Proof of The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator

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

Conventions. Throughout, $n$ denotes a natural number, $\gamma_{n}$ abbreviates the diagonal Gaussian measure $\gamma_{c^{(n)}}$ on $\mathbb{R}^{n}$ of Variance Sequences and Their Truncations §truncations, and $\rho_{n}$ is the diagonal Gaussian density with variances $c^{(n)}$. By Diagonal Gaussian Measures on Euclidean Space §measure, $\gamma_{n}$ is the measure with density $\rho_{n}$ with respect to Lebesgue measure $\lambda_{n}$ on $\mathcal{B}(\mathbb{R}^{n})$; so by claim 3 of Image Measures, Measures with Densities, and Change of Variables every Borel $f:\mathbb{R}^{n}\to[0,\infty]$ satisfies $\int f\,d\gamma_{n}=\int f\rho_{n}\,d\lambda_{n}$ in $[0,\infty]$, and a Borel $f:\mathbb{R}^{n}\to\mathbb{R}$ is $\gamma_{n}$-integrable exactly when $f\rho_{n}$ is $\lambda_{n}$-integrable, with the same identity. We call this the density rule. Integrals $\int\dots\,dz$ over $\mathbb{R}^{n}$ are taken with respect to $\lambda_{n}$. For real $v>0$, $\gamma_{(v)}$ denotes the diagonal Gaussian measure on $\mathbb{R}^{1}$ with variance vector $(v)$, which is the measure written $\gamma_{v}$ in Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity, and $\rho_{(v)}$ is its density. For $\mathbf{a},\mathbf{b}\in\mathbb{R}^{n}$ put $\mathbf{a}\odot\mathbf{b}=(\mathbf{a}_{1}\mathbf{b}_{1},\dots,\mathbf{a}_{n}\mathbf{b}_{n})$; if every $\mathbf{b}_{k}$ is nonzero, $\mathbf{b}^{-1}=(\mathbf{b}_{1}^{-1},\dots,\mathbf{b}_{n}^{-1})$; $\mathbf{1}=(1,\dots,1)$; and for $k\in[n]$, $\delta^{(k)}\in\mathbb{R}^{n}$ is the point with $k$-th entry $1$ and all other entries $0$. For real $r\ge0$ put $\eta(r)=(\eta_{1}(r),\dots,\eta_{n}(r))$ and $\xi(r)=(\xi_{1}(r),\dots,\xi_{n}(r))$, with the numbers $\eta_{k}(r)$, $\xi_{k}(r)$ of The Mehler Maps and the Mehler Semigroup with Noise Weights on Continuous Cylindrical Functions of Polynomial Growth §map; by that clause $0<\eta_{k}(r)\le1$, $0\le\xi_{k}(r)\le1$ and $\eta_{k}(r)^{2}+\xi_{k}(r)^{2}=1$. We write $\Theta_{n}=\max\{\theta_{1},\dots,\theta_{n}\}$, with the rates of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §rates.

A function $\varphi:\mathbb{R}^{n}\to\mathbb{R}$ is called of polynomial growth with constants $(B_{\varphi},q_{\varphi})$ if it is continuous, $B_{\varphi}\ge0$ is real, $q_{\varphi}\in\mathbb{N}\cup\{0\}$, and $|\varphi(y)|\le B_{\varphi}(1+\lVert y\rVert^{q_{\varphi}})$ for every $y\in\mathbb{R}^{n}$. By Continuous Cylindrical Functions of Polynomial Growth on a Hilbert Space §class and Continuous Cylindrical Functions of Polynomial Growth on a Hilbert Space §representation, $(n,\psi,B,q)$ is a representation of $F$ exactly when $\psi$ is of polynomial growth with constants $(B,q)$ and $F=\psi\circ p_{n}$. For such $\varphi$ and real $r\ge0$ we put

$$\varphi_{r}(u)=\int_{\mathbb{R}^{n}}\varphi\bigl(\eta(r)\odot u+\xi(r)\odot z\bigr)\,\gamma_{n}(dz)\qquad(u\in\mathbb{R}^{n}),$$

which is well defined by Step 9 below; for $\varphi=\psi$ and $r=t$ this is the function $\psi_{t}$ of claim 2.

Step 1 (Elementary inequalities). Let $a,b\ge0$ be real and $q,q',Q\in\mathbb{N}\cup\{0\}$, with $a^{0}=1$. (E1) If $q\le Q$, then $a^{q}\le1+a^{Q}$: for $a\le1$, $a^{q}\le1$, and for $a\ge1$, $a^{q}\le a^{Q}$. (E2) $1+a^{q}\le2(1+a^{2})^{q}$, since each of $1$ and $a^{q}$ is at most $(1+a^{2})^{q}$: for $a\le1$ because $a^{q}\le1$, and for $a\ge1$ because $a^{q}\le a^{2q}\le(1+a^{2})^{q}$. (E3) $(a+b)^{q}\le2^{q}(a^{q}+b^{q})$ and $1+(a+b)^{q}\le2^{q}(1+a^{q})(1+b^{q})$: for $q=0$ these read $1\le2$ and $2\le4$; for $q\ge1$, with $m=\max\{a,b\}$, $(a+b)^{q}\le(2m)^{q}=2^{q}m^{q}\le2^{q}(a^{q}+b^{q})$, whence $1+(a+b)^{q}\le2^{q}(1+a^{q}+b^{q}+a^{q}b^{q})=2^{q}(1+a^{q})(1+b^{q})$. (E4) $(1+a^{q})(1+a^{q'})\le4(1+a^{q+q'})$, since by (E1) each of $1$, $a^{q}$, $a^{q'}$, $a^{q+q'}$ is at most $1+a^{q+q'}$; also $1+a^{q}\le2(1+a^{Q})$ for $q\le Q$, by (E1). (E5) If $p\ge1$ is real, $m\in\mathbb{N}$ and $p\le m$, then $a^{p}\le1+a^{m}$, for the power $a^{p}$ of Measure Spaces and the Lebesgue Integral: Standing Notation §powers. Indeed $1^{p}=1$, because $1^{p}=(1\cdot1)^{p}=1^{p}1^{p}$ and $1^{p}>0$ by Properties of Real Powers of Nonnegative Real Numbers §product and Properties of Real Powers of Nonnegative Real Numbers §values; so for $a\le1$, $a^{p}\le1$ by Properties of Real Powers of Nonnegative Real Numbers §monotone, while for $a\ge1$ and $p<m$, $a^{m}=a^{p}a^{m-p}\ge a^{p}1^{m-p}=a^{p}$ by Properties of Real Powers of Nonnegative Real Numbers §exponents, Properties of Real Powers of Nonnegative Real Numbers §monotone and Properties of Real Powers of Nonnegative Real Numbers §agreement, the case $p=m$ being Properties of Real Powers of Nonnegative Real Numbers §agreement. Finally, for $\mathbf{a},y\in\mathbb{R}^{n}$ with $|\mathbf{a}_{k}|\le1$ for every $k$, $\lVert\mathbf{a}\odot y\rVert\le\lVert y\rVert$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square and claim 5 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities; and $|y_{k}|\le\lVert y\rVert$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate.

Step 2 (Moments and the transfer rule). For $N\in\mathbb{N}\cup\{0\}$ the function $y\mapsto(1+|y|^{2})^{N}$ on $X$ is continuous, hence Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space, and

$$K_{N}=\int_{X}(1+|y|^{2})^{N}\,\gamma_{c}(dy)<\infty .$$

Indeed, put $\bar{c}=\sum_{k=1}^{\infty}c_{k}>0$, so that $c_{k}\le\bar{c}$ for every $k$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, and $\alpha=1/(4\bar{c})$. The function $u\mapsto(1+u/\alpha)^{N}$ is a polynomial function, so claim 1 of The Exponential Function Dominates Every Polynomial Function (with $\varepsilon=1$, multiplying by $\exp(u)>0$ and using claims 1 and 2 of Basic Properties of the Exponential Function) gives a real $u_{0}\ge1$ with $(1+u/\alpha)^{N}<\exp(u)$ for $u\ge u_{0}$, while $(1+u/\alpha)^{N}\le(1+u_{0}/\alpha)^{N}$ for $0\le u\le u_{0}$; with $u=\alpha|y|^{2}$ and $C_{N}=(1+u_{0}/\alpha)^{N}$ this gives $(1+|y|^{2})^{N}\le C_{N}+\exp(\alpha|y|^{2})$ for every $y\in X$. The function $y\mapsto\exp(\alpha|y|^{2})$ is Borel and $\gamma_{c}$-integrable by Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §exponential, applied with this $\alpha$ and with $1/2$ in place of the number written $\theta$ there, since $2\alpha c_{k}\le1/2$; constants are integrable by claim 6 of Borel Measurability and Bounded Integration on a Metric Space, so $K_{N}<\infty$ by Linearity and Monotonicity of the Lebesgue Integral §nonnegative. The coordinate map $p_{n}$ is linear with $\lVert p_{n}(x)\rVert\le|x|$ by Orthonormal Expansions in a Real Hilbert Space §bessel, hence Lipschitz, continuous and Borel; and $(p_{n})_{\#}\gamma_{c}=\gamma_{n}$ by Diagonal Gaussian Measures on a Hilbert Space §measure, push-forwards being the image measures of Image Measures, Measures with Densities, and Change of Variables by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward. So claim 2 of Image Measures, Measures with Densities, and Change of Variables gives the transfer rule: $\int_{\mathbb{R}^{n}}g\,d\gamma_{n}=\int_{X}g(p_{n}x)\,\gamma_{c}(dx)$ for every Borel $g:\mathbb{R}^{n}\to[0,\infty]$, and a Borel $g:\mathbb{R}^{n}\to\mathbb{R}$ is $\gamma_{n}$-integrable exactly when $g\circ p_{n}$ is $\gamma_{c}$-integrable, with the same identity. In particular

$$\int_{\mathbb{R}^{n}}(1+\lVert z\rVert^{2})^{N}\,\gamma_{n}(dz)=\int_{X}(1+\lVert p_{n}y\rVert^{2})^{N}\,\gamma_{c}(dy)\le K_{N},$$

and therefore, by (E2), $\int(1+\lVert z\rVert^{m})\,\gamma_{n}(dz)\le2K_{m}$ for every $m\in\mathbb{N}\cup\{0\}$, and $\int\lVert z\rVert^{2}\,\gamma_{n}(dz)\le K_{1}$. Since the $k$-th entry of $p_{n}(y)$ is $y_{k}$, Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §coordinates and the transfer rule show that for $j,k\in[n]$ the functions $z\mapsto z_{k}$ and $z\mapsto z_{j}z_{k}$ are $\gamma_{n}$-integrable, with $\int z_{k}\,\gamma_{n}(dz)=0$, $\int z_{k}^{2}\,\gamma_{n}(dz)=c_{k}$, and $\int z_{j}z_{k}\,\gamma_{n}(dz)=0$ for $j\ne k$.

