings was shown in Step 5, $(\langle F_{n},\xi_{n}\rangle_{\nu_{n}})_{n}$ converges to $\langle\nabla\varphi(\mu),\xi_{\mu}\rangle_{\mu}$. Together with Steps 4 and 6, the sequence

$$a_{n}:=\rho\,u(\mu_{n})+H(\mu_{n},p_{n})+\frac{\sigma^{2}}{2}\langle F_{n},\xi_{n}\rangle_{\nu_{n}}$$

converges to $L:=\rho\,u(\mu)+H(\mu,\nabla\varphi(\mu))+\frac{\sigma^{2}}{2}\langle\nabla\varphi(\mu),\xi_{\mu}\rangle_{\mu}$. As $\frac{\sigma^{2}\delta}{4}I_{n}\ge0$, (5.2) gives $a_{n}\le C_{K}\delta$ for $n\ge n_{0}$. Claim 1 of Order Properties of Limits of Real Sequences, applied to the tail $(a_{n_{0}+n})_{n}$, which converges to $L$ by A Subsequence of a Convergent Sequence Has the Same Limit (with the strictly increasing sequence $n\mapsto n_{0}+n$), and the constant sequence $C_{K}\delta$, gives

$$\rho\,u(\mu)+H\bigl(\mu,\nabla\varphi(\mu)\bigr)+\frac{\sigma^{2}}{2}\bigl\langle\nabla\varphi(\mu),\xi_{\mu}\bigr\rangle_{\mu}\le C_{K}\delta\le\delta\,c\bigl(\lVert\nabla\varphi(\mu)\rVert_{\mu}\bigr).$$

Since $\varphi$, $\delta$ and $\mu$ were arbitrary, $u$ is an entropy-penalised viscosity subsolution of $(\mathrm{E})$, with the $\delta_{0}$ and $c$ of Step 0. This proves claim 1.

Step 8 (Supersolutions). Let $v$ be a lower semicontinuous viscosity supersolution of $(\mathrm{E})$; we use the same $\delta_{0}$ and $c$. Fix $\varphi$ and $\delta$ as in Step 2, and $\mu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ with $G'(\nu)>G'(\mu)$ for every $\nu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ with $\nu\ne\mu$, where $G'(\nu):=v(\nu)-\varphi(\nu)+\delta\,\mathrm{Ent}(\nu)$. Steps 2 to 7 are repeated with the following changes, everything else being unchanged.

In Step 2, $\psi_{n}:=\varphi\circ S_{s_{n}}-\delta\,\mathrm{Ent}\circ S_{s_{n}}$; by Laplacian Test Functions on the Torus: Uniqueness of the Gradient Field, a Criterion, and Linearity of the Test Classes §linear with the coefficient $-\delta$ it is a Laplacian test function with gradient field $g^{n}-\delta h^{n}$, so $\nabla\psi_{n}(\mu_{n})=p_{n}-\delta z_{n}$ and, by the same argument as for (2.1), $\mathcal{L}\psi_{n}(\nu)=-\langle\nabla\varphi(S_{s_{n}}\nu),\xi_{S_{s_{n}}\nu}\rangle_{S_{s_{n}}\nu}+\delta\,\mathcal{I}(S_{s_{n}}\nu)$. The function $v-\psi_{n}$ is lower semicontinuous, and $\mu_{n}$ is chosen as a minimiser of it by Step 1(a): $v(\nu)-\psi_{n}(\nu)\ge v(\mu_{n})-\psi_{n}(\mu_{n})$ for every $\nu$.

In Step 3, (3.1) becomes $v(\mu_{n})-\varphi(\nu_{n})+\delta\,\mathrm{Ent}(\nu_{n})\le v(\mu)-\varphi(S_{s_{n}}\mu)+\delta\,\mathrm{Ent}(S_{s_{n}}\mu)\le v(\mu)-\varphi(S_{s_{n}}\mu)+\delta\,\mathrm{Ent}(\mu)$, and with a real $V\le v$ on $\mathcal{P}(\mathbb{T}^{d})$ (Step 1(a)) one gets $\mathrm{Ent}(\nu_{n})\le\delta^{-1}(v(\mu)-V+m'-m)+\mathrm{Ent}(\mu)$.

