Proof of Integrating a Semiconvex Viscosity Subsolution of the Penalty-Drift Equation against a Measure of Finite Relative Fisher Information

Each result cited below is universally quantified over the data in its own statement, and is applied to the data named at the point of use.

Conventions. Elementary order and arithmetic manipulations of real numbers, for instance $|\max(s,0)-\max(t,0)|\le|s-t|$, $s\le|s|$, and the fact that a real number $c$ with $c\le st$ for a fixed positive $s$ and every positive $t$ satisfies $c\le0$ (otherwise $t=c(2s)^{-1}$ gives $c\le c/2$), are covered by Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic in an Ordered Field. For $x,z\in\mathbb{R}^{d}$ we use $\lVert x\rVert^{2}=x\cdot x=\sum_{i=1}^{d}x_{i}^{2}$, $|x_{i}|\le\lVert x\rVert$, $\lVert sx\rVert=|s|\lVert x\rVert$ for $s\in\mathbb{R}$, the triangle inequality and $d_{E}(x,z)=\lVert x-z\rVert$, which are claims 1, 4, 5, 6 and 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$; with the bilinearity and symmetry of the dot product they give $\lVert x\rVert\le\sum_{i=1}^{d}|x_{i}|$, $|x\cdot z|\le\bigl(\sum_{i=1}^{d}|x_{i}|\bigr)\lVert z\rVert$, and, from $0\le\lVert x\mp z\rVert^{2}=\lVert x\rVert^{2}\mp2\,x\cdot z+\lVert z\rVert^{2}$, the inequalities $2|x\cdot z|\le\lVert x\rVert^{2}+\lVert z\rVert^{2}$ and $\lVert x+z\rVert^{2}\le2\lVert x\rVert^{2}+2\lVert z\rVert^{2}$. For $d=1$ the Euclidean norm is the absolute value, by claim 1 there. $B(x,r)$ is the open ball of Euclidean Space and Lebesgue Measure: Standing Notation §space, and a set $O\subseteq\mathbb{R}^{d}$ is open exactly when every $x\in O$ has an $r>0$ with $B(x,r)\subseteq O$. By Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces, $\mathcal{B}(\mathbb{R}^{d})$ is the Borel $\sigma$-algebra of the metric space $(\mathbb{R}^{d},d_{E})$, and it contains every open subset of $\mathbb{R}^{d}$ by Euclidean Space and Lebesgue Measure: Standing Notation §borel; as a $\sigma$-algebra it contains the union and the intersection of every countable family of its members. We write $\mathbb{Q}_{>0}$ for the set of positive rational numbers; $\mathbb{Q}^{d}$ is countable by claim 3 of The Integers and the Rational Numbers are Countable, and $\mathbb{Q}_{>0}$ and every subset of $\mathbb{Q}^{d}$ are countable by claim 3 of Basic Properties of Countable Sets (and claim 2 of The Integers and the Rational Numbers are Countable). Continuity of $w$ on $D$ means, by Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §extrema, that for every $x\in D$ and every $\varepsilon>0$ there is $\delta>0$ with $|w(y)-w(x)|<\varepsilon$ for every $y\in D$ with $\lVert y-x\rVert<\delta$.

Step 0 (a Borel criterion). Let $O\subseteq\mathbb{R}^{d}$ be open and let $G:\mathbb{R}^{d}\to\mathbb{R}$ satisfy $G(x)=0$ for $x\notin O$ and have the property that for every $x\in O$ and every $\varepsilon>0$ there is $\delta>0$ with $|G(y)-G(x)|<\varepsilon$ for every $y\in O$ with $\lVert y-x\rVert<\delta$. We show that $G$ is Borel. Put $G^{+}=\max(G,0)$ and $G^{-}=\max(-G,0)$, pointwise, so that $G=G^{+}+(-1)G^{-}$. We claim that $G^{+}$ is lower semicontinuous on $\mathbb{R}^{d}$ for the metric $d_{E}$. Let $x\in\mathbb{R}^{d}$ and $\varepsilon>0$. If $x\notin O$, then $G^{+}(x)=0$ and $G^{+}(x)-\varepsilon<0\le G^{+}(y)$ for every $y$. If $x\in O$, choose $\delta_{1}$ as in the hypothesis and $r>0$ with $B(x,r)\subseteq O$, and put $\delta=\min(\delta_{1},r)$; every $y$ with $\lVert y-x\rVert<\delta$ lies in $O$ and satisfies $|G^{+}(y)-G^{+}(x)|\le|G(y)-G(x)|<\varepsilon$, hence $G^{+}(x)-\varepsilon<G^{+}(y)$. The function $-G$ satisfies the same hypotheses as $G$, so $G^{-}=\max(-G,0)$ is lower semicontinuous on $\mathbb{R}^{d}$ as well. By claim 5 of Borel Measurability and Bounded Integration on a Metric Space, $G^{+}$ and $G^{-}$ are Borel, and so is $G$ by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions.