Step 3 (Measurability). Equip $\mathbb{R}^{n}\times\mathbb{R}^{n}$ with the product metric of two copies of $(\mathbb{R}^{n},d_{E})$; by Euclidean Space is a Separable Metric Space §separable and The Borel Sigma-Algebra of a Product of Two Separable Metric Spaces is the Product Sigma-Algebra §product its Borel $\sigma$-algebra is $\mathcal{B}(\mathbb{R}^{n})\otimes\mathcal{B}(\mathbb{R}^{n})$, on which the product measures of Existence and Uniqueness of the Product Measure are defined. A map between two of the spaces $\mathbb{R}$, $\mathbb{R}^{n}$, $\mathbb{R}^{n}\times\mathbb{R}^{n}$ each of whose entries is a constant plus a fixed real linear combination of the entries of the argument satisfies a Lipschitz bound, since by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate each entry of the difference of two arguments is at most their distance; so such an affine map is continuous, hence Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space. Continuous real functions are Borel by the same claim, composites of Borel maps are Borel by claim 4 there, and sums, products, absolute values, maxima and positive and negative parts of measurable real functions are measurable by Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and Integrable Function and the Lebesgue Integral. Every function on $\mathbb{R}^{n}$ or on $\mathbb{R}^{n}\times\mathbb{R}^{n}$ integrated below is built in this way from Borel functions on $\mathbb{R}^{n}$ and affine maps (for instance $(y,z)\mapsto\varphi(\mathbf{a}\odot y+\mathbf{b}\odot z)$ with $\varphi$ Borel), and is therefore Borel; we use this without further comment. The measures $\lambda_{n}$, $\gamma_{n}$, $\gamma_{c}$ are $\sigma$-finite ($\lambda_{n}$ by Lebesgue Measure on Euclidean Space is Sigma-Finite §sigma-finite, the others being probability measures), so Tonelli and Fubini Theorems applies to their products. A composite of continuous maps is continuous by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §on-subset. Continuity at every point in the sense of clause 1 of C^k Maps on a Euclidean Open Set (continuity at a point) coincides with continuity for the Euclidean distances in the sense of Continuous Map Between Metric Spaces: by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §distance and Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square, the sums of squares occurring in the former are the squares of the corresponding distances, and for real $a,b\ge0$ one has $a^{2}<b^{2}$ exactly when $a<b$, by claim 5 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities (the square root being strictly increasing, with $\sqrt{a^{2}}=a$); so the two notions are used interchangeably.

Step 4 (Partial derivatives after an affine substitution). Let $f:\mathbb{R}^{n}\to\mathbb{R}$, $\mathbf{a},\mathbf{b}\in\mathbb{R}^{n}$, $g(v)=f(\mathbf{a}+\mathbf{b}\odot v)$, $k\in[n]$ and $v\in\mathbb{R}^{n}$, and suppose that the partial derivative $\partial_{k}f(w)$ exists at $w=\mathbf{a}+\mathbf{b}\odot v$. Then $\partial_{k}g(v)$ exists and equals $\mathbf{b}_{k}\partial_{k}f(w)$. Indeed, for real $h\ne0$, $\mathbf{a}+\mathbf{b}\odot(v+h\delta^{(k)})=w+\mathbf{b}_{k}h\,\delta^{(k)}$. If $\mathbf{b}_{k}=0$, the difference quotient of $g$ is $0=\mathbf{b}_{k}\partial_{k}f(w)$ for every $h$. If $\mathbf{b}_{k}\ne0$ and $\varepsilon>0$, let $\delta>0$ be as in Partial Derivative on a Euclidean Open Set for $f$ at $w$ and $\varepsilon/|\mathbf{b}_{k}|$; for $0<|h|<\delta/|\mathbf{b}_{k}|$ the number $h'=\mathbf{b}_{k}h$ satisfies $0<|h'|<\delta$, and

$$\Bigl|\frac{g(v+h\delta^{(k)})-g(v)}{h}-\mathbf{b}_{k}\partial_{k}f(w)\Bigr|=|\mathbf{b}_{k}|\,\Bigl|\frac{f(w+h'\delta^{(k)})-f(w)}{h'}-\partial_{k}f(w)\Bigr|<\varepsilon .$$

Since $v\mapsto\mathbf{a}+\mathbf{b}\odot v$ is continuous (Step 3) and a real multiple of a function of class $C^{1}$ is of class $C^{1}$ (directly from Partial Derivative on a Euclidean Open Set), clauses 1 to 4 of C^k Maps on a Euclidean Open Set give: if $f$ is of class $C^{1}$ on $\mathbb{R}^{n}$, so is $g$, with $\partial_{k}g(v)=\mathbf{b}_{k}\partial_{k}f(\mathbf{a}+\mathbf{b}\odot v)$; and if $f$ is of class $C^{2}$, then $\partial_{k}g$ is $\mathbf{b}_{k}$ times the function obtained in the same way from the $C^{1}$ function $\partial_{k}f$, so $g$ is of class $C^{2}$ with $\partial_{l}\partial_{k}g(v)=\mathbf{b}_{l}\mathbf{b}_{k}\partial_{l}\partial_{k}f(\mathbf{a}+\mathbf{b}\odot v)$. Consequently, if $f$ and its first (and second) partial derivatives are bounded, so are those of $g$; when $|\mathbf{b}_{k}|\le1$ for every $k$, with the same bounds.

Step 5 (Affine change of variables). Let $\mathbf{a},\mathbf{b}\in\mathbb{R}^{n}$ with $\mathbf{b}_{k}>0$ for every $k$, and $\beta=\prod_{k=1}^{n}\mathbf{b}_{k}>0$. Then for every Borel $f:\mathbb{R}^{n}\to[0,\infty]$,

$$\int f(\mathbf{a}+\mathbf{b}\odot z)\,dz=\beta^{-1}\int f(w)\,dw\qquad\text{in }[0,\infty].$$

Indeed, $\Phi(z)=\mathbf{a}+\mathbf{b}\odot z$ is a bijection of $\mathbb{R}^{n}$ with inverse $\Phi^{-1}(w)=\mathbf{b}^{-1}\odot(w-\mathbf{a})$. The coordinate functions $v\mapsto v_{j}$ are continuous with constant partial derivatives $0$ or $1$, directly from Partial Derivative on a Euclidean Open Set; so by Step 4 the components of $\Phi$ and of $\Phi^{-1}$ are of class $C^{1}$, and their Jacobian matrices are, at every point, the diagonal matrices with diagonal entries $\mathbf{b}_{k}$, respectively $\mathbf{b}_{k}^{-1}$. The first is symmetric and positive definite, since $x\cdot(D\Phi\,x)=\sum_{k}\mathbf{b}_{k}x_{k}^{2}>0$ for $x\ne0$; the second is its inverse matrix; and $\det D\Phi=\beta$ by The Determinant of a Triangular Matrix is the Product of its Diagonal Entries, a diagonal matrix being lower triangular. So Change of Variables for the Lebesgue Integral under a Continuously Differentiable Bijection of Euclidean Space with Symmetric Positive Definite Jacobian Matrix, and the Density of a Push-Forward §integrals gives $\int(f\circ\Phi)\,\beta\,dz=\int f\,dw$, and the claim follows by Linearity and Monotonicity of the Lebesgue Integral §nonnegative.

Step 6 (The coefficients). Let $k\in\mathbb{N}$. (i) For real $y\ge0$, $0\le1-\exp(-y)\le y$: by claim 6 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities, $\exp(-y)\le1$ and $\exp(y)-1\le y\exp(y)$, and multiplying the latter by $\exp(-y)>0$ gives $1-\exp(-y)\le y$, since $\exp(y)\exp(-y)=\exp(0)=1$ by claim 1 of Basic Properties of the Exponential Function. (ii) For real $y>0$, $\bigl|\frac{1-\exp(-y)}{y}-1\bigr|\le y\exp(y)$: by (i), $\exp(-y)\ge1-y$, and by claim 6 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities, $y\le\exp(y)-1\le y+y^{2}\exp(y)$; as $1-\exp(-y)=\exp(-y)(\exp(y)-1)$ and $0<\exp(-y)\le1$, this gives $(1-y)y\le1-\exp(-y)\le y+y^{2}\exp(y)$, and $y\le y\exp(y)$. (iii) Hence, for real $h>0$, $0\le1-\eta_{k}(h)\le\theta_{k}h$ and $0\le\xi_{k}(h)^{2}=1-\exp(-2\theta_{k}h)\le2\theta_{k}h$, and

$$\Bigl|\frac{1-\eta_{k}(h)}{h}-\theta_{k}\Bigr|\le\theta_{k}^{2}h\exp(\theta_{k}h),\qquad\Bigl|\frac{\xi_{k}(h)^{2}}{h}-2\theta_{k}\Bigr|\le4\theta_{k}^{2}h\exp(2\theta_{k}h),$$

by (i) and (ii) with $y=\theta_{k}h$, respectively $y=2\theta_{k}h$; for $0<h\le1$ the right-hand sides are at most $\theta_{k}^{2}\exp(\theta_{k})h$ and $4\theta_{k}^{2}\exp(2\theta_{k})h$, $\exp$ being increasing by claim 4 of Basic Properties of the Exponential Function. (iv) If $(r_{m})_{m\in\mathbb{N}}$ is a sequence in $[0,\infty)$ converging to $r$, then $\eta_{k}(r_{m})\to\eta_{k}(r)$ and $\xi_{k}(r_{m})\to\xi_{k}(r)$, by claims 3(f) and 3(e) of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities. (v) $\eta_{k}(0)=1$ and $\xi_{k}(0)=0$, since $\exp(0)=1$; for $r>0$, $\exp(-2\theta_{k}r)<\exp(0)=1$ by claim 4 of Basic Properties of the Exponential Function, so $\xi_{k}(r)>0$ by claim 5 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities. (vi) For real $s,t\ge0$, claim 1 of Basic Properties of the Exponential Function gives $\eta_{k}(s)\eta_{k}(t)=\eta_{k}(s+t)$, $\eta_{k}(t)^{2}=\exp(-2\theta_{k}t)$, and

$$\eta_{k}(t)^{2}\xi_{k}(s)^{2}+\xi_{k}(t)^{2}=\exp(-2\theta_{k}t)\bigl(1-\exp(-2\theta_{k}s)\bigr)+1-\exp(-2\theta_{k}t)=1-\exp(-2\theta_{k}(s+t))=\xi_{k}(s+t)^{2}.$$

Step 7 (Gaussian reflection). Let $\mathbf{a},\mathbf{b}\in\mathbb{R}^{n}$ with $\mathbf{b}_{k}>0$ and $\mathbf{a}_{k}^{2}+\mathbf{b}_{k}^{2}=1$ for every $k\in[n]$, and let $S:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}\times\mathbb{R}^{n}$ be the affine map $S(y,z)=(\mathbf{a}\odot y+\mathbf{b}\odot z,\ \mathbf{b}\odot y-\mathbf{a}\odot z)$.