In Step 4, lower semicontinuity of $v$ at $\bar\mu$ gives $v(\mu_{m_{J}})>v(\bar\mu)-\varepsilon$, and with the same estimates for $\varphi$ and $\mathrm{Ent}$ the modified (3.1) gives $G'(\bar\mu)-(2+\delta)\varepsilon<G'(\mu)+\varepsilon$; hence $G'(\bar\mu)\le G'(\mu)$ and $\bar\mu=\mu$ by strict minimality. Then $v(\mu_{n})>v(\mu)-\varepsilon$ for large $n$ by lower semicontinuity, while the modified (3.1) gives $v(\mu_{n})\le v(\mu)-\varphi(S_{s_{n}}\mu)+\varphi(\nu_{n})+\delta\,\mathrm{Ent}(\mu)-\delta\,\mathrm{Ent}(\nu_{n})<v(\mu)+(2+\delta)\varepsilon$; so $(v(\mu_{n}))$ converges to $v(\mu)$.

In Step 5, Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space §super applied to $\psi_{n}$ and the minimiser $\mu_{n}$ gives

$$\rho\,v(\mu_{n})+H(\mu_{n},p_{n}-\delta z_{n})+\frac{\sigma^{2}}{2}\langle F_{n},\xi_{n}\rangle_{\nu_{n}}-\frac{\sigma^{2}\delta}{2}\,I_{n}\ge0.$$

The field $-z_{n}$ lies in $T_{\mu_{n}}$ by The Torus Tangent Space: Linearity of the Periodic Calculus, Closed Subspace, and Representation of Bounded Functionals on Gradients §subspace and $\lVert-z_{n}\rVert_{\mu_{n}}^{2}=\lVert z_{n}\rVert_{\mu_{n}}^{2}\le I_{n}$, so the score-perturbation bound with $\zeta=-z_{n}$ gives $H(\mu_{n},p_{n}-\delta z_{n})\le H(\mu_{n},p_{n})+\frac{\sigma^{2}\delta}{4}I_{n}+C_{K}\delta$ for $n\ge n_{0}$, and therefore

$$\rho\,v(\mu_{n})+H(\mu_{n},p_{n})+\frac{\sigma^{2}}{2}\langle F_{n},\xi_{n}\rangle_{\nu_{n}}-\frac{\sigma^{2}\delta}{4}\,I_{n}\ge-C_{K}\delta\qquad(n\ge n_{0}).$$

In Step 6, $M\ge0$ is chosen with $C_{K}\delta+\rho\,v(\mu_{n})+H(\mu_{n},p_{n})\le M$ for every $n$, and the last display gives $\frac{\sigma^{2}\delta}{4}I_{n}\le M+\frac{\sigma^{2}}{2}B\sqrt{I_{n}}$, that is $a\,t_{n}^{2}\le M+b\,t_{n}$, for $n\ge n_{0}$; the bound $I_{n}\le C$ follows as before.

In Step 7, since $-\frac{\sigma^{2}\delta}{4}I_{n}\le0$, the sequence $a_{n}:=\rho\,v(\mu_{n})+H(\mu_{n},p_{n})+\frac{\sigma^{2}}{2}\langle F_{n},\xi_{n}\rangle_{\nu_{n}}$ satisfies $a_{n}\ge-C_{K}\delta$ for $n\ge n_{0}$. As before, $\mu\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$ and $(a_{n})$ converges to the left-hand side below, and claim 1 of Order Properties of Limits of Real Sequences, applied to the tail $(a_{n_{0}+n})_{n}$, which converges to the same limit by A Subsequence of a Convergent Sequence Has the Same Limit (with the strictly increasing sequence $n\mapsto n_{0}+n$), and the constant sequence $-C_{K}\delta$, gives

$$\rho\,v(\mu)+H\bigl(\mu,\nabla\varphi(\mu)\bigr)+\frac{\sigma^{2}}{2}\bigl\langle\nabla\varphi(\mu),\xi_{\mu}\bigr\rangle_{\mu}\ge-C_{K}\delta\ge-\delta\,c\bigl(\lVert\nabla\varphi(\mu)\rVert_{\mu}\bigr).$$

Hence $v$ is an entropy-penalised viscosity supersolution of $(\mathrm{E})$, which proves claim 2.