Step 1 (clause 1). Assume that $w$ is continuous on $D$.

(a) The map $\bar{w}$ is Borel by Step 0 applied with $O=D$ and $G=\bar{w}$.

(b) Translated differences. For $h\in\mathbb{R}^{d}$ let $O_{h}=\{x\in D:x+h\in D\}$. The set $\{z\in\mathbb{R}^{d}:z+h\in D\}$ is open by Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §translation (with $V=D$ and $b=h$), so $O_{h}$, its intersection with $D$, is open by Metric Open Sets Form a Topology; note $O_{0_{\mathbb{R}^{d}}}=D$. Let $F_{h}:\mathbb{R}^{d}\to\mathbb{R}$ be given by $F_{h}(x)=w(x+h)-w(x)$ for $x\in O_{h}$ and $F_{h}(x)=0$ otherwise. Let $x\in O_{h}$ and $\varepsilon>0$; continuity of $w$ at $x$ and at $x+h$ gives $\delta>0$ with $|w(y)-w(x)|<\varepsilon/2$ and $|w(z)-w(x+h)|<\varepsilon/2$ for all $y,z\in D$ with $\lVert y-x\rVert<\delta$ and $\lVert z-(x+h)\rVert<\delta$; for $y\in O_{h}$ with $\lVert y-x\rVert<\delta$ we may take $z=y+h$, and obtain $|F_{h}(y)-F_{h}(x)|<\varepsilon$. By Step 0, $F_{h}$ is Borel.

(c) A Borel description of $E_{w}$. For $\delta\in\mathbb{Q}_{>0}$ let $H(\delta)=\{h\in\mathbb{Q}^{d}:\lVert h\rVert<\delta\}$, a countable set containing $0_{\mathbb{R}^{d}}$. For $\varepsilon,\delta\in\mathbb{Q}_{>0}$ and $q\in\mathbb{Q}^{d}$ put

The function $\Phi=|F_{h}-c|$, where $c$ is the constant function with value $q\cdot h$, is Borel by (b) and claims 1, 2 and 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; the interval $[0,\varepsilon\lVert h\rVert]$ is closed in $\mathbb{R}$, hence belongs to $\mathcal{B}(\mathbb{R})$ by claims 1 and 2 of Borel Measurability and Bounded Integration on a Metric Space; so $S_{h}(\varepsilon,q)=O_{h}\cap\Phi^{-1}([0,\varepsilon\lVert h\rVert])\in\mathcal{B}(\mathbb{R}^{d})$, and $S(\varepsilon,\delta,q)\in\mathcal{B}(\mathbb{R}^{d})$, $H(\delta)$ being countable. Consequently

We show $E'=E_{w}$.

$E_{w}\subseteq E'$. Let $x\in E_{w}$ and $p=Dw(x)$, so that $w$ is differentiable at $x$ with a derivative matrix $J$ satisfying $Jh=p\cdot h$. Let $\varepsilon\in\mathbb{Q}_{>0}$. There is $\delta_{0}>0$ such that every $h$ with $0<\lVert h\rVert<\delta_{0}$ satisfies $x+h\in D$ and $|w(x+h)-w(x)-p\cdot h|\le\tfrac{\varepsilon}{2}\lVert h\rVert$. By claims 1 and 2 of The Rational Numbers are Dense in the Real Numbers choose $\delta\in\mathbb{Q}$ with $0<\delta<\delta_{0}$ and $q\in\mathbb{Q}^{d}$ with $|q_{i}-p_{i}|<\varepsilon/(2d)$ for $i\in[d]$. Let $h\in H(\delta)$. If $h=0_{\mathbb{R}^{d}}$, then $x\in D=O_{h}$ and $|F_{h}(x)-q\cdot h|=0$. Otherwise $0<\lVert h\rVert<\delta_{0}$ by claim 3 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, so $x\in O_{h}$ and

Thus $x\in S(\varepsilon,\delta,q)$; as $\varepsilon$ was arbitrary, $x\in E'$.