(a) For every measurable $f:\mathbb{R}^{n}\times\mathbb{R}^{n}\to[0,\infty)$,

$$\int\Bigl(\int f(S(y,z))\,\gamma_{n}(dz)\Bigr)\gamma_{n}(dy)=\int\Bigl(\int f(y,z)\,\gamma_{n}(dz)\Bigr)\gamma_{n}(dy)\qquad\text{in }[0,\infty],$$

the inner integrands and the inner integrals, as functions of $y$, being measurable by Tonelli and Fubini Theorems applied to $\gamma_{n}\otimes\gamma_{n}$.

(b) If $f:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ is measurable and $|f(y,z)|\le K(1+\lVert y\rVert^{m})(1+\lVert z\rVert^{m})$ for some real $K\ge0$ and $m\in\mathbb{N}\cup\{0\}$, then for every $y$ the functions $z\mapsto f(y,z)$ and $z\mapsto f(S(y,z))$ are $\gamma_{n}$-integrable, the functions $y\mapsto\int f(y,z)\,\gamma_{n}(dz)$ and $y\mapsto\int f(S(y,z))\,\gamma_{n}(dz)$ are $\gamma_{n}$-integrable, and the two iterated integrals of (a) are equal real numbers.

Proof of (a). Let $\beta=\prod_{k}\mathbf{b}_{k}$. For all $y,z$ and $k$, $(\mathbf{a}_{k}y_{k}+\mathbf{b}_{k}z_{k})^{2}+(\mathbf{b}_{k}y_{k}-\mathbf{a}_{k}z_{k})^{2}=y_{k}^{2}+z_{k}^{2}$; so the weighted squares of The Diagonal Gaussian Density on Euclidean Space and Its Notation §scaling satisfy $|S_{1}(y,z)|^{2}_{c^{(n)}}+|S_{2}(y,z)|^{2}_{c^{(n)}}=|y|^{2}_{c^{(n)}}+|z|^{2}_{c^{(n)}}$, where $S=(S_{1},S_{2})$, and The Diagonal Gaussian Density on Euclidean Space and Its Notation §density with claim 1 of Basic Properties of the Exponential Function gives

$$\rho_{n}(S_{1}(y,z))\,\rho_{n}(S_{2}(y,z))=\rho_{n}(y)\,\rho_{n}(z).\qquad(7.1)$$

Put $g(w,v)=\rho_{n}(w)\rho_{n}(v)f(w,v)$, measurable and nonnegative, $\rho_{n}$ being Borel and positive by The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §regularity. By the density rule (twice), Linearity and Monotonicity of the Lebesgue Integral §nonnegative (to move the factor $\rho_{n}(y)$ inside) and (7.1), the left-hand side of (a) equals $\int\bigl(\int g(S(y,z))\,dz\bigr)dy$. Fix $y$ and put $k_{y}(w)=g\bigl(w,\mathbf{b}^{-1}\odot(y-\mathbf{a}\odot w)\bigr)$. Since $\mathbf{b}_{k}^{2}=1-\mathbf{a}_{k}^{2}$, for every $z$

$$\mathbf{b}^{-1}\odot\bigl(y-\mathbf{a}\odot(\mathbf{a}\odot y+\mathbf{b}\odot z)\bigr)=\mathbf{b}^{-1}\odot(\mathbf{b}\odot\mathbf{b}\odot y-\mathbf{a}\odot\mathbf{b}\odot z)=\mathbf{b}\odot y-\mathbf{a}\odot z,$$

that is, $k_{y}(\mathbf{a}\odot y+\mathbf{b}\odot z)=g(S(y,z))$; so Step 5 gives $\int g(S(y,z))\,dz=\beta^{-1}\int k_{y}(w)\,dw$. With the measurable function $\kappa(y,w)=g\bigl(w,\mathbf{b}^{-1}\odot(y-\mathbf{a}\odot w)\bigr)$ on $\mathbb{R}^{n}\times\mathbb{R}^{n}$, Linearity and Monotonicity of the Lebesgue Integral §nonnegative and Tonelli and Fubini Theorems for $\lambda_{n}\otimes\lambda_{n}$ give that the left-hand side equals $\beta^{-1}\int\bigl(\int\kappa(y,w)\,dy\bigr)dw$. Fix $w$; for every $v$, $\kappa(\mathbf{a}\odot w+\mathbf{b}\odot v,w)=g(w,v)$, so Step 5 (with $\mathbf{a}\odot w$ in place of $\mathbf{a}$) gives $\int g(w,v)\,dv=\beta^{-1}\int\kappa(y,w)\,dy$. Hence the left-hand side equals $\int\bigl(\int g(w,v)\,dv\bigr)dw=\int\bigl(\int f(w,v)\rho_{n}(v)\,dv\bigr)\rho_{n}(w)\,dw$, which is the right-hand side by the density rule and Linearity and Monotonicity of the Lebesgue Integral §nonnegative.

Proof of (b). Since $|\mathbf{a}_{k}|,|\mathbf{b}_{k}|\le1$, Step 1 and Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §triangle give $\lVert S_{1}(y,z)\rVert,\lVert S_{2}(y,z)\rVert\le\lVert y\rVert+\lVert z\rVert$, so by (E3) and (E4)

$$|f(S(y,z))|\le K\,2^{2m}(1+\lVert y\rVert^{m})^{2}(1+\lVert z\rVert^{m})^{2}\le2^{2m+4}K(1+\lVert y\rVert^{2m})(1+\lVert z\rVert^{2m}),$$

and by (E4) also $|f(y,z)|\le4K(1+\lVert y\rVert^{2m})(1+\lVert z\rVert^{2m})$. Let $h$ be $f$ or $f\circ S$, so $|h(y,z)|\le K'(1+\lVert y\rVert^{2m})(1+\lVert z\rVert^{2m})$ with $K'=2^{2m+4}K$. Then $h^{+},h^{-}\le|h|$, and by Step 2 and Linearity and Monotonicity of the Lebesgue Integral §nonnegative, $\int|h(y,z)|\,\gamma_{n}(dz)\le2K'K_{2m}(1+\lVert y\rVert^{2m})<\infty$ for every $y$, so $z\mapsto h(y,z)$ is integrable; the functions $y\mapsto\int h^{\pm}(y,z)\,\gamma_{n}(dz)$ are measurable (Tonelli and Fubini Theorems) with integrals at most $4K'K_{2m}^{2}<\infty$; hence $y\mapsto\int h(y,z)\,\gamma_{n}(dz)=\int h^{+}(y,z)\,\gamma_{n}(dz)-\int h^{-}(y,z)\,\gamma_{n}(dz)$ is integrable, and its integral is the difference of the iterated integrals of $h^{+}$ and $h^{-}$, by Integrable Function and the Lebesgue Integral and Linearity and Monotonicity of the Lebesgue Integral §integrable. Since $(f\circ S)^{\pm}=f^{\pm}\circ S$, (a) applied to $f^{+}$ and to $f^{-}$ gives the claim.

Step 8 (Products of one-dimensional integrals). Let $g_{1},\dots,g_{n}:\mathbb{R}\to\mathbb{R}$ be Borel with $\int|g_{k}|\,d\gamma_{(c_{k})}<\infty$ for every $k$. Then $z\mapsto\prod_{k=1}^{n}g_{k}(z_{k})$ is Borel and $\gamma_{n}$-integrable on $\mathbb{R}^{n}$, and

$$\int_{\mathbb{R}^{n}}\prod_{k=1}^{n}g_{k}(z_{k})\,\gamma_{n}(dz)=\prod_{k=1}^{n}\int_{\mathbb{R}}g_{k}\,d\gamma_{(c_{k})} .$$

The function is Borel because the coordinate maps $z\mapsto z_{k}$ are measurable by claims 1 and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets (Step 3). We argue by induction on $n$. For $n=1$, $\gamma_{1}=\gamma_{(c_{1})}$. Let $n\ge2$ and identify $\mathbb{R}^{n}$ with $\mathbb{R}^{n-1}\times\mathbb{R}$, $z=(z',s)$, as in Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$; by claim 1 there, Lebesgue Measure on $\mathbb{R}^n$ and claim 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, $\mathcal{B}(\mathbb{R}^{n})=\mathcal{B}(\mathbb{R}^{n-1})\otimes\mathcal{B}(\mathbb{R})$ and $\lambda_{n}=\lambda_{n-1}\otimes\lambda_{1}$, both factors being $\sigma$-finite. Since $|z|^{2}_{c^{(n)}}=|z'|^{2}_{c^{(n-1)}}+s^{2}/c_{n}$ and the normalising sum of The Diagonal Gaussian Density on Euclidean Space and Its Notation §density for $c^{(n)}$ is the sum of those for $c^{(n-1)}$ and for $(c_{n})$, claim 1 of Basic Properties of the Exponential Function gives $\rho_{n}(z)=\rho_{n-1}(z')\rho_{(c_{n})}(s)$. Put $a(z')=\prod_{k<n}g_{k}(z'_{k})\rho_{n-1}(z')$ and $b(s)=g_{n}(s)\rho_{(c_{n})}(s)$, so that $\prod_{k}g_{k}(z_{k})\rho_{n}(z)=a(z')b(s)$. By the induction hypothesis, applied to $g_{1},\dots,g_{n-1}$ and to $|g_{1}|,\dots,|g_{n-1}|$, and the density rule, $a$ is $\lambda_{n-1}$-integrable with $\int a\,d\lambda_{n-1}=\prod_{k<n}\int g_{k}\,d\gamma_{(c_{k})}$; likewise $b$ is $\lambda_{1}$-integrable with $\int b\,d\lambda_{1}=\int g_{n}\,d\gamma_{(c_{n})}$. For each choice of signs, the function $(z',s)\mapsto a^{\pm}(z')b^{\pm}(s)$ is measurable for $\mathcal{B}(\mathbb{R}^{n-1})\otimes\mathcal{B}(\mathbb{R})$ (the projections being measurable by Product Sigma-Algebra), and Tonelli and Fubini Theorems with Linearity and Monotonicity of the Lebesgue Integral §nonnegative gives $\int a^{\pm}b^{\pm}\,d\lambda_{n}=\int a^{\pm}\,d\lambda_{n-1}\int b^{\pm}\,d\lambda_{1}<\infty$. As $|ab|=(a^{+}+a^{-})(b^{+}+b^{-})$ and $ab=a^{+}b^{+}-a^{+}b^{-}-a^{-}b^{+}+a^{-}b^{-}$, the function $ab$ is $\lambda_{n}$-integrable and, by Linearity and Monotonicity of the Lebesgue Integral §integrable and Integrable Function and the Lebesgue Integral, $\int ab\,d\lambda_{n}=\bigl(\int a^{+}-\int a^{-}\bigr)\bigl(\int b^{+}-\int b^{-}\bigr)=\int a\,d\lambda_{n-1}\int b\,d\lambda_{1}$. The density rule turns this into the claim.

