Proof of A Bound on the Gibbs Ornstein-Uhlenbeck Functional by Noise Gradients Gives Finite Fisher Information Relative to the Gibbs Measure

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

Real arithmetic, the order of $\mathbb{R}$ and absolute values are those of The Real Numbers: Standing Notation and Background §background, used without further mention. Every square root is the nonnegative one of Existence and Uniqueness of the Nonnegative Square Root, and for nonnegative reals $s,t$ we use that $s\le t$ if and only if $s^{2}\le t^{2}$, the weak form (claim 2) of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; in particular $(s+t)^{1/2}\le s^{1/2}+t^{1/2}$, since $(s^{1/2}+t^{1/2})^{2}\ge s+t$. For real $s,t$ we also use $st\le\tfrac14s^{2}+t^{2}$, $(s+t)^{2}\le2s^{2}+2t^{2}$ and $|s|\le1+s^{2}$, which follow from $(\tfrac12s-t)^{2}\ge0$, $(s-t)^{2}\ge0$ and $(|s|-1)^{2}\ge0$.

Data. Write $V=v\circ p_{d}$ with head dimension $d$, profile $v$ and semiconvexity constant $K\ge0$, as in Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §admissible. Fix $b$ as in Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §below and $C$ as in Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §slope; by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §translation, $C\ge0$, and $L=C^{1/2}(|b+1|+2)\ge0$ is the number defined there. Fix $C'$ as in Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §curvature for $\varepsilon=(2\beta)^{-1}$, and $C''$ as there for $\varepsilon=1$. For $u\in\mathbb{R}^{d}$ put $|\nabla_{a}v(u)|^{2}=\sum_{k=1}^{d}a_{k}(\partial_{k}v(u))^{2}\ge0$, with square root $|\nabla_{a}v(u)|$; $\lVert u\rVert$ is the Euclidean norm, with $\lVert u\rVert^{2}=\sum_{k=1}^{d}u_{k}^{2}$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square. For $x\in X$ and $k\in[d]$, $x_{k}$ is the $k$-th entry of $p_{d}(x)$ and $\partial_{k}V(x)=\partial_{k}v(p_{d}(x))$ by Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient, so that $|\nabla_{a}V(x)|_{a}^{2}=|\nabla_{a}v(p_{d}(x))|^{2}$ by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity. Put $\sigma=\sum_{k=1}^{d}a_{k}$ and $\kappa_{0}=\sum_{k=1}^{d}a_{k}c_{k}^{-2}$, positive real numbers ($a_{k}$ and $c_{k}$ are positive by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights and A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian).

Step 0 (Integrability and measurability). By Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, $p_{d}$ is Borel and $\lVert p_{d}(x)-p_{d}(0_{X})\rVert\le|x-0_{X}|$, where $p_{d}(0_{X})=0_{\mathbb{R}^{d}}$; so $\lVert p_{d}(x)\rVert^{2}\le|x|^{2}$, and since $\mu\in\mathcal{P}_{2}(X)$ the monotonicity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative gives

$$m:=\int_{X}\lVert p_{d}(x)\rVert^{2}\,\mu(dx)\le M_{2}(\mu)<\infty,$$

with $M_{2}$ the second moment of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §moment. For $k\in[d]$, $|x_{k}|\le1+x_{k}^{2}\le1+\lVert p_{d}(x)\rVert^{2}$, so $x\mapsto|x_{k}|$ is integrable with respect to $\mu$; and $\partial_{k}V$ is integrable with respect to $\mu$ by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §integrable, $V$ being integrable. Hence the functions

$$S_{1}(x)=\sum_{k=1}^{d}a_{k}\,|\partial_{k}V(x)|,\qquad S_{2}(x)=\sum_{k=1}^{d}a_{k}c_{k}^{-1}\,|x_{k}|$$

are integrable with respect to $\mu$ by Linearity and Monotonicity of the Lebesgue Integral §integrable. Every function on $X$ formed below is obtained from the coordinates $x_{k}$, the continuous functions $\partial_{k}V$ of Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity, and functions $f\circ p_{d}$ with $f$ continuous on $\mathbb{R}^{d}$, by finitely many sums, products and absolute values; it is therefore Borel by claims 3 and 4 of Borel Measurability and Bounded Integration on a Metric Space and claims 2 to 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, a function of class $C^{k}$ on $\mathbb{R}^{d}$ being continuous by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. Whenever such a function is bounded pointwise in absolute value by an integrable function (or by a constant, constants being integrable by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space), it is integrable, by the monotonicity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative and the definition of integrability; this remark is used without repetition.