$E'\subseteq E_{w}$. Let $x\in E'$. For $\varepsilon=1$ there are $\delta,q$ with $x\in S(1,\delta,q)\subseteq S_{0_{\mathbb{R}^{d}}}(1,q)\subseteq O_{0_{\mathbb{R}^{d}}}=D$, so $x\in D$; choose $r>0$ with $B(x,r)\subseteq D$. For every natural number $j\ge1$ choose $\delta_{j}\in\mathbb{Q}_{>0}$ and $q_{j}\in\mathbb{Q}^{d}$ with $x\in S(1/j,\delta_{j},q_{j})$, and put $\eta_{j}=\min(\delta_{j},r)$ and $s_{j}=\sum_{i=1}^{d}|(q_{j})_{i}|$. We claim:

Indeed $x+h\in B(x,r)\subseteq D$. Suppose that $\gamma=|w(x+h)-w(x)-q_{j}\cdot h|-\tfrac1j\lVert h\rVert>0$. Continuity of $w$ at $x+h$ gives $\delta'>0$ with $|w(z)-w(x+h)|<\gamma/3$ for $z\in D$ with $\lVert z-(x+h)\rVert<\delta'$. Put $c=\min\bigl(\delta',\eta_{j}-\lVert h\rVert,\gamma/(3(1+s_{j}))\bigr)>0$ and choose, by claim 2 of The Rational Numbers are Dense in the Real Numbers, $h'\in\mathbb{Q}^{d}$ with $|h'_{i}-h_{i}|<c/d$ for $i\in[d]$, so that $\lVert h'-h\rVert<c$. Then $\lVert h'\rVert<\lVert h\rVert+c\le\eta_{j}\le\delta_{j}$, so $h'\in H(\delta_{j})$ and $x\in S_{h'}(1/j,q_{j})$, that is $|w(x+h')-w(x)-q_{j}\cdot h'|\le\tfrac1j\lVert h'\rVert$. On the other hand $x+h'\in D$ with $\lVert(x+h')-(x+h)\rVert<\delta'$, so $|w(x+h')-w(x+h)|<\gamma/3$; moreover $|q_{j}\cdot(h'-h)|\le s_{j}\lVert h'-h\rVert<\gamma/3$ and $\tfrac1j\lVert h'\rVert\le\tfrac1j\lVert h\rVert+\lVert h'-h\rVert<\tfrac1j\lVert h\rVert+\gamma/3$. Hence

a contradiction. This proves $(\ast)$.

Next let $j,k\ge1$ and $v=q_{j}-q_{k}$. If $v\ne0_{\mathbb{R}^{d}}$, choose a real $s$ with $0<s<\min(\eta_{j},\eta_{k})$ and put $h=(s\lVert v\rVert^{-1})v$, so that $\lVert h\rVert=s$ and $v\cdot h=s\lVert v\rVert$. Subtracting the two instances of $(\ast)$ gives $s\lVert v\rVert=|v\cdot h|\le(\tfrac1j+\tfrac1k)s$. Hence in all cases $\lVert q_{j}-q_{k}\rVert\le\tfrac1j+\tfrac1k$, and therefore $|(q_{j})_{i}-(q_{k})_{i}|\le\tfrac1j+\tfrac1k$ for each $i\in[d]$. By the Archimedean property, for each $i$ the sequence $((q_{m+1})_{i})_{m\in\mathbb{N}}$ is a Cauchy sequence, so it converges to a real number $p_{i}$ by Every Cauchy Sequence of Real Numbers Converges; let $p=(p_{1},\dots,p_{d})$. For $j\ge1$ and $i\in[d]$ we have $|(q_{j})_{i}-p_{i}|\le\tfrac1j$: otherwise $\gamma=|(q_{j})_{i}-p_{i}|-\tfrac1j>0$, and choosing $k\ge1$ with $\tfrac1k<\gamma/2$ and $|(q_{k})_{i}-p_{i}|<\gamma/2$ we would get $|(q_{j})_{i}-(q_{k})_{i}|\ge|(q_{j})_{i}-p_{i}|-|(q_{k})_{i}-p_{i}|>\tfrac1j+\tfrac{\gamma}{2}>\tfrac1j+\tfrac1k$. Consequently, by $(\ast)$, for $j\ge1$ and $\lVert h\rVert<\eta_{j}$,