Step 9 (The integrals $\varphi_{r}$). Let $\varphi$ be of polynomial growth with constants $(B_{\varphi},q)$ and let $r\ge0$. For $u,z\in\mathbb{R}^{n}$, Step 1 and Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §triangle give $\lVert\eta(r)\odot u+\xi(r)\odot z\rVert\le\lVert u\rVert+\lVert z\rVert$, so by (E3)

$$|\varphi(\eta(r)\odot u+\xi(r)\odot z)|\le B_{\varphi}\bigl(1+(\lVert u\rVert+\lVert z\rVert)^{q}\bigr)\le2^{q}B_{\varphi}(1+\lVert u\rVert^{q})(1+\lVert z\rVert^{q}).\qquad(9.1)$$

The function $z\mapsto\varphi(\eta(r)\odot u+\xi(r)\odot z)$ is continuous (Step 3), hence Borel, and by (9.1) and Step 2 it is $\gamma_{n}$-integrable; so $\varphi_{r}(u)$ is defined, and by Linearity and Monotonicity of the Lebesgue Integral §integrable and Step 2

$$|\varphi_{r}(u)|\le B'_{\varphi}(1+\lVert u\rVert^{q}),\qquad B'_{\varphi}=2^{q+1}B_{\varphi}K_{q}.\qquad(9.2)$$

$\varphi_{r}$ is continuous: let $(u_{m})$ converge to $u$ in $\mathbb{R}^{n}$; there is a real $R$ with $\lVert u_{m}\rVert\le R$ for every $m$ (take $R$ larger than $\lVert u\rVert+1$ and than the finitely many $\lVert u_{m}\rVert$ with $\lVert u_{m}-u\rVert>1$). For every $z$, the points $\eta(r)\odot u_{m}+\xi(r)\odot z$ converge to $\eta(r)\odot u+\xi(r)\odot z$, their distance being at most $\lVert u_{m}-u\rVert$ by Step 1, so the integrands converge by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential; by (9.1) they are dominated by the integrable function $z\mapsto2^{q}B_{\varphi}(1+R^{q})(1+\lVert z\rVert^{q})$. So $\varphi_{r}(u_{m})\to\varphi_{r}(u)$ by Dominated Convergence Theorem, and $\varphi_{r}$ is continuous at $u$ by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §criterion. Thus $\varphi_{r}$ is of polynomial growth with constants $(B'_{\varphi},q)$. Next, for fixed $u$:

$$r\mapsto\varphi_{r}(u)\ \text{is continuous on }[0,\infty).\qquad(9.3)$$

Indeed, if $(r_{m})$ is a sequence in $[0,\infty)$ converging to $r$, then by Step 6(iv) every entry of $\eta(r_{m})\odot u+\xi(r_{m})\odot z$ converges to the corresponding entry of $\eta(r)\odot u+\xi(r)\odot z$, so these points converge in $\mathbb{R}^{n}$ (by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square and claim 3(e) of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities), the integrands converge by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential, they are dominated by $2^{q}B_{\varphi}(1+\lVert u\rVert^{q})(1+\lVert z\rVert^{q})$ by (9.1), and Dominated Convergence Theorem and Continuity Between Metric Spaces is Equivalent to Sequential Continuity §criterion (with $A=[0,\infty)$) apply. Finally:

$$(\alpha_{1}\varphi+\alpha_{2}\chi)_{r}=\alpha_{1}\varphi_{r}+\alpha_{2}\chi_{r},\qquad\varphi_{r}=\varphi\ \text{if }\varphi\text{ is constant},\qquad\varphi_{0}=\varphi,\qquad(9.4)$$

for $\chi$ also of polynomial growth and real $\alpha_{1},\alpha_{2}$: the function $\alpha_{1}\varphi+\alpha_{2}\chi$ is continuous and, by (E4), of polynomial growth with constants $(2(|\alpha_{1}|B_{\varphi}+|\alpha_{2}|B_{\chi}),\max\{q,q_{\chi}\})$, and the first identity is Linearity and Monotonicity of the Lebesgue Integral §integrable; the second is claim 6 of Borel Measurability and Bounded Integration on a Metric Space, $\gamma_{n}$ being a probability measure; and the third holds because $\eta(0)=\mathbf{1}$ and $\xi(0)=0$ (Step 6(v)), so that the integrand of $\varphi_{0}(u)$ is the constant $\varphi(u)$.

Step 10 (Cylindrical reduction). Let $G\in\mathcal{F}C_{\mathrm{pol}}(X)$ have representation $(n,\varphi,B_{\varphi},q)$, let $r\ge0$ and $x\in X$, and put $u=p_{n}(x)$. Then $P_{r}G(x)=\varphi_{r}(u)$. Indeed, by The Mehler Maps and the Mehler Semigroup with Noise Weights on Continuous Cylindrical Functions of Polynomial Growth §map the $k$-th coordinate of $M_{r}(x,y)$ is $\eta_{k}(r)x_{k}+\xi_{k}(r)y_{k}$, so $p_{n}(M_{r}(x,y))=\eta(r)\odot u+\xi(r)\odot p_{n}(y)$ and $G(M_{r}(x,y))=g(p_{n}(y))$ for the Borel function $g(z)=\varphi(\eta(r)\odot u+\xi(r)\odot z)$, which is $\gamma_{n}$-integrable by Step 9. By the transfer rule of Step 2, the integral defining $P_{r}G(x)$ in The Mehler Maps and the Mehler Semigroup with Noise Weights on Continuous Cylindrical Functions of Polynomial Growth §semigroup is $\int_{X}g(p_{n}y)\,\gamma_{c}(dy)=\int g\,d\gamma_{n}=\varphi_{r}(u)$.

Step 11 (Smoothing of bounded $C^{1}$ functions). Let $\varphi\in C^{1}_{b}(\mathbb{R}^{n})$ (Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded), let $C_{0},C_{1}\ge0$ be real with $|\varphi|\le C_{0}$ and $|\partial_{k}\varphi|\le C_{1}$ for $k\in[n]$, and let $r\ge0$. Then $\varphi$ and each $\partial_{k}\varphi$ are of polynomial growth with constants $(C_{0},0)$ and $(C_{1},0)$, being continuous by clause 1 of C^k Maps on a Euclidean Open Set; and $\varphi_{r}\in C^{1}_{b}(\mathbb{R}^{n})$, with $|\varphi_{r}|\le C_{0}$ and, for $k\in[n]$,

$$\partial_{k}(\varphi_{r})=\eta_{k}(r)\,(\partial_{k}\varphi)_{r},\qquad|\partial_{k}(\varphi_{r})|\le C_{1}.$$

Proof. Let $G(v)=\varphi(\eta(r)\odot v)$; by Step 4, $G$ is of class $C^{1}$ with $\partial_{k}G(v)=\eta_{k}(r)\partial_{k}\varphi(\eta(r)\odot v)$, so $|G|\le C$ and $|\partial_{k}G|\le C$ with $C=\max\{C_{0},C_{1}\}$. Let $T(z)=-\eta(r)^{-1}\odot\xi(r)\odot z$, a Borel map (Step 3), and $\mu=T_{\#}\gamma_{n}$, a probability measure on $\mathcal{B}(\mathbb{R}^{n})$ by claim 1 of Image Measures, Measures with Densities, and Change of Variables. For $u\in\mathbb{R}^{n}$ and a bounded Borel $H:\mathbb{R}^{n}\to\mathbb{R}$, claim 2 of Image Measures, Measures with Densities, and Change of Variables and claim 6 of Borel Measurability and Bounded Integration on a Metric Space give $\int H(u-x)\,\mu(dx)=\int H(u-T(z))\,\gamma_{n}(dz)$, and $\eta(r)\odot(u-T(z))=\eta(r)\odot u+\xi(r)\odot z$. Hence, in the notation of The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral, $(G*\mu)(u)=\varphi_{r}(u)$ and $((\partial_{k}G)*\mu)(u)=\eta_{k}(r)(\partial_{k}\varphi)_{r}(u)$. By The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral §derivative, $\varphi_{r}=G*\mu$ is of class $C^{1}$ with $\partial_{k}(\varphi_{r})=(\partial_{k}G)*\mu=\eta_{k}(r)(\partial_{k}\varphi)_{r}$. By The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral §continuous, applied to $G$ with the bound $C_{0}$ and to $v\mapsto\partial_{k}\varphi(\eta(r)\odot v)$ with the bound $C_{1}$, $|\varphi_{r}|\le C_{0}$ and $|(\partial_{k}\varphi)_{r}|\le C_{1}$; and $0<\eta_{k}(r)\le1$.

Claim 1. Common length. Let $(n,\psi,B,q)$ represent $F$ and let $N\ge n$ be a natural number. If $N>n$, let $\pi:\mathbb{R}^{N}\to\mathbb{R}^{n}$ be the coordinate projection $\mathrm{pr}^{n,N-n}_{1}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, which sends $y$ to $(y_{1},\dots,y_{n})$ and satisfies $\lVert\pi(y)\rVert\le\lVert y\rVert$; it is affine, hence continuous (Step 3). Then $p_{n}=\pi\circ p_{N}$, so $F=(\psi\circ\pi)\circ p_{N}$, and $\psi\circ\pi$ is continuous with $|\psi(\pi(y))|\le B(1+\lVert y\rVert^{q})$; thus $(N,\psi\circ\pi,B,q)$ is a representation of $F$. Consequently any two elements of $\mathcal{F}C_{\mathrm{pol}}(X)$ have representations with the same first entry, the larger of two given ones.

Algebraic closure. Let $(n,\psi,B,q)$ and $(n,\chi,B_{G},q_{G})$ represent $F$ and $G$, let $\alpha_{1}\in\mathbb{R}$, and put $Q=\max\{q,q_{G}\}$. Sums, products and real multiples of continuous functions are continuous (Step 3), and by (E4), $|\psi+\chi|\le2(B+B_{G})(1+\lVert y\rVert^{Q})$, $|\psi\chi|\le4BB_{G}(1+\lVert y\rVert^{q+q_{G}})$ and $|\alpha_{1}\psi|\le|\alpha_{1}|B(1+\lVert y\rVert^{q})$. So $F+G=(\psi+\chi)\circ p_{n}$, $FG=(\psi\chi)\circ p_{n}$ and $\alpha_{1}F=(\alpha_{1}\psi)\circ p_{n}$ belong to $\mathcal{F}C_{\mathrm{pol}}(X)$. The constant function with value $\alpha_{1}$ is $\varphi\circ p_{1}$ with $\varphi$ the constant function $\alpha_{1}$ on $\mathbb{R}^{1}$, so $(1,\varphi,|\alpha_{1}|,0)$ represents it.