Step 1 (Bounds on the profile). Put $U(u)=v(u)+b+1$ for $u\in\mathbb{R}^{d}$; then $U(u)\ge1$ by Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §below. A constant function on $\mathbb{R}^{d}$ is smooth by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence of class $C^{2}$ by Smooth Map on a Euclidean Open Set, and its partial derivatives vanish, its difference quotients in Partial Derivative on a Euclidean Open Set being $0$; so by claims 1 and 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, $U$ is of class $C^{2}$ on $\mathbb{R}^{d}$, with $\partial_{k}U=\partial_{k}v$ and $\partial_{j}\partial_{k}U=\partial_{j}\partial_{k}v$ for $j,k\in[d]$.

Slope. Since $v(u)=U(u)-(b+1)$ and $U(u)\ge1$, $|v(u)|\le U(u)+|b+1|\le(1+|b+1|)U(u)$, so $0<1+|v(u)|\le(|b+1|+2)U(u)$. By Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §slope and $C\ge0$, $|\nabla_{a}v(u)|^{2}\le C(1+|v(u)|)^{2}\le C(|b+1|+2)^{2}U(u)^{2}=(L\,U(u))^{2}$, whence

$$|\nabla_{a}v(u)|\le L\,U(u),\qquad a_{k}^{1/2}|\partial_{k}v(u)|\le|\nabla_{a}v(u)|\quad(k\in[d]),\tag{1.1}$$

the second because $a_{k}(\partial_{k}v(u))^{2}$ is one of the nonnegative terms of $|\nabla_{a}v(u)|^{2}$.

Hessian. Let $u\in\mathbb{R}^{d}$ and put $T(u)=\sum_{i=1}^{d}\bigl(a_{i}\,\partial_{i}\partial_{i}v(u)+K\bigr)$. Applying Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §semiconvex with the vector $h$ whose only nonzero entry is $h_{j}=a_{j}^{1/2}$ gives $a_{j}\partial_{j}\partial_{j}v(u)\ge-K$, so each term of $T(u)$ is nonnegative. For $j\ne k$ in $[d]$, apply it with $h_{j}=a_{j}^{1/2}$, $h_{k}=\pm a_{k}^{1/2}$ and all other entries $0$; since $\partial_{j}\partial_{k}v(u)=\partial_{k}\partial_{j}v(u)$ by claim 1 of Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian, this gives $a_{j}\partial_{j}\partial_{j}v(u)+a_{k}\partial_{k}\partial_{k}v(u)\pm2(a_{j}a_{k})^{1/2}\partial_{j}\partial_{k}v(u)\ge-2K$, whence $2(a_{j}a_{k})^{1/2}|\partial_{j}\partial_{k}v(u)|\le(a_{j}\partial_{j}\partial_{j}v(u)+K)+(a_{k}\partial_{k}\partial_{k}v(u)+K)\le T(u)$. For $j=k$, the number $y=a_{j}\partial_{j}\partial_{j}v(u)$ satisfies $-K\le y$ and $y+K\le T(u)$, so $|y|\le T(u)+K$. With Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §curvature for $\varepsilon=1$, therefore, for all $j,k\in[d]$,

$$(a_{j}a_{k})^{1/2}|\partial_{j}\partial_{k}v(u)|\le T(u)+K,\qquad T(u)\le|\nabla_{a}v(u)|^{2}+C''\bigl(1+\lVert u\rVert^{2}\bigr)+dK.\tag{1.2}$$

Step 2 (The truncations). Let $p\in\mathbb{N}$, read as a positive real number, and let $J=(0,\infty)$, an open interval, which is an open subset of $\mathbb{R}^{1}$ (for $s\in J$ the ball of radius $s$ about $s$ lies in $J$). Define $G_{p}:J\to\mathbb{R}$ by $G_{p}(s)=ps(p+s)^{-1}=p-p^{2}(p+s)^{-1}$. The function $s\mapsto p+s$ is differentiable at every point of $J$ with derivative $1$, its difference quotients in Derivative at an Interior Point being $1$, and it does not vanish on $J$. Hence, by claim 1 of Reciprocal Rule for One-Dimensional Derivatives and claims 2 and 3 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives (writing $(p+s)^{-2}$ and $(p+s)^{-3}$ as products of copies of $(p+s)^{-1}$), $G_{p}$, $G_{p}'$ and $G_{p}''$ are differentiable at every point of $J$, with