Given $\varepsilon>0$, choose $j\ge1$ with $(1+d)/j\le\varepsilon$; then every $h$ with $0<\lVert h\rVert<\eta_{j}$ satisfies $x+h\in D$ and $|w(x+h)-w(x)-Jh|\le\varepsilon\lVert h\rVert$, where $J$ is the real matrix with one row and entries $J_{1i}=p_{i}$, so that $Jh=p\cdot h$. Thus $w$ is differentiable at $x$ with derivative matrix $J$, and $x\in E_{w}$. Hence $E_{w}=E'\in\mathcal{B}(\mathbb{R}^{d})$.

(d) The map $\nabla w$ is Borel. Fix $i\in[d]$, let $e_{i}\in\mathbb{R}^{d}$ have $i$th coordinate $1$ and all others $0$, and for $m\in\mathbb{N}$ put $t_{m}=(m+1)^{-1}$ and $G_{i,m}=(m+1)\,\mathbf{1}_{E_{w}}F_{t_{m}e_{i}}$, which is Borel by (b), (c) and claims 1, 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. Let $x\in E_{w}$ and $p=Dw(x)$, and let $\varepsilon>0$. Differentiability gives $\delta>0$ with $x+h\in D$ and $|w(x+h)-w(x)-p\cdot h|\le\tfrac{\varepsilon}{2}\lVert h\rVert$ whenever $0<\lVert h\rVert<\delta$; choose $m_{0}$ with $t_{m_{0}}<\delta$. For $m\ge m_{0}$ the point $h=t_{m}e_{i}$ has $0<\lVert h\rVert=t_{m}<\delta$, so $x\in O_{h}$, $p\cdot h=t_{m}p_{i}$ and

Hence $(G_{i,m}(x))_{m\in\mathbb{N}}$ converges to $p_{i}$, the $i$th coordinate of $\nabla w(x)$. For $x\notin E_{w}$ every $G_{i,m}(x)$ is $0$, which is the $i$th coordinate of $\nabla w(x)=0_{\mathbb{R}^{d}}$. By claim 5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions the $i$th coordinate of $\nabla w$ is Borel, and by claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, as recorded in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, $\nabla w$ is Borel. This proves clause 1.

Step 2 (clause 2). Assume that $w$ is continuous on $D$ and let $\mu\in\mathcal{D}_{U,a}$. By the definition of $\mathcal{D}_{U,a}$ the Borel map $\bar{U}$ is integrable with respect to $\mu$, hence so is $|\bar{U}|$, by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and the definition of integrability. Let $h_{0}=A+B|\bar{U}|+B|p_{0}|$, an integrable function by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space and claim 2 of Linearity and Monotonicity of the Lebesgue Integral, with $\int h_{0}\,d\mu=A+B\int|\bar{U}|\,d\mu+B|p_{0}|$. For $x\in E_{w}$ we have $\lVert\nabla w(x)\rVert^{2}=\lVert Dw(x)\rVert^{2}\le A+B(U(x)-p_{0})\le A+B|U(x)|+B|p_{0}|=h_{0}(x)$, since $B\ge0$ and $\bar{U}(x)=U(x)$; for $x\notin E_{w}$, $\lVert\nabla w(x)\rVert^{2}=0\le h_{0}(x)$, since $A,B\ge0$. The map $\lVert\nabla w\rVert^{2}$ is Borel, being the composition of $\nabla w$ (Step 1) with the Borel map $x\mapsto\lVert x\rVert^{2}$ of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs, by claim 4 of Borel Measurability and Bounded Integration on a Metric Space. By claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and since the two readings of the integral of the nonnegative integrable $h_{0}$ agree by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures,

Hence the class of $\nabla w$ lies in $L^{2}(\mu;\mathbb{R}^{d})$ by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, and $\lVert\nabla w\rVert^{2}$, a nonnegative Borel function with finite integral, is integrable. The maps $\bar{w}$ (Step 1) and $\bar{g}$ (by hypothesis) are Borel and bounded, $w$ and $\tilde{g}$ being bounded, so they are integrable by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures; the combination $\lambda\bar{w}+\tfrac{\theta'}{2}\lVert\nabla w\rVert^{2}-\bar{g}$ is integrable by claim 2 of Linearity and Monotonicity of the Lebesgue Integral. This proves clause 2.