Cylindrical classes. If $\varphi\in\mathcal{F}C^{1}_{b}(X)$ has representation $(n,\psi)$, then $\psi\in C^{1}_{b}(\mathbb{R}^{n})$ is continuous by clause 1 of C^k Maps on a Euclidean Open Set and bounded by some real $B\ge0$ by Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded, so $(n,\psi,B,0)$ represents $\varphi$. Every element of $\mathcal{F}C^{2}_{b}(X)$ belongs to $\mathcal{F}C^{1}_{b}(X)$ by The Ornstein-Uhlenbeck Operator with Noise Weights on Bounded C^2 Cylindrical Functions §second-derivatives. For $\alpha\in\mathcal{A}$ with length bound $n$, Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §cylindrical gives $H_{\alpha}=h_{\alpha}\circ p_{n}$ with $h_{\alpha}(y)=\prod_{k=1}^{n}H^{c_{k}}_{\alpha_{k}}(y_{k})$ of class $C^{2}$, hence continuous, and $|h_{\alpha}(y)|\le M(1+\lVert y\rVert^{|\alpha|})$, where $M\ge0$ because $0\le|h_{\alpha}(0)|\le M(1+\lVert0\rVert^{|\alpha|})$ and $1+\lVert0\rVert^{|\alpha|}>0$; so $(n,h_{\alpha},M,|\alpha|)$ represents $H_{\alpha}$.

Continuity and integrability. With $(n,\psi,B,q)$ representing $F$, $F=\psi\circ p_{n}$ is a composite of continuous maps (Step 2), hence continuous, and Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space. Let $p\ge1$ be real and $m\in\mathbb{N}$ with $p\le m$. Since $\lVert p_{n}x\rVert\le|x|$, (E2) gives $|F(x)|\le2B(1+|x|^{2})^{q}$; so by Properties of Real Powers of Nonnegative Real Numbers §monotone, Properties of Real Powers of Nonnegative Real Numbers §product, Properties of Real Powers of Nonnegative Real Numbers §agreement and (E5) applied to $(1+|x|^{2})^{q}\ge1$,

$$|F(x)|^{p}\le(2B)^{p}\bigl((1+|x|^{2})^{q}\bigr)^{p}\le2(2B)^{p}(1+|x|^{2})^{qm}.$$

The function $|F|^{p}$ is measurable by Power-Integrable Functions and the p-Seminorm §measurable-power, and Step 2 with Linearity and Monotonicity of the Lebesgue Integral §nonnegative gives $\int_{X}|F|^{p}\,d\gamma_{c}\le2(2B)^{p}K_{qm}<\infty$; so $F\in L^{p}(\gamma_{c})$ by Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §lebesgue. The same argument shows that every Borel $g:\mathbb{R}^{n}\to\mathbb{R}$ with $|g(z)|\le C(1+\lVert z\rVert^{q})$ satisfies $\int|g|^{p}\,d\gamma_{n}\le2(2C)^{p}K_{qm}$, by Step 2.

Claim 2. Let $(n,\psi,B,q)$ represent $F$. By Step 9 (with $\varphi=\psi$) the integrand is integrable, $\psi_{t}$ is continuous, and $|\psi_{r}(u)|\le B'(1+\lVert u\rVert^{q})$ for every real $r\ge0$ and $u\in\mathbb{R}^{n}$, with $B'=2^{q+1}BK_{q}$, which depends only on $B$, $q$ and $c$. By Step 10, $P_{t}F=\psi_{t}\circ p_{n}$; so $(n,\psi_{t},B',q)$ is a representation of $P_{t}F\in\mathcal{F}C_{\mathrm{pol}}(X)$. By (9.4), $P_{0}F=\psi_{0}\circ p_{n}=\psi\circ p_{n}=F$. For $F,G\in\mathcal{F}C_{\mathrm{pol}}(X)$ and real $\alpha_{1},\alpha_{2}$, $\alpha_{1}F+\alpha_{2}G\in\mathcal{F}C_{\mathrm{pol}}(X)$ by claim 1, and for $x\in X$ the functions $y\mapsto F(M_{t}(x,y))$ and $y\mapsto G(M_{t}(x,y))$ are $\gamma_{c}$-integrable by The Mehler Maps and the Mehler Semigroup with Noise Weights on Continuous Cylindrical Functions of Polynomial Growth §semigroup; so $P_{t}(\alpha_{1}F+\alpha_{2}G)(x)=\alpha_{1}P_{t}F(x)+\alpha_{2}P_{t}G(x)$ by Linearity and Monotonicity of the Lebesgue Integral §integrable. For the constant function with value $\alpha_{1}$, its image under $P_{t}$ has the value $\int_{X}\alpha_{1}\,\gamma_{c}(dy)=\alpha_{1}$ at every $x$, by claim 6 of Borel Measurability and Bounded Integration on a Metric Space, $\gamma_{c}$ being a probability measure.

Claim 3. If $s=0$ or $t=0$, the claim follows from $P_{0}G=G$ for every $G\in\mathcal{F}C_{\mathrm{pol}}(X)$ (claim 2). Let $s,t>0$ and let $(n,\psi,B,q)$ represent $F$. By claim 2, $(n,\psi_{t},B',q)$ represents $P_{t}F$, so $P_{s}(P_{t}F)=(\psi_{t})_{s}\circ p_{n}$ by Step 10, and it suffices to show $(\psi_{t})_{s}(u)=\psi_{s+t}(u)$ for $u\in\mathbb{R}^{n}$. Put $\sigma=\xi(s+t)$, whose entries are positive by Step 6(v), $\mathbf{a}=\sigma^{-1}\odot\eta(t)\odot\xi(s)$ and $\mathbf{b}=\sigma^{-1}\odot\xi(t)$; then $\mathbf{b}_{k}>0$ by Step 6(v) and $\mathbf{a}_{k}^{2}+\mathbf{b}_{k}^{2}=1$ by Step 6(vi). Since $\eta(t)\odot\eta(s)=\eta(s+t)$ by Step 6(vi), for all $y,z\in\mathbb{R}^{n}$

$$\eta(t)\odot\bigl(\eta(s)\odot u+\xi(s)\odot y\bigr)+\xi(t)\odot z=\eta(s+t)\odot u+\sigma\odot(\mathbf{a}\odot y+\mathbf{b}\odot z).$$

Let $\phi(w)=\psi(\eta(s+t)\odot u+\sigma\odot w)$ and $f(y,z)=\phi(y)$; by (9.1), $|f(y,z)|\le2^{q}B(1+\lVert u\rVert^{q})(1+\lVert y\rVert^{q})(1+\lVert z\rVert^{q})$. With $S$ the map of Step 7 for these $\mathbf{a},\mathbf{b}$, $f(S(y,z))=\phi(\mathbf{a}\odot y+\mathbf{b}\odot z)$, so by the displayed identity, Step 7(b) and claim 6 of Borel Measurability and Bounded Integration on a Metric Space,

$$(\psi_{t})_{s}(u)=\int\Bigl(\int f(S(y,z))\,\gamma_{n}(dz)\Bigr)\gamma_{n}(dy)=\int\Bigl(\int\phi(y)\,\gamma_{n}(dz)\Bigr)\gamma_{n}(dy)=\int\phi\,d\gamma_{n}=\psi_{s+t}(u).$$

Claim 4. Let $(n,\psi,B,q)$ represent $F$. By claims 1 and 2 and the transfer rule, $\int_{X}P_{t}F\,d\gamma_{c}=\int\psi_{t}\,d\gamma_{n}$ and $\int_{X}F\,d\gamma_{c}=\int\psi\,d\gamma_{n}$, all these integrals existing. For $t=0$ there is nothing to prove. For $t>0$, apply Step 7(b) with $\mathbf{a}=\eta(t)$ and $\mathbf{b}=\xi(t)$ (admissible by Step 6(v) and The Mehler Maps and the Mehler Semigroup with Noise Weights on Continuous Cylindrical Functions of Polynomial Growth §map) to $f(y,z)=\psi(y)$, for which $|f(y,z)|\le B(1+\lVert y\rVert^{q})(1+\lVert z\rVert^{q})$ and $f(S(y,z))=\psi(\eta(t)\odot y+\xi(t)\odot z)$:

$$\int\psi_{t}\,d\gamma_{n}=\int\Bigl(\int f(S(y,z))\,\gamma_{n}(dz)\Bigr)\gamma_{n}(dy)=\int\Bigl(\int\psi(y)\,\gamma_{n}(dz)\Bigr)\gamma_{n}(dy)=\int\psi\,d\gamma_{n},$$

the last step by claim 6 of Borel Measurability and Bounded Integration on a Metric Space.

Claim 5. For $x\in X$, $P_{t}F(x)$ is the integral of the $\gamma_{c}$-integrable function $y\mapsto F(M_{t}(x,y))$ (The Mehler Maps and the Mehler Semigroup with Noise Weights on Continuous Cylindrical Functions of Polynomial Growth §semigroup), which is nonnegative; so $P_{t}F(x)\ge\int_{X}0\,d\gamma_{c}=0$ by Linearity and Monotonicity of the Lebesgue Integral §integrable.

Claim 6. Let $p\ge1$ be real. For $t=0$ there is nothing to prove; let $t>0$ and let $(n,\psi,B,q)$ represent $F$. Fix $u\in\mathbb{R}^{n}$ and let $g(z)=\psi(\eta(t)\odot u+\xi(t)\odot z)$, which by (9.1) and the last sentence of claim 1 is $\gamma_{n}$-integrable and satisfies $\int|g|^{p}\,d\gamma_{n}<\infty$. We show

$$|\psi_{t}(u)|^{p}\le\int|g|^{p}\,d\gamma_{n}.\qquad(6.1)$$

For $p=1$ this is Linearity and Monotonicity of the Lebesgue Integral §integrable. For $p>1$ let $p'$ be its conjugate exponent; the constant function $1$ has $p'$-seminorm $(\int1^{p'}d\gamma_{n})^{1/p'}=1$ (Step 1 and claim 6 of Borel Measurability and Bounded Integration on a Metric Space), so Linearity and Monotonicity of the Lebesgue Integral §integrable and Hoelder's Inequality, for Two and for Finitely Many Factors §holder give $|\psi_{t}(u)|\le\int|g|\,d\gamma_{n}\le\lVert g\rVert_{p}$, with the seminorm of Power-Integrable Functions and the p-Seminorm §seminorm, and (6.1) follows by Properties of Real Powers of Nonnegative Real Numbers §monotone and Properties of Real Powers of Nonnegative Real Numbers §inverse. Now apply Step 7(a) with $\mathbf{a}=\eta(t)$, $\mathbf{b}=\xi(t)$ to the nonnegative measurable function $f(y,z)=|\psi(y)|^{p}$ (Power-Integrable Functions and the p-Seminorm §measurable-power), for which $f(S(y,z))=|\psi(\eta(t)\odot y+\xi(t)\odot z)|^{p}$. By claim 2, the transfer rule, (6.1), Linearity and Monotonicity of the Lebesgue Integral §nonnegative, Step 7(a) and claim 6 of Borel Measurability and Bounded Integration on a Metric Space,

$$\int_{X}|P_{t}F|^{p}\,d\gamma_{c}=\int|\psi_{t}|^{p}\,d\gamma_{n}\le\int\Bigl(\int|\psi(\eta(t)\odot y+\xi(t)\odot z)|^{p}\,\gamma_{n}(dz)\Bigr)\gamma_{n}(dy)=\int|\psi|^{p}\,d\gamma_{n}=\int_{X}|F|^{p}\,d\gamma_{c}.$$

Taking powers with exponent $1/p$ (Properties of Real Powers of Nonnegative Real Numbers §monotone) gives $\lVert P_{t}F\rVert_{p}\le\lVert F\rVert_{p}$, by The Lebesgue Space of Power-Integrable Functions §norm and Power-Integrable Functions and the p-Seminorm §seminorm.