$$G_{p}'(s)=p^{2}(p+s)^{-2},\qquad G_{p}''(s)=-2p^{2}(p+s)^{-3},$$

and all three are continuous at every point of $J$ relative to $J$ by Differentiability at an Interior Point Implies Continuity There; on $\mathbb{R}^{1}$ the Euclidean distance is $|s-s'|$, so this is continuity in the sense of clause 1 of C^k Maps on a Euclidean Open Set. As the partial derivative of Partial Derivative on a Euclidean Open Set with $n=1$ is the derivative of Derivative at an Interior Point, $G_{p}$ is of class $C^{2}$ on $J$ in the sense of C^k Maps on a Euclidean Open Set, with $\partial_{1}G_{p}=G_{p}'$ and $\partial_{1}\partial_{1}G_{p}=G_{p}''$. For $s\ge1$ put $\alpha=p(p+s)^{-1}$, so that $\alpha$ and $1-\alpha=s(p+s)^{-1}$ lie in $(0,1)$; then $G_{p}(s)=p(1-\alpha)$, $G_{p}'(s)=\alpha^{2}$, $G_{p}'(s)\,s=p\,\alpha(1-\alpha)$, $G_{p}'(s)\,s^{2}=p^{2}(1-\alpha)^{2}$ and $|G_{p}''(s)|\,s^{2}=2p\,\alpha(1-\alpha)^{2}$, so

$$0<G_{p}(s)\le p,\quad0<G_{p}'(s)\le1,\quad G_{p}'(s)\,s\le p,\quad G_{p}'(s)\,s^{2}\le p^{2},\quad G_{p}''(s)<0,\quad|G_{p}''(s)|\,s^{2}\le2p.\tag{2.1}$$

As $U$ takes values in $[1,\infty)\subseteq J$, we may put $F_{p}=G_{p}\circ U$, $w_{p}=G_{p}'\circ U$ and $z_{p}=G_{p}''\circ U$ on $\mathbb{R}^{d}$; note $w_{p}(u)=(1+p^{-1}U(u))^{-2}$.

Monotonicity in $p$. Fix $u$. If $p\le p'$ in $\mathbb{N}$, then $0<p'^{-1}U(u)\le p^{-1}U(u)$, so $1\le1+p'^{-1}U(u)\le1+p^{-1}U(u)$; squaring, and using that reciprocals reverse the order of positive numbers (Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares §reciprocal), $w_{p}(u)\le w_{p'}(u)$. Moreover $p^{-1}\to0$ as $p\to\infty$, by The Archimedean Property of the Real Numbers and Limit of a Sequence of Real Numbers, so $w_{p}(u)\to1$ by Arithmetic of Limits of Real Sequences.

Step 3 (The test functions). Apply Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball with $q=d$: fix a function $\chi:\mathbb{R}^{d}\to\mathbb{R}$ as in that lemma, and for positive $r$ let $\chi_{r}(u)=\chi(r^{-1}u)$ ($u\in\mathbb{R}^{d}$) be its cutoffs, as defined there; let $M_{1},M_{2}\ge0$ be the constants of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff. We use $r\in\mathbb{N}$, read as a positive real number, so $r^{-2}\le r^{-1}$. By that clause, $\chi_{r}$ is smooth, hence of class $C^{2}$ by Smooth Map on a Euclidean Open Set; $0\le\chi_{r}\le1$; $\chi_{r}(u)=1$ if $\lVert u\rVert\le r$ and $\chi_{r}(u)=0$ if $\lVert u\rVert\ge2r$; and $|\partial_{i}\chi_{r}|\le M_{1}r^{-1}$, $|\partial_{j}\partial_{i}\chi_{r}|\le M_{2}r^{-2}$ on $\mathbb{R}^{d}$ for $i,j\in[d]$.

For $p,r\in\mathbb{N}$ let $g=g_{p,r}=F_{p}\,\chi_{r}:\mathbb{R}^{d}\to\mathbb{R}$. By claim 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$, applied to $U:\mathbb{R}^{d}\to J$ and $G_{p}$, both of class $C^{2}$, $F_{p}$ is of class $C^{2}$ on $\mathbb{R}^{d}$, and so is $g$ by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set. By claim 1 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$, applied to $G_{p}$ and to $G_{p}'$ (both of class $C^{1}$ on $J$), $\partial_{k}F_{p}=w_{p}\,\partial_{k}v$ and $\partial_{j}w_{p}=z_{p}\,\partial_{j}v$; so the product rule (claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set) gives, for $j,k\in[d]$, on $\mathbb{R}^{d}$,

$$\partial_{k}g=w_{p}\,\partial_{k}v\,\chi_{r}+F_{p}\,\partial_{k}\chi_{r},\tag{3.1}$$ $$\partial_{j}\partial_{k}g=\bigl(z_{p}\,\partial_{j}v\,\partial_{k}v+w_{p}\,\partial_{j}\partial_{k}v\bigr)\chi_{r}+w_{p}\,\partial_{k}v\,\partial_{j}\chi_{r}+w_{p}\,\partial_{j}v\,\partial_{k}\chi_{r}+F_{p}\,\partial_{j}\partial_{k}\chi_{r}.\tag{3.2}$$