Step 3 (set-up for clauses 3 and 4). Put $\sigma=1$ under the hypotheses of clause 3 and $\sigma=-1$ under those of clause 4, and let $v:D\to\mathbb{R}$, $v(x)=\sigma w(x)$. In both cases $v$ is semiconvex on $D$ with constant $K\ge0$, and $w(x)=\sigma v(x)$ for $x\in D$ since $\sigma\sigma=1$. By Local Lipschitz Bound and Continuity for a Semiconvex Function on an Open Convex Set §continuity (with $S=D$) $v$ is continuous on $D$, hence so is $w=\sigma v$, and clauses 1 and 2 apply. By Viscosity Subsolution and Supersolution of a Second-Order Equation, read through The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space §equation and with local extrema taken as in Local Maximum of a Function Relative to a Subset of a Metric Space and Local Minimum of a Function Relative to a Subset of a Metric Space in the metric space $(\mathbb{R}^{d},d_{E})$, we have:

(V) for every $\psi:D\to\mathbb{R}$ of class $C^{2}$ on $D$ and every $y\in D$ such that $\sigma(w(z)-\psi(z))\le\sigma(w(y)-\psi(y))$ for all $z\in D$ with $\lVert z-y\rVert<\delta$, for some $\delta>0$, one has $\sigma F(y,w(y),D\psi(y),D^{2}\psi(y))\le0$.

For $\sigma=1$ the hypothesis of (V) says that $w-\psi$ has a local maximum at $y$ relative to $D$, and the conclusion is the subsolution inequality; for $\sigma=-1$ it says that $w-\psi$ has a local minimum at $y$ relative to $D$, and the conclusion is the supersolution inequality $0\le F(y,w(y),D\psi(y),D^{2}\psi(y))$.

Fix $\mu\in\mathcal{D}^{\Sigma}_{U,a}$. By the definition of $\mathcal{D}^{\Sigma}_{U,a}$ and of $\mathcal{D}_{U,a}$, $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, $\mu(D)=1$, $\int\lVert\nabla U\rVert^{2}\,d\mu<\infty$, and $\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ has finite entropy, so that $\mu$ is absolutely continuous by Basic Properties of the Entropy on the Wasserstein Space: Comparison with the Gaussian Relative Entropy, Lower Bound, Translation Invariance, Absolute Continuity, Closed Sublevel Sets and Lower Semicontinuity §absolutely-continuous. Let $\xi_{\mu}$ be its score.

Step 4 (convexification and a Borel set of full measure). Let $\varphi:D\to\mathbb{R}$, $\varphi(x)=v(x)+\tfrac{K}{2}\lVert x\rVert^{2}$; it is convex on $D$ by Semiconvex Function on a Convex Subset of $\mathbb{R}^n$. By The Points of Twice Differentiability of a Convex Function: a Borel Set of Full Measure, and Borel Measurability of the Gradient and Hessian on It §full, applied with $n=d$, with $D$ in place of its open convex set and $\varphi$ in place of $f$, there is $T\in\mathcal{B}(\mathbb{R}^{d})$ with $T\subseteq D$ and $\lambda_{d}(D\setminus T)=0$ such that $\varphi$ is twice differentiable at every point of $T$. Since $D\setminus T\in\mathcal{B}(\mathbb{R}^{d})$ and $\mu$ is absolutely continuous, $\mu(D\setminus T)=0$; by additivity of the measure $\mu$, $\mu(T)=\mu(D)-\mu(D\setminus T)=1$, and $\mu(\mathbb{R}^{d}\setminus T)=0$. For $y\in T$ write $p_{\varphi}(y)\in\mathbb{R}^{d}$ and $H_{\varphi}(y)\in\mathcal{S}(d)$ for the first-order coefficient and the Hessian of $\varphi$ at $y$, as in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §twice-differentiable.