Claim 7. By claims 1 and 2, $G\,P_{t}F$ and $F\,P_{t}G$ belong to $\mathcal{F}C_{\mathrm{pol}}(X)$, hence are $\gamma_{c}$-integrable. If $t=0$ both equal $FG$. Let $t>0$. By claim 1 choose representations $(n,\psi,B,q)$ of $F$ and $(n,\chi,B_{G},q_{G})$ of $G$ with the same $n$. By claim 2 and the transfer rule, $\int_{X}G\,P_{t}F\,d\gamma_{c}=\int\chi\,\psi_{t}\,d\gamma_{n}$ and $\int_{X}F\,P_{t}G\,d\gamma_{c}=\int\psi\,\chi_{t}\,d\gamma_{n}$. Let $f(y,z)=\chi(y)\,\psi(\eta(t)\odot y+\xi(t)\odot z)$; by (9.1) and (E4), $|f(y,z)|\le K(1+\lVert y\rVert^{m})(1+\lVert z\rVert^{m})$ with $m=q+q_{G}$ and $K=2^{q+3}BB_{G}$. Let $S$ be the map of Step 7 for $\mathbf{a}=\eta(t)$, $\mathbf{b}=\xi(t)$. Since $\eta_{k}(t)^{2}+\xi_{k}(t)^{2}=1$,

$$\eta(t)\odot\bigl(\eta(t)\odot y+\xi(t)\odot z\bigr)+\xi(t)\odot\bigl(\xi(t)\odot y-\eta(t)\odot z\bigr)=y,$$

so $f(S(y,z))=\chi(\eta(t)\odot y+\xi(t)\odot z)\,\psi(y)$. Taking the factors $\psi(y)$, respectively $\chi(y)$, out of the inner integrals (Linearity and Monotonicity of the Lebesgue Integral §integrable), Step 7(b) gives

$$\int\psi\,\chi_{t}\,d\gamma_{n}=\int\Bigl(\int f(S(y,z))\,\gamma_{n}(dz)\Bigr)\gamma_{n}(dy)=\int\Bigl(\int f(y,z)\,\gamma_{n}(dz)\Bigr)\gamma_{n}(dy)=\int\chi\,\psi_{t}\,d\gamma_{n}.$$

Claim 8. Let $\alpha\in\mathcal{A}$ have length bound $n$ (Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order §multi-indices). By claim 1, $(n,h_{\alpha},M,|\alpha|)$ represents $H_{\alpha}$, where $h_{\alpha}(y)=\prod_{k=1}^{n}H^{c_{k}}_{\alpha_{k}}(y_{k})$, so $P_{t}H_{\alpha}=(h_{\alpha})_{t}\circ p_{n}$ by claim 2. Fix $u\in\mathbb{R}^{n}$ and put $g_{k}(\tau)=H^{c_{k}}_{\alpha_{k}}(\eta_{k}(t)u_{k}+\xi_{k}(t)\tau)$ for $\tau\in\mathbb{R}$ and $k\in[n]$. Since $c_{k}>0$ and $\eta_{k}(t)^{2}+\xi_{k}(t)^{2}=1$, Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §mehler, applied with $v=c_{k}$, $r=\eta_{k}(t)$, $s=\xi_{k}(t)$ and $u_{k}$ in place of the number written $t$ there, shows that $g_{k}$ is Borel and $\gamma_{(c_{k})}$-integrable (so $\int|g_{k}|\,d\gamma_{(c_{k})}<\infty$ by Integrable Function and the Lebesgue Integral) with $\int g_{k}\,d\gamma_{(c_{k})}=\eta_{k}(t)^{\alpha_{k}}H^{c_{k}}_{\alpha_{k}}(u_{k})$. The integrand of $(h_{\alpha})_{t}(u)$ is $z\mapsto\prod_{k}g_{k}(z_{k})$, so Step 8 gives

$$(h_{\alpha})_{t}(u)=\prod_{k=1}^{n}\eta_{k}(t)^{\alpha_{k}}H^{c_{k}}_{\alpha_{k}}(u_{k})=\exp\Bigl(-t\sum_{k=1}^{n}\alpha_{k}\theta_{k}\Bigr)h_{\alpha}(u)=\exp(-t\,\theta\cdot\alpha)\,h_{\alpha}(u),$$

using $\eta_{k}(t)^{\alpha_{k}}=\exp(-\alpha_{k}\theta_{k}t)$ and $\prod_{k}\exp(b_{k})=\exp(\sum_{k}b_{k})$, both from claim 1 of Basic Properties of the Exponential Function (with $\exp(b)^{0}=1=\exp(0)$), and Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order §order with the sequence $\theta$. Hence $P_{t}H_{\alpha}=\exp(-t\,\theta\cdot\alpha)\,h_{\alpha}\circ p_{n}=\exp(-t\,\theta\cdot\alpha)\,H_{\alpha}$.

Claim 9. Let $F\in\mathcal{F}C^{1}_{b}(X)$ have representation $(n,\psi)$, $\psi\in C^{1}_{b}(\mathbb{R}^{n})$, and let $C_{0},C_{1}$ bound $|\psi|$ and the $|\partial_{k}\psi|$, $k\in[n]$. By claim 1, $(n,\psi,C_{0},0)$ represents $F$, so $P_{t}F=\psi_{t}\circ p_{n}$ by claim 2, and $\psi_{t}\in C^{1}_{b}(\mathbb{R}^{n})$ by Step 11; hence $P_{t}F\in\mathcal{F}C^{1}_{b}(X)$ by Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical, with representation $(n,\psi_{t})$. Let $k\in\mathbb{N}$. If $k\le n$, Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial gives $\partial_{k}F=(\partial_{k}\psi)\circ p_{n}$, where $\partial_{k}\psi$ is of polynomial growth with constants $(C_{1},0)$ (Step 11); so $\partial_{k}F\in\mathcal{F}C_{\mathrm{pol}}(X)$ and $P_{t}(\partial_{k}F)=(\partial_{k}\psi)_{t}\circ p_{n}$ by claim 2. Applying Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial to $P_{t}F$ with representation $(n,\psi_{t})$, and Step 11,

$$\partial_{k}(P_{t}F)=(\partial_{k}\psi_{t})\circ p_{n}=\eta_{k}(t)\,(\partial_{k}\psi)_{t}\circ p_{n}=\exp(-\theta_{k}t)\,P_{t}(\partial_{k}F).$$

If $k>n$, the same lemma gives $\partial_{k}F=0$ and $\partial_{k}(P_{t}F)=0$; the constant $0$ lies in $\mathcal{F}C_{\mathrm{pol}}(X)$ and $P_{t}0=0$ by claims 1 and 2, so both sides of the identity vanish.

Claim 10. Let $t>0$, let $F$ be bounded, and let $(n,\psi,B,q)$ represent $F$. For $u\in\mathbb{R}^{n}$ the point $p_{n}^{*}(u)=\sum_{k=1}^{n}u_{k}e_{k}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates satisfies $p_{n}(p_{n}^{*}(u))=u$, by linearity of the inner product and orthonormality of $(e_{k})$ (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space); so $\psi(u)=F(p_{n}^{*}(u))$, and $|\psi|\le M_{0}$ for some real $M_{0}\ge0$.

Standardisation. For $0<s\le1$ let $g_{s}$ be the Gaussian weight of The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails, read with $n$ in place of the dimension written $q$ there, so $g_{1}(w)=\varkappa\exp(-\lVert w\rVert^{2}/2)$, where $\varkappa$ is the unique positive real number with $\int g_{1}\,dw=1$ (written $c_{1}$ there). Let $\varsigma=(\sqrt{c_{1}},\dots,\sqrt{c_{n}})$ and $\beta_{0}=\prod_{k}\sqrt{c_{k}}$. Since $(\sqrt{c_{k}}\,w_{k})^{2}/c_{k}=w_{k}^{2}$, The Diagonal Gaussian Density on Euclidean Space and Its Notation §density, Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square and claim 1 of Basic Properties of the Exponential Function give $\beta_{0}\rho_{n}(\varsigma\odot w)=C\exp(-\lVert w\rVert^{2}/2)$ for a real $C>0$ independent of $w$. By the density rule and Step 5 (with $\mathbf{a}=0$, $\mathbf{b}=\varsigma$), for every Borel $\phi:\mathbb{R}^{n}\to[0,\infty)$,

$$\int\phi\,d\gamma_{n}=\int\phi\rho_{n}\,dz=\int\phi(\varsigma\odot w)\,C\exp(-\lVert w\rVert^{2}/2)\,dw .$$

With $\phi=1$ the left side is $1$, so $C=\varkappa$ by the uniqueness just quoted; hence $\int\phi\,d\gamma_{n}=\int\phi(\varsigma\odot w)g_{1}(w)\,dw$, and, splitting $\phi=\phi^{+}-\phi^{-}$ (Integrable Function and the Lebesgue Integral, Linearity and Monotonicity of the Lebesgue Integral §integrable), the same holds for every $\gamma_{n}$-integrable Borel $\phi$, the right-hand integrand being then $\lambda_{n}$-integrable. (10.1)

Reduction to a Gaussian smoothing. Let $D=\xi(t)\odot\varsigma$, whose entries are positive by Step 6(v), $E=D^{-1}\odot\eta(t)$, and $H(v)=\psi(D\odot v)$, a continuous, hence Borel, function with $|H|\le M_{0}$. For $u,w\in\mathbb{R}^{n}$, $\eta(t)\odot u+\xi(t)\odot\varsigma\odot w=D\odot(E\odot u+w)$, so by (10.1)

$$\psi_{t}(u)=\int H(E\odot u+w)\,g_{1}(w)\,dw=\int H(E\odot u-x)\,g_{1}(x)\,dx=H_{1}(E\odot u),$$

where the second equality is claim 3 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$ with $a=0$, applied to the integrable function $w\mapsto H(E\odot u+w)g_{1}(w)$, together with $g_{1}(-x)=g_{1}(x)$ (The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §derivatives), and $H_{1}$ is the smoothing of $H$ of Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence §smoothing with $s=1$.