Bounds. Let $u\in\mathbb{R}^{d}$ and write $U=U(u)$. By (1.1) and (2.1), $w_{p}|\partial_{k}v|\le a_{k}^{-1/2}w_{p}LU\le a_{k}^{-1/2}Lp$, $|z_{p}|\,|\partial_{j}v|\,|\partial_{k}v|\le(a_{j}a_{k})^{-1/2}|z_{p}|L^{2}U^{2}\le(a_{j}a_{k})^{-1/2}2pL^{2}$, and

$$w_{p}(u)\,|\nabla_{a}v(u)|^{2}\le w_{p}(u)\,L^{2}U^{2}\le L^{2}p^{2}.\tag{3.3}$$

If $\chi_{r}(u)\ne0$, then $\lVert u\rVert<2r$, and by (1.2), $0<w_{p}\le1$ and (3.3), $w_{p}(a_{j}a_{k})^{1/2}|\partial_{j}\partial_{k}v|\chi_{r}\le w_{p}(T(u)+K)\le w_{p}|\nabla_{a}v(u)|^{2}+|C''|(1+4r^{2})+(d+1)K\le L^{2}p^{2}+|C''|(1+4r^{2})+(d+1)K$; if $\chi_{r}(u)=0$ this term vanishes. With $0<F_{p}\le p$ and the bounds on $\chi_{r}$ and its derivatives, (3.1) and (3.2) show that $g$, all $\partial_{k}g$ and all $\partial_{j}\partial_{k}g$ are bounded on $\mathbb{R}^{d}$ (with bounds depending on $p$ and $r$). Hence $g\in C^{2}_{b}(\mathbb{R}^{d})$ by Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded, and the hypothesis of the lemma with $n=d$ gives, $\beta$ being positive,

$$\beta\,L^{a,V}_{\mu}(g_{p,r})\le\beta\,\bigl|L^{a,V}_{\mu}(g_{p,r})\bigr|\le\beta R\,\lVert\nabla_{a}(g_{p,r}\circ p_{d})\rVert_{\mu}.\tag{3.4}$$

Step 4 (A lower bound for the functional). Fix $p,r\in\mathbb{N}$ and write $g=g_{p,r}$. For $x\in X$ put $u=p_{d}(x)$. By The Gibbs Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates §functional with $n=d$ and the linearity of Linearity and Monotonicity of the Lebesgue Integral §integrable, $L^{a,V}_{\mu}(g)=\int_{X}\Phi\,d\mu$, where the integrable function $\Phi$ is

$$\Phi(x)=\sum_{k=1}^{d}a_{k}\Bigl(\Bigl(\frac{u_{k}}{c_{k}}+\frac{\partial_{k}v(u)}{\beta}\Bigr)\partial_{k}g(u)-\partial_{k}\partial_{k}g(u)\Bigr).$$

Inserting (3.1) and (3.2) with $j=k$, and suppressing the argument $u$,

$$\beta\Phi=w_{p}\chi_{r}|\nabla_{a}v|^{2}+\beta w_{p}\chi_{r}\sum_{k=1}^{d}\frac{a_{k}}{c_{k}}u_{k}\,\partial_{k}v-\beta w_{p}\chi_{r}\sum_{k=1}^{d}a_{k}\,\partial_{k}\partial_{k}v-\beta z_{p}\chi_{r}|\nabla_{a}v|^{2}+E,$$ $$E=F_{p}\sum_{k=1}^{d}a_{k}\,\partial_{k}v\,\partial_{k}\chi_{r}+\beta F_{p}\sum_{k=1}^{d}\frac{a_{k}}{c_{k}}u_{k}\,\partial_{k}\chi_{r}-2\beta w_{p}\sum_{k=1}^{d}a_{k}\,\partial_{k}v\,\partial_{k}\chi_{r}-\beta F_{p}\sum_{k=1}^{d}a_{k}\,\partial_{k}\partial_{k}\chi_{r}.$$