Step 5 (second-order data of $w$ on $T$). Fix $y\in T$. Let $Q_{K}:\mathbb{R}^{d}\to\mathbb{R}$, $Q_{K}(z)=\tfrac12\,z\cdot((KI_{d})z)=\tfrac{K}{2}\lVert z\rVert^{2}$. By Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic (with $M=KI_{d}\in\mathcal{S}(d)$, $q=0_{\mathbb{R}^{d}}$ and $c=0$) its restriction to $D$ is of class $C^{2}$ on $D$ with gradient $Kz$ and Hessian $KI_{d}$ at $z\in D$, so by Basic Properties of Twice Differentiability at a Point §c2 it is twice differentiable at $y$ with first-order coefficient $Ky$ and Hessian $KI_{d}$. Since $v=\varphi-Q_{K}$ on $D$, Sums, Differences and Scalar Multiples of Functions Twice Differentiable at a Point §difference shows that $v$ is twice differentiable at $y$ with first-order coefficient $p_{\varphi}(y)-Ky$ and Hessian $H_{\varphi}(y)-KI_{d}$, and Sums, Differences and Scalar Multiples of Functions Twice Differentiable at a Point §multiple (with the scalar $\sigma$) shows that $w=\sigma v$ is twice differentiable at $y$ with first-order coefficient and Hessian

By Basic Properties of Twice Differentiability at a Point §gradient, $w$ is differentiable at $y$ and $p_{w}$ is its gradient $Dw(y)$; hence $y\in E_{w}$ and $\nabla w(y)=p_{w}$. Multiplying by $\sigma$ and using $\sigma\sigma=1$, the linearity of the trace (claim 1 of Basic Properties of the Trace) and $\operatorname{tr}(I_{d})=d$,

Step 6 (the pointwise inequality). We show that for every $y\in T$

Fix $y\in T$ and a real $t>0$, and let $M_{t}=X_{w}+\sigma tI_{d}$, which lies in $\mathcal{S}(d)$ by Second-Order Equations on Euclidean Open Sets §matrices. Let $Q_{t}:\mathbb{R}^{d}\to\mathbb{R}$, $Q_{t}(z)=\tfrac12\,z\cdot(M_{t}z)+p_{w}\cdot z+w(y)$. By Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic and Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §translation (with $V=\mathbb{R}^{d}$ and $b=-y$, so that $V-b=\mathbb{R}^{d}$), the map $z\mapsto Q_{t}(z-y)$ is of class $C^{2}$ on $\mathbb{R}^{d}$ with gradient $M_{t}(z-y)+p_{w}$ and Hessian $M_{t}$ at $z$; by Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives its restriction $\psi_{t}$ to $D$ is of class $C^{2}$ on $D$ with the same gradient and Hessian at points of $D$. In particular $D\psi_{t}(y)=p_{w}$, $D^{2}\psi_{t}(y)=M_{t}$ and $\psi_{t}(y)=w(y)$, and for $h\in\mathbb{R}^{d}$ with $y+h\in D$, since $M_{t}h=X_{w}h+\sigma t\,h$,

By twice differentiability of $w$ at $y$ (Step 5), applied with $\varepsilon=t/2$, there is $\delta>0$ such that every $h$ with $\lVert h\rVert<\delta$ satisfies $y+h\in D$ and $|w(y+h)-w(y)-p_{w}\cdot h-\tfrac12h\cdot(X_{w}h)|\le\tfrac{t}{2}\lVert h\rVert^{2}$. For such $h$, using $\sigma\sigma=1$,

Every $z\in D$ with $\lVert z-y\rVert<\delta$ is $y+h$ with $h=z-y$, so (V) applies to $\psi_{t}$ and $y$ and gives $\sigma F(y,w(y),p_{w},M_{t})\le0$. By The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space §operator, with $\tfrac{\kappa}{2}=a$, $\operatorname{tr}(M_{t})=\operatorname{tr}(X_{w})+\sigma td$ (claim 1 of Basic Properties of the Trace), $\sigma\sigma=1$ and (5.2),

Thus the real number $c=\sigma[\lambda w(y)+\tfrac{\theta'}{2}\lVert p_{w}\rVert^{2}+DU(y)\cdot p_{w}-\tilde{g}(y)]-a(\operatorname{tr}H_{\varphi}(y)-Kd)$, which does not depend on $t$, satisfies $c\le(ad)t$ for every $t>0$; as $ad>0$, $c\le0$. Since $p_{w}=\nabla w(y)$, this is (6.1).