$H_{1}\in C^{2}_{b}(\mathbb{R}^{n})$. Fix $y\in\mathbb{R}^{n}$. By The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §convolution with $s=\tfrac12$ and the evenness of $g_{1/2}$ and $g_{1}$, $g_{1}(y-z)=\int g_{1/2}(w-y)\,g_{1/2}(w-z)\,dw$ for every $z$. The functions $H^{+},H^{-}$ are Borel with values in $[0,M_{0}]$. Applying Tonelli and Fubini Theorems for $\lambda_{n}\otimes\lambda_{n}$ to the nonnegative measurable function $(z,w)\mapsto g_{1/2}(w-y)\,g_{1/2}(w-z)\,H^{\pm}(z)$, with Linearity and Monotonicity of the Lebesgue Integral §nonnegative, and then Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence §smoothing (second form) with $s=\tfrac12$,

$$\int g_{1}(y-z)\,H^{\pm}(z)\,dz=\int g_{1/2}(w-y)\Bigl(\int g_{1/2}(w-z)\,H^{\pm}(z)\,dz\Bigr)dw=\int g_{1/2}(y-w)\,(H^{\pm})_{1/2}(w)\,dw,$$

all these quantities being finite by the same claim (the functions $(H^{\pm})_{1/2}$ are bounded by $M_{0}$ and Borel, being continuous by Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence §kernel-derivative). Subtracting the two identities and using Linearity and Monotonicity of the Lebesgue Integral §integrable (so that $(H^{+})_{1/2}-(H^{-})_{1/2}=H_{1/2}$) gives $H_{1}(y)=\int g_{1/2}(y-w)\,H_{1/2}(w)\,dw=(H_{1/2})_{1/2}(y)$, the smoothing with $s=\tfrac12$ of the Borel function $H_{1/2}$, which is bounded by $M_{0}$. By Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence §kernel-derivative, $H_{1/2}$ is of class $C^{1}$ with $|\partial_{i}H_{1/2}|\le M_{0}\sqrt{n}\sqrt{2}$, so $H_{1/2}\in C^{1}_{b}(\mathbb{R}^{n})$. By Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence §function-derivative, applied with $H_{1/2}$ in place of $H$, $\partial_{i}H_{1}=(\partial_{i}H_{1/2})_{1/2}$, the smoothing of the bounded Borel function $\partial_{i}H_{1/2}$; by Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence §smoothing and Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence §kernel-derivative, applied with $\partial_{i}H_{1/2}$ in place of $H$, this function is bounded and of class $C^{1}$ with bounded partial derivatives. Since $H_{1}$ is of class $C^{1}$ and bounded (same claims with $s=1$), clause 2 of C^k Maps on a Euclidean Open Set shows that $H_{1}$ is of class $C^{2}$, and $H_{1}\in C^{2}_{b}(\mathbb{R}^{n})$ by Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded.

Conclusion. Since $\psi_{t}(u)=H_{1}(E\odot u)$, Step 4 (with $\mathbf{a}=0$, $\mathbf{b}=E$) shows that $\psi_{t}$ is of class $C^{2}$ with $\partial_{k}\psi_{t}(u)=E_{k}\partial_{k}H_{1}(E\odot u)$ and $\partial_{l}\partial_{k}\psi_{t}(u)=E_{l}E_{k}\partial_{l}\partial_{k}H_{1}(E\odot u)$, all bounded; so $\psi_{t}\in C^{2}_{b}(\mathbb{R}^{n})$, and $P_{t}F=\psi_{t}\circ p_{n}\in\mathcal{F}C^{2}_{b}(X)$ by claim 2 and Bounded C^2 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical.

Claim 11. Let $(n,\psi,B,q)$ represent $F$, let $x\in X$ and $u=p_{n}(x)$. By claim 2, $P_{r}F(x)=\psi_{r}(u)$ for every real $r\ge0$, and $r\mapsto\psi_{r}(u)$ is continuous on $[0,\infty)$ by (9.3); this is the first assertion. Let $p\ge1$ be real, $m\in\mathbb{N}$ with $p\le m$, and let $(r_{m'})_{m'\in\mathbb{N}}$ be a sequence in $[0,\infty)$ converging to $t$. The functions $f_{m'}=|P_{r_{m'}}F-P_{t}F|^{p}$ on $X$ are measurable (Power-Integrable Functions and the p-Seminorm §measurable-power) and converge to $0$ at every point, by the first assertion and Properties of Real Powers of Nonnegative Real Numbers §continuity. By claim 2, $|P_{r_{m'}}F(x')-P_{t}F(x')|\le2B'(1+\lVert p_{n}x'\rVert^{q})\le4B'(1+|x'|^{2})^{q}$ for $x'\in X$, using (E2); so, as in claim 1, $f_{m'}(x')\le2(4B')^{p}(1+|x'|^{2})^{qm}$, which is $\gamma_{c}$-integrable by Step 2. By Dominated Convergence Theorem, $\int_{X}f_{m'}\,d\gamma_{c}\to0$. Now suppose that $\lVert P_{r}F-P_{t}F\rVert_{p}$ did not tend to $0$ as $r\to t$ over $r\ge0$, the difference belonging to $L^{p}(\gamma_{c})$ by claims 1 and 2. Then there would be $\varepsilon>0$ and, for each $m'\in\mathbb{N}$, a real $r_{m'}\ge0$ with $|r_{m'}-t|<1/m'$ and $\lVert P_{r_{m'}}F-P_{t}F\rVert_{p}\ge\varepsilon$, that is, $\int_{X}f_{m'}\,d\gamma_{c}\ge\varepsilon^{p}$ by Power-Integrable Functions and the p-Seminorm §seminorm, Properties of Real Powers of Nonnegative Real Numbers §monotone and Properties of Real Powers of Nonnegative Real Numbers §inverse; since $r_{m'}\to t$, this contradicts what was just shown.

Claim 12. Let $F\in\mathcal{F}C^{2}_{b}(X)$, $F=\psi\circ p_{n}$ with $\psi\in C^{2}_{b}(\mathbb{R}^{n})$ (Bounded C^2 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical), and let $M_{0},M_{1},M_{2}\ge0$ be real bounds of $|\psi|$, of the $|\partial_{k}\psi|$ and of the $|\partial_{l}\partial_{k}\psi|$ ($k,l\in[n]$). By clause 2 of C^k Maps on a Euclidean Open Set, each $\partial_{k}\psi$ is of class $C^{1}$, and it is bounded with bounded partial derivatives, so $\partial_{k}\psi\in C^{1}_{b}(\mathbb{R}^{n})$; and $(n,\psi,M_{0},0)$ represents $F$ (claim 1).

(a) $P_{t}F\in\mathcal{F}C^{2}_{b}(X)$. By Step 11 applied to $\psi$ and to each $\partial_{k}\psi$, $\psi_{t}$ is of class $C^{1}$ with $\partial_{k}\psi_{t}=\eta_{k}(t)(\partial_{k}\psi)_{t}$, and $(\partial_{k}\psi)_{t}$ is of class $C^{1}$ with $\partial_{l}(\partial_{k}\psi)_{t}=\eta_{l}(t)(\partial_{l}\partial_{k}\psi)_{t}$, these functions being bounded by $M_{0}$, $M_{1}$ and $M_{1}$ (Step 11), and $|(\partial_{l}\partial_{k}\psi)_{t}|\le M_{2}$ by Linearity and Monotonicity of the Lebesgue Integral §integrable. Hence each $\partial_{k}\psi_{t}$ is of class $C^{1}$, $\psi_{t}$ is of class $C^{2}$ by clause 2 of C^k Maps on a Euclidean Open Set, with

$$\partial_{l}\partial_{k}\psi_{t}=\eta_{l}(t)\eta_{k}(t)\,(\partial_{l}\partial_{k}\psi)_{t},\qquad(12.1)$$

and $\psi_{t}\in C^{2}_{b}(\mathbb{R}^{n})$. Since $P_{t}F=\psi_{t}\circ p_{n}$ (claim 2), $P_{t}F\in\mathcal{F}C^{2}_{b}(X)$.

(b) The operator in cylindrical form. For $\varphi\in C^{2}_{b}(\mathbb{R}^{n})$ put $\Lambda\varphi(v)=\sum_{k=1}^{n}a_{k}\bigl(\partial_{k}\partial_{k}\varphi(v)-c_{k}^{-1}v_{k}\,\partial_{k}\varphi(v)\bigr)$. Then $L^{a}(\varphi\circ p_{n})=(\Lambda\varphi)\circ p_{n}$: for $k\le n$, Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial gives $\partial_{k}(\varphi\circ p_{n})=(\partial_{k}\varphi)\circ p_{n}$, which belongs to $\mathcal{F}C^{1}_{b}(X)$ with representation $(n,\partial_{k}\varphi)$ by The Ornstein-Uhlenbeck Operator with Noise Weights on Bounded C^2 Cylindrical Functions §second-derivatives, so the same lemma gives $\partial_{k}\partial_{k}(\varphi\circ p_{n})=(\partial_{k}\partial_{k}\varphi)\circ p_{n}$; since $x_{k}$ is the $k$-th entry of $p_{n}(x)$, The Ornstein-Uhlenbeck Operator with Noise Weights on Bounded C^2 Cylindrical Functions §operator (with the pair $(n,\varphi)$) gives the formula. The function $\Lambda\psi$ is continuous, and by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate, $|\Lambda\psi(v)|\le C_{\Lambda}(1+\lVert v\rVert)$ with $C_{\Lambda}=\sum_{k}a_{k}(M_{2}+c_{k}^{-1}M_{1})$. So $(n,\Lambda\psi,C_{\Lambda},1)$ represents $L^{a}F=(\Lambda\psi)\circ p_{n}$; thus $L^{a}F\in\mathcal{F}C_{\mathrm{pol}}(X)$, $P_{t}(L^{a}F)=(\Lambda\psi)_{t}\circ p_{n}$ by claim 2, and $L^{a}(P_{t}F)=(\Lambda\psi_{t})\circ p_{n}$ by (a).

(c) Commutation. Fix $u\in\mathbb{R}^{n}$, write $\eta_{k}=\eta_{k}(t)$, $\xi_{k}=\xi_{k}(t)$ and $w(z)=\eta(t)\odot u+\xi(t)\odot z$. For $k\in[n]$ let $f_{k}(z)=\partial_{k}\psi(w(z))$; by Step 4, $f_{k}$ is of class $C^{1}$ with $\partial_{i}f_{k}(z)=\xi_{i}\,\partial_{i}\partial_{k}\psi(w(z))$, so $|\partial_{i}f_{k}(z)|\le M_{2}\le M_{2}(1+\lVert z\rVert)$. Hence Gaussian Integration by Parts for Functions of Finitely Many Coordinates Relative to a Diagonal Gaussian Measure on a Hilbert Space §coordinate, applied with $f_{k}$ in place of $\psi$ and $i=k$, together with the transfer rule of Step 2 (the $k$-th entry of $p_{n}(x)$ being $x_{k}$), gives

$$\int z_{k}\,f_{k}(z)\,\gamma_{n}(dz)=c_{k}\int\partial_{k}f_{k}\,d\gamma_{n}=c_{k}\,\xi_{k}\,(\partial_{k}\partial_{k}\psi)_{t}(u).$$

The $k$-th entry of $w(z)$ is $\eta_{k}u_{k}+\xi_{k}z_{k}$, and the functions $z\mapsto\partial_{k}\partial_{k}\psi(w(z))$, $f_{k}$ and $z\mapsto z_{k}f_{k}(z)$ are $\gamma_{n}$-integrable (the first two being bounded, the third bounded by $M_{1}|z_{k}|$; Step 2). So by Linearity and Monotonicity of the Lebesgue Integral §integrable, $1-\xi_{k}^{2}=\eta_{k}^{2}$, Step 11 and (12.1),

$$(\Lambda\psi)_{t}(u)=\sum_{k=1}^{n}a_{k}\Bigl((\partial_{k}\partial_{k}\psi)_{t}(u)-\frac{\eta_{k}u_{k}}{c_{k}}(\partial_{k}\psi)_{t}(u)-\frac{\xi_{k}}{c_{k}}\int z_{k}f_{k}(z)\,\gamma_{n}(dz)\Bigr)=\sum_{k=1}^{n}a_{k}\Bigl(\eta_{k}^{2}(\partial_{k}\partial_{k}\psi)_{t}(u)-\frac{u_{k}}{c_{k}}\,\eta_{k}(\partial_{k}\psi)_{t}(u)\Bigr)=\Lambda\psi_{t}(u).$$

By (b), $L^{a}(P_{t}F)=P_{t}(L^{a}F)$.