We bound the terms from below. (i) $-\beta z_{p}\chi_{r}|\nabla_{a}v|^{2}\ge0$, since $z_{p}<0$ by (2.1), $\chi_{r}\ge0$ and $\beta>0$. (ii) For each $k$, with $s=a_{k}^{1/2}|\partial_{k}v|$ and $t=\beta a_{k}^{1/2}c_{k}^{-1}|u_{k}|$, $\beta a_{k}c_{k}^{-1}|u_{k}|\,|\partial_{k}v|=st\le\tfrac14a_{k}(\partial_{k}v)^{2}+\beta^{2}a_{k}c_{k}^{-2}u_{k}^{2}$, and $u_{k}^{2}\le\lVert u\rVert^{2}$; summing over $k$ and multiplying by $w_{p}\chi_{r}\in[0,1]$, the second term of $\beta\Phi$ is at least $-\tfrac14w_{p}\chi_{r}|\nabla_{a}v|^{2}-\beta^{2}\kappa_{0}\lVert u\rVert^{2}$. (iii) By the choice of $C'$, $\sum_{k=1}^{d}a_{k}\partial_{k}\partial_{k}v(u)\le(2\beta)^{-1}|\nabla_{a}v(u)|^{2}+C'(1+\lVert u\rVert^{2})$; multiplying by $-\beta w_{p}\chi_{r}\le0$, and using $w_{p}\chi_{r}\in[0,1]$, the third term of $\beta\Phi$ is at least $-\tfrac12w_{p}\chi_{r}|\nabla_{a}v|^{2}-\beta|C'|(1+\lVert u\rVert^{2})$. (iv) Since $0<F_{p}\le p$, $0<w_{p}\le1$, $|u_{k}|=|x_{k}|$, $\partial_{k}v(u)=\partial_{k}V(x)$ and $r^{-2}\le r^{-1}$, the bounds on the derivatives of $\chi_{r}$ give $|E|\le e_{r}(x)$, where

$$e_{r}(x)=r^{-1}\Bigl(M_{1}(p+2\beta)\,S_{1}(x)+\beta pM_{1}\,S_{2}(x)+\beta pM_{2}\,\sigma\Bigr).$$

Adding, $\beta\Phi(x)\ge\tfrac14w_{p}(u)\chi_{r}(u)|\nabla_{a}v(u)|^{2}-Q(x)-e_{r}(x)$, with $Q(x)=\beta^{2}\kappa_{0}\lVert p_{d}(x)\rVert^{2}+\beta|C'|(1+\lVert p_{d}(x)\rVert^{2})$. By Step 0, $Q$ and $e_{r}$ are integrable, with

$$\int_{X}Q\,d\mu=D:=(\beta^{2}\kappa_{0}+\beta|C'|)\,m+\beta|C'|,\qquad\int_{X}e_{r}\,d\mu=\frac{A_{p}}{r},\quad A_{p}:=M_{1}(p+2\beta)\!\int_{X}\!S_{1}\,d\mu+\beta pM_{1}\!\int_{X}\!S_{2}\,d\mu+\beta pM_{2}\sigma,$$

by Linearity and Monotonicity of the Lebesgue Integral §integrable and claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space; $D$ does not depend on $p$ or $r$, and $A_{p}$ does not depend on $r$. The function $x\mapsto w_{p}(p_{d}(x))\chi_{r}(p_{d}(x))|\nabla_{a}v(p_{d}(x))|^{2}$ lies between $0$ and $L^{2}p^{2}$ by (3.3), so it is integrable; let $Y_{p,r}$ be its integral. The linearity and monotonicity of Linearity and Monotonicity of the Lebesgue Integral §integrable now give