Step 7 (the Laplacian comparison). Apply The Score Paired with the Identity, and the Laplacian of a Convex Potential Bounded by its Pairing with the Score §comparison with its open convex set $G=D$ (for which $\mu(D)=1$), the convex function $\varphi$ and the Borel set $T\subseteq D$ with $\mu(T)=1$ at every point of which $\varphi$ is twice differentiable. The maps $g_{\varphi}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ and $\Delta_{\varphi}:\mathbb{R}^{d}\to\mathbb{R}$ of that clause are Borel, with $g_{\varphi}(y)=p_{\varphi}(y)$ and $\Delta_{\varphi}(y)=\operatorname{tr}H_{\varphi}(y)$ for $y\in T$ and $g_{\varphi}=0_{\mathbb{R}^{d}}$, $\Delta_{\varphi}=0$ off $T$. By (5.1), for $y\in T$, $\lVert g_{\varphi}(y)\rVert^{2}=\lVert\sigma\nabla w(y)+Ky\rVert^{2}\le2\lVert\nabla w(y)\rVert^{2}+2K^{2}\lVert y\rVert^{2}$, and the same bound holds trivially off $T$. The second moment $\int\lVert x\rVert^{2}\,\mu(dx)$ is finite, as recorded in the preamble of The Score Paired with the Identity, and the Laplacian of a Convex Potential Bounded by its Pairing with the Score, so claim 1 of Linearity and Monotonicity of the Lebesgue Integral and Step 2 give $\int\lVert g_{\varphi}\rVert^{2}\,d\mu\le2\int\lVert\nabla w\rVert^{2}\,d\mu+2K^{2}\int\lVert x\rVert^{2}\,\mu(dx)<\infty$. That clause therefore yields that $\Delta_{\varphi}$ is integrable with respect to $\mu$ and

The Borel map $\sigma\nabla w+K\,\mathrm{id}$ agrees with $g_{\varphi}$ on $T$, and $\mu(T)=1$, so the two maps have the same class in $L^{2}(\mu;\mathbb{R}^{d})$ by The Space of Square-Integrable Random Vectors §classes, read as in Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu; by the vector operations defined there this class is $\sigma\nabla w+K\,\mathrm{id}$, formed in $L^{2}(\mu;\mathbb{R}^{d})$. By the bilinearity of $\langle\cdot,\cdot\rangle_{\mu}$ (The Space of Square-Integrable Random Vectors §inner-product) and The Score Paired with the Identity, and the Laplacian of a Convex Potential Bounded by its Pairing with the Score §identity, $\langle\xi_{\mu},g_{\varphi}\rangle_{\mu}=\sigma\langle\xi_{\mu},\nabla w\rangle_{\mu}+K\langle\xi_{\mu},\mathrm{id}\rangle_{\mu}=\sigma\langle\xi_{\mu},\nabla w\rangle_{\mu}-Kd$. Hence

Step 8 (integration and conclusion). Let $H_{1}=\lambda\bar{w}+\tfrac{\theta'}{2}\lVert\nabla w\rVert^{2}-\bar{g}$, integrable by Step 2, and $H_{2}=\nabla U\cdot\nabla w=\sum_{i=1}^{d}(\nabla U)_{i}(\nabla w)_{i}$, which is Borel by claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets and claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, $\nabla U$ being Borel as recorded in The Relative Free Energy and the Relative Score of a Probability Measure for a Potential on an Open Set and $\nabla w$ by Step 1. Since $|H_{2}|\le\tfrac12\lVert\nabla U\rVert^{2}+\tfrac12\lVert\nabla w\rVert^{2}$, with $\int\lVert\nabla U\rVert^{2}\,d\mu<\infty$ (Step 3) and $\int\lVert\nabla w\rVert^{2}\,d\mu<\infty$ (Step 2), $H_{2}$ is integrable by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and $\int H_{2}\,d\mu=\langle\nabla U,\nabla w\rangle_{\mu}$ by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu. The function