(d) An expansion. Write $\Theta=\Theta_{n}$. For real $h$ with $0<h\le1$ and $v,z\in\mathbb{R}^{n}$ put $d_{h}(v,z)=(\eta(h)-\mathbf{1})\odot v+\xi(h)\odot z$, so that $\eta(h)\odot v+\xi(h)\odot z=v+d_{h}(v,z)$. By Step 6(iii), $|\eta_{k}(h)-1|\le\Theta h$ and $\xi_{k}(h)^{2}\le2\Theta h$, so by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square and claim 5 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities

$$\lVert d_{h}(v,z)\rVert^{2}\le2\Theta^{2}h^{2}\lVert v\rVert^{2}+4\Theta h\lVert z\rVert^{2}.\qquad(12.2)$$

As $\gamma_{n}$ is a probability measure, $\psi_{h}(v)-\psi(v)=\int\bigl(\psi(v+d_{h}(v,z))-\psi(v)\bigr)\gamma_{n}(dz)$. With $d=d_{h}(v,z)$ define

$$R_{h}(v,z)=\psi(v+d)-\psi(v)-\sum_{i=1}^{n}\partial_{i}\psi(v)\,d_{i}-\frac12\sum_{i,j=1}^{n}\partial_{j}\partial_{i}\psi(v)\,d_{i}d_{j}.$$

By Step 2, $\int d_{i}\,\gamma_{n}(dz)=(\eta_{i}(h)-1)v_{i}$ and $\int d_{i}d_{j}\,\gamma_{n}(dz)=(\eta_{i}(h)-1)(\eta_{j}(h)-1)v_{i}v_{j}+\epsilon_{ij}\xi_{i}(h)^{2}c_{i}$, where $\epsilon_{ij}=1$ if $i=j$ and $0$ otherwise. Hence, $R_{h}(v,\cdot)$ being integrable as a combination of integrable functions,

$$\frac{\psi_{h}(v)-\psi(v)}{h}=\sum_{i}\frac{\eta_{i}(h)-1}{h}\,v_{i}\,\partial_{i}\psi(v)+\frac12\sum_{i,j}\frac{(\eta_{i}(h)-1)(\eta_{j}(h)-1)}{h}\,v_{i}v_{j}\,\partial_{j}\partial_{i}\psi(v)+\frac12\sum_{i}\frac{\xi_{i}(h)^{2}}{h}\,c_{i}\,\partial_{i}\partial_{i}\psi(v)+\frac1h\int R_{h}(v,z)\,\gamma_{n}(dz).\qquad(12.3)$$

By Multivariate Taylor Expansion with Uniform Second-Order Remainder, part (iii) with $\bar{\varepsilon}=2M_{2}$ (the segment lying in $W=\mathbb{R}^{n}$),

$$|R_{h}(v,z)|\le n\,M_{2}\,\lVert d_{h}(v,z)\rVert^{2},\qquad(12.4)$$

and by part (ii), integrated with Linearity and Monotonicity of the Lebesgue Integral §integrable, (12.2) and Step 2, $\bigl|\psi_{h}(v)-\psi(v)-\sum_{i}(\eta_{i}(h)-1)v_{i}\partial_{i}\psi(v)\bigr|\le\frac12nM_{2}(2\Theta^{2}h^{2}\lVert v\rVert^{2}+4\Theta hK_{1})$; with $|v_{i}|\le\lVert v\rVert\le1+\lVert v\rVert^{2}$ and $h\le1$ this gives

$$|\psi_{h}(v)-\psi(v)|\le C_{*}\,h\,(1+\lVert v\rVert^{2}),\qquad C_{*}=n\Theta M_{1}+nM_{2}\Theta^{2}+2nM_{2}\Theta K_{1}.\qquad(12.5)$$

(e) Convergence of difference quotients. Let $(h_{m})$ be a sequence in $(0,1]$ converging to $0$ and $(v_{m})$ a sequence in $\mathbb{R}^{n}$ converging to $v$. Then $(\psi_{h_{m}}(v_{m})-\psi(v_{m}))/h_{m}\to\Lambda\psi(v)$. Indeed, use (12.3) with $(h_{m},v_{m})$. By Step 6(iii), $(\eta_{i}(h_{m})-1)/h_{m}\to-\theta_{i}$ and $\xi_{i}(h_{m})^{2}/h_{m}\to2\theta_{i}$, and $|(\eta_{i}(h_{m})-1)(\eta_{j}(h_{m})-1)|/h_{m}\le\Theta^{2}h_{m}\to0$; the entries of $v_{m}$ converge to those of $v$; and $\partial_{i}\psi$, $\partial_{j}\partial_{i}\psi$ are continuous (Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential). So the first three terms of (12.3) converge to $\sum_{i}\bigl(-\theta_{i}v_{i}\partial_{i}\psi(v)+\theta_{i}c_{i}\partial_{i}\partial_{i}\psi(v)\bigr)=\Lambda\psi(v)$, because $\theta_{i}c_{i}=a_{i}$ and $\theta_{i}=a_{i}/c_{i}$ by Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §rates. For the last term let $V$ be a real bound of the $\lVert v_{m}\rVert$, and let $\varepsilon>0$. By continuity of the finitely many functions $\partial_{j}\partial_{i}\psi$ at $v$ there is $\varrho>0$ with $|\partial_{j}\partial_{i}\psi(y)-\partial_{j}\partial_{i}\psi(v)|\le\varepsilon/2$ whenever $\lVert y-v\rVert<\varrho$. Fix $z$ and put $d_{m}=d_{h_{m}}(v_{m},z)$; by (12.2), $\lVert d_{m}\rVert^{2}\le(2\Theta^{2}V^{2}+4\Theta\lVert z\rVert^{2})h_{m}\to0$, so there is $m_{0}$ with $\lVert v_{m}-v\rVert<\varrho/2$ and $\lVert d_{m}\rVert<\varrho/2$ for $m\ge m_{0}$. For such $m$ every point of the segment from $v_{m}$ to $v_{m}+d_{m}$ lies within $\varrho$ of $v$, so $|\partial_{j}\partial_{i}\psi(y)-\partial_{j}\partial_{i}\psi(v_{m})|\le\varepsilon$ on it, and part (iii) of Multivariate Taylor Expansion with Uniform Second-Order Remainder with $\bar{\varepsilon}=\varepsilon$ gives $|R_{h_{m}}(v_{m},z)|/h_{m}\le\frac12n\varepsilon(2\Theta^{2}V^{2}+4\Theta\lVert z\rVert^{2})$. As $\varepsilon$ was arbitrary, $R_{h_{m}}(v_{m},z)/h_{m}\to0$ for every $z$. By (12.4) and (12.2), $|R_{h_{m}}(v_{m},z)|/h_{m}\le nM_{2}(2\Theta^{2}V^{2}+4\Theta\lVert z\rVert^{2})$, which is $\gamma_{n}$-integrable by Step 2; so Dominated Convergence Theorem gives $\frac{1}{h_{m}}\int R_{h_{m}}(v_{m},z)\,\gamma_{n}(dz)\to0$.

(f) The derivative. Let $x\in X$, $u=p_{n}(x)$, and let $h\ne0$ be real with $t+h\ge0$ and $|h|\le1$; put $s_{h}=\min\{t,t+h\}\ge0$. If $h>0$, claim 3 (with $t$, $h$ in place of $s$, $t$) and the linearity in claim 2 give $P_{t+h}F-P_{t}F=P_{t}(P_{h}F)-P_{t}F=P_{t}(P_{h}F-F)$; if $h<0$, claim 3 gives $P_{t}F=P_{t+h}(P_{|h|}F)$, so $P_{t+h}F-P_{t}F=-P_{t+h}(P_{|h|}F-F)$. In both cases $P_{t+h}F-P_{t}F=\frac{h}{|h|}P_{s_{h}}(P_{|h|}F-F)$. The function $Q_{h}=(\psi_{|h|}-\psi)/|h|$ is continuous and, by (12.5), of polynomial growth with constants $(C_{*},2)$; as $P_{|h|}F-F=|h|\,Q_{h}\circ p_{n}$ by claim 2, Step 10 and (9.4) give

$$\frac{P_{t+h}F(x)-P_{t}F(x)}{h}=\int Q_{h}\bigl(\eta(s_{h})\odot u+\xi(s_{h})\odot z\bigr)\,\gamma_{n}(dz).\qquad(12.6)$$

Let $(h_{m})$ be a sequence of such numbers converging to $0$. Then $s_{h_{m}}\to t$, and for each $z$ the points $v_{m}(z)=\eta(s_{h_{m}})\odot u+\xi(s_{h_{m}})\odot z$ converge to $v(z)=\eta(t)\odot u+\xi(t)\odot z$ by Step 6(iv); so (e), applied with $|h_{m}|$ in place of $h_{m}$, gives $Q_{h_{m}}(v_{m}(z))\to\Lambda\psi(v(z))$. By (12.5), Step 1 and claim 5 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities, $|Q_{h_{m}}(v_{m}(z))|\le C_{*}(1+(\lVert u\rVert+\lVert z\rVert)^{2})\le C_{*}(1+2\lVert u\rVert^{2}+2\lVert z\rVert^{2})$, which is $\gamma_{n}$-integrable by Step 2. By Dominated Convergence Theorem, the quotients (12.6) along $(h_{m})$ converge to $\int\Lambda\psi(v(z))\,\gamma_{n}(dz)=(\Lambda\psi)_{t}(u)=P_{t}(L^{a}F)(x)$, by (b). Finally, if the limit in claim 12 failed, there would be $\varepsilon>0$ and, for each $m\in\mathbb{N}$, a real $h_{m}\ne0$ with $|h_{m}|<1/m$ and $t+h_{m}\ge0$ whose difference quotient differs from $P_{t}(L^{a}F)(x)$ by at least $\varepsilon$; as $h_{m}\to0$, this contradicts what was just shown. Hence

$$\lim_{h\to0}\frac{P_{t+h}F(x)-P_{t}F(x)}{h}=P_{t}(L^{a}F)(x).$$