$$\beta\,L^{a,V}_{\mu}(g_{p,r})\ge\tfrac14Y_{p,r}-D-\frac{A_{p}}{r}.\tag{4.1}$$

Step 5 (An upper bound for the gradient norm). Since $g\in C^{2}_{b}(\mathbb{R}^{d})$ is of class $C^{1}$ by claim 2 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, with $g$ and its first partial derivatives bounded, $g\in C^{1}_{b}(\mathbb{R}^{d})$ (Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded) and $(d,g)$ is a representation of $g\circ p_{d}$. By The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient and Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial, $|\nabla_{a}(g\circ p_{d})(x)|_{a}^{2}=\sum_{k=1}^{d}a_{k}(\partial_{k}g(u))^{2}$ with $u=p_{d}(x)$, and by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, $\lVert\nabla_{a}(g\circ p_{d})\rVert_{\mu}^{2}=\int_{X}|\nabla_{a}(g\circ p_{d})|_{a}^{2}\,d\mu$. By (3.1), $(\partial_{k}g)^{2}\le2w_{p}^{2}\chi_{r}^{2}(\partial_{k}v)^{2}+2F_{p}^{2}(\partial_{k}\chi_{r})^{2}\le2w_{p}(\partial_{k}v)^{2}+2p^{2}M_{1}^{2}r^{-2}$, using $0<w_{p}\le1$, $0\le\chi_{r}\le1$ and $0<F_{p}\le p$. Multiplying by $a_{k}$, summing, and integrating,

$$\lVert\nabla_{a}(g_{p,r}\circ p_{d})\rVert_{\mu}^{2}\le2X_{p}+2\sigma p^{2}M_{1}^{2}r^{-2},\qquad X_{p}:=\int_{X}w_{p}(p_{d}(x))\,|\nabla_{a}v(p_{d}(x))|^{2}\,\mu(dx)\le L^{2}p^{2},$$

the integrand of $X_{p}$ lying between $0$ and $L^{2}p^{2}$ by (3.3). Taking square roots,

$$\lVert\nabla_{a}(g_{p,r}\circ p_{d})\rVert_{\mu}\le(2X_{p})^{1/2}+(2\sigma)^{1/2}pM_{1}r^{-1}.\tag{5.1}$$

Step 6 (Removing the cutoff). Fix $p$. By (3.4), (4.1) and (5.1), for every $r\in\mathbb{N}$,

$$\tfrac14Y_{p,r}\le D+\beta R\,(2X_{p})^{1/2}+\frac{B_{p}}{r},\qquad B_{p}:=A_{p}+\beta R\,(2\sigma)^{1/2}pM_{1},\tag{6.1}$$

where $R\ge0$ was used. For each $x\in X$ there is, by The Archimedean Property of the Real Numbers, an $r_{0}\in\mathbb{N}$ with $\lVert p_{d}(x)\rVert\le r_{0}$, and then $\chi_{r}(p_{d}(x))=1$ for all $r\ge r_{0}$; so the integrands of $Y_{p,r}$ converge pointwise, as $r\to\infty$, to that of $X_{p}$, and they are bounded by the integrable constant $L^{2}p^{2}$. By claim 3 of Dominated Convergence Theorem, $Y_{p,r}\to X_{p}$. Also $B_{p}/r\to0$ by The Archimedean Property of the Real Numbers and Limit of a Sequence of Real Numbers. Letting $r\to\infty$ in (6.1), by Arithmetic of Limits of Real Sequences and the comparison claim (claim 1) of Order Properties of Limits of Real Sequences,

$$\tfrac14X_{p}\le D+\beta R\,(2X_{p})^{1/2}.\tag{6.2}$$

Step 7 (Square-integrability of the noise gradient of $V$). Put $y=X_{p}^{1/2}\ge0$ and $c_{1}=4\beta R\,2^{1/2}\ge0$; (6.2) reads $y^{2}\le4D+c_{1}y$. We claim $y\le c_{1}+2D^{1/2}$. Otherwise $y>c_{1}+2D^{1/2}\ge0$, so $y>0$, $y\ge2D^{1/2}$, and $y^{2}>(c_{1}+2D^{1/2})y=c_{1}y+2D^{1/2}y\ge c_{1}y+4D$, a contradiction. Hence, for every $p\in\mathbb{N}$,

$$X_{p}\le M_{*}:=(c_{1}+2D^{1/2})^{2},$$

a number not depending on $p$. By Step 2, for each $x$ the sequence $p\mapsto w_{p}(p_{d}(x))|\nabla_{a}v(p_{d}(x))|^{2}$ of nonnegative Borel functions is nondecreasing and converges to $|\nabla_{a}v(p_{d}(x))|^{2}=|\nabla_{a}V(x)|_{a}^{2}$ (Arithmetic of Limits of Real Sequences); a nondecreasing convergent real sequence has its limit as least upper bound (each term is at most the limit by claim 1 of Order Properties of Limits of Real Sequences applied to the tails, and every upper bound dominates the limit by the same claim). So Monotone Convergence Theorem gives

$$\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu=\sup_{p\in\mathbb{N}}X_{p}\le M_{*}<\infty.\tag{7.1}$$

By Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §tangent, the class of $\nabla_{a}V$ lies in $L^{2}(\mu;X^{a})$ and belongs to $T^{a}_{\mu}$; write $\lVert\nabla_{a}V\rVert_{\mu}$ for its norm.

Step 8 (Splitting the Gibbs functional). Let $n\in\mathbb{N}$ and $g\in C^{2}_{b}(\mathbb{R}^{n})$, and put $\psi=g\circ p_{n}$. As in Step 5, $(n,g)$ is a representation of $\psi\in\mathcal{F}C^{1}_{b}(X)$, and by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient and Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial the $k$-th coordinate of $\nabla_{a}\psi(x)\in X^{a}$ is $a_{k}\partial_{k}g(p_{n}(x))$ for $k\le n$ and $0$ for $k>n$. By Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity, applied with $h=\nabla_{a}\psi(x)$, and since $\partial_{k}V=0$ for $k>d$ (Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient),

$$\langle\nabla_{a}V(x),\nabla_{a}\psi(x)\rangle_{a}=\sum_{k=1}^{n}a_{k}\,\partial_{k}V(x)\,\partial_{k}g(p_{n}(x))\qquad(x\in X).\tag{8.1}$$

(The clause cited gives $\sum_{k=1}^{d}\partial_{k}V(x)\,h_{k}$, with $h_{k}$ the $k$-th coordinate of $\nabla_{a}\psi(x)$; this sum and the sum in (8.1) both reduce to the sum over $k\le\min(d,n)$: the terms with $k>d$ vanish because $\partial_{k}V=0$, and those with $k>n$ because $h_{k}=0$.) Each summand is integrable with respect to $\mu$ by the preamble of The Gibbs Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates, so the function (8.1) is integrable by Linearity and Monotonicity of the Lebesgue Integral §integrable. The noise Ornstein-Uhlenbeck functional $L^{a}_{\mu}(g)$ of The Noise Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates §functional is defined, $\mu$ being in $\mathcal{P}_{2}(X)$, and for each $k\in[n]$ the integrand of the $k$-th term of The Gibbs Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates §functional is the integrand of the $k$-th term of The Noise Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates §functional plus $\beta^{-1}\partial_{k}V\,(\partial_{k}g\circ p_{n})$. By the linearity of Linearity and Monotonicity of the Lebesgue Integral §integrable and (8.1),

$$L^{a,V}_{\mu}(g)=L^{a}_{\mu}(g)+\frac{1}{\beta}\int_{X}\langle\nabla_{a}V,\nabla_{a}\psi\rangle_{a}\,d\mu=L^{a}_{\mu}(g)+\frac{1}{\beta}\,\langle\nabla_{a}V,\nabla_{a}\psi\rangle_{\mu},\tag{8.2}$$

the second equality by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, both $\nabla_{a}V$ (Step 7) and $\nabla_{a}\psi$ (The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient) being square-integrable with respect to $\mu$; here $\langle\cdot,\cdot\rangle_{\mu}$ is the inner product of $L^{2}(\mu;X^{a})$.

Step 9 (Finite Fisher information relative to $\gamma_{c}$). $L^{2}(\mu;X^{a})$ is a real Hilbert space by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, so The Cauchy-Schwarz Inequality in a Real Inner Product Space gives $|\langle\nabla_{a}V,\nabla_{a}\psi\rangle_{\mu}|\le\lVert\nabla_{a}V\rVert_{\mu}\lVert\nabla_{a}\psi\rVert_{\mu}$. With (8.2) and the hypothesis,

$$\bigl|L^{a}_{\mu}(g)\bigr|\le\bigl|L^{a,V}_{\mu}(g)\bigr|+\beta^{-1}\bigl|\langle\nabla_{a}V,\nabla_{a}\psi\rangle_{\mu}\bigr|\le\bigl(R+\beta^{-1}\lVert\nabla_{a}V\rVert_{\mu}\bigr)\lVert\nabla_{a}(g\circ p_{n})\rVert_{\mu}$$

for every $n\in\mathbb{N}$ and every $g\in C^{2}_{b}(\mathbb{R}^{n})$, with the nonnegative constant $R+\beta^{-1}\lVert\nabla_{a}V\rVert_{\mu}$. Since $\mu\in\mathcal{P}_{2}(X)$, A Bound on the Noise Ornstein-Uhlenbeck Functional by Noise Gradients Gives Finite Weighted Fisher Information §fisher shows that $\mu$ has a relative score $(\zeta_{k})_{k\in\mathbb{N}}$ with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$. Let $Z^{a}_{\mu}$ be its noise score field (The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §field).

Step 10 (The relative score with respect to the Gibbs measure). Now $\mu\in\mathcal{P}_{2}(X)$, $V$ is integrable with respect to $\mu$, $\mu$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$ (Step 9), and $\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu<\infty$ by (7.1). By Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §splitting, $\mu$ has a relative score with respect to $\gamma^{V}_{\beta}$ and finite Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$, which is the first assertion of the lemma. By Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §field, the element $\Xi=\beta Z^{a}_{\mu}+\nabla_{a}V$ of $L^{2}(\mu;X^{a})$ satisfies

$$\lVert\Xi\rVert_{\mu}^{2}=\beta^{2}\,\mathcal{I}_{a}(\mu\,|\,\gamma^{V}_{\beta}).\tag{10.1}$$

Moreover $Z^{a}_{\mu}\in T^{a}_{\mu}$ by The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §tangent, $\nabla_{a}V\in T^{a}_{\mu}$ by Step 7, and $T^{a}_{\mu}$ is a linear subspace of $L^{2}(\mu;X^{a})$ by Linearity of the Noise Gradient, and the Noise Tangent Space is a Closed Linear Subspace §subspace; so $\Xi\in T^{a}_{\mu}$.

Step 11 (The bound $\mathcal{I}_{a}(\mu\,|\,\gamma^{V}_{\beta})\le R^{2}$). Let $n\in\mathbb{N}$, $g\in C^{2}_{b}(\mathbb{R}^{n})$ and $\psi=g\circ p_{n}$. By bilinearity of the inner product, The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §pairing-functional and (8.2),

$$\langle\Xi,\nabla_{a}\psi\rangle_{\mu}=\beta\langle Z^{a}_{\mu},\nabla_{a}\psi\rangle_{\mu}+\langle\nabla_{a}V,\nabla_{a}\psi\rangle_{\mu}=\beta L^{a}_{\mu}(g)+\langle\nabla_{a}V,\nabla_{a}\psi\rangle_{\mu}=\beta L^{a,V}_{\mu}(g),$$

so by the hypothesis

$$\langle\Xi,\nabla_{a}\psi\rangle_{\mu}\le\bigl|\langle\Xi,\nabla_{a}\psi\rangle_{\mu}\bigr|\le\beta R\,\lVert\nabla_{a}\psi\rVert_{\mu}.\tag{11.1}$$

Since $\Xi\in T^{a}_{\mu}$, Noise Gradients of Bounded C^2 Cylindrical Functions Are Dense in the Noise Tangent Space §density yields $\psi_{j}\in\mathcal{F}C^{2}_{b}(X)$, $j\in\mathbb{N}$, with $\lVert\nabla_{a}\psi_{j}-\Xi\rVert_{\mu}\to0$; that is, $\nabla_{a}\psi_{j}\to\Xi$ in the metric space $L^{2}(\mu;X^{a})$, whose distance is $\lVert\cdot-\cdot\rVert_{\mu}$ (Real Hilbert Space §topology). By Bounded C^2 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical, $\psi_{j}=g_{j}\circ p_{n_{j}}$ with $n_{j}\in\mathbb{N}$ and $g_{j}\in C^{2}_{b}(\mathbb{R}^{n_{j}})$, so (11.1) holds for each $\psi_{j}$. By The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity, $\langle\nabla_{a}\psi_{j},\Xi\rangle_{\mu}\to\langle\Xi,\Xi\rangle_{\mu}=\lVert\Xi\rVert_{\mu}^{2}$ and $\lVert\nabla_{a}\psi_{j}\rVert_{\mu}\to\lVert\Xi\rVert_{\mu}$. Letting $j\to\infty$ in (11.1), by Arithmetic of Limits of Real Sequences and claim 1 of Order Properties of Limits of Real Sequences, $\lVert\Xi\rVert_{\mu}^{2}\le\beta R\,\lVert\Xi\rVert_{\mu}$. If $\lVert\Xi\rVert_{\mu}>0$, dividing gives $\lVert\Xi\rVert_{\mu}\le\beta R$; if $\lVert\Xi\rVert_{\mu}=0$, then $\lVert\Xi\rVert_{\mu}\le\beta R$ as $\beta R\ge0$. Hence $\lVert\Xi\rVert_{\mu}^{2}\le\beta^{2}R^{2}$, and by (10.1), dividing by $\beta^{2}>0$,

$$\mathcal{I}_{a}(\mu\,|\,\gamma^{V}_{\beta})\le R^{2}.\qquad\blacksquare$$