The Mehler Maps and the Mehler Semigroup with Noise Weights on Continuous Cylindrical Functions of Polynomial Growth

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the rates $\theta_{k}=a_{k}/c_{k}$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §rates, and with the set $\mathcal{F}C_{\mathrm{pol}}(X)$ of continuous cylindrical functions of polynomial growth and their representations. $\exp$ is the exponential function, with the properties of Basic Properties of the Exponential Function, and $\sqrt{\cdot}$ is the nonnegative square root. $X\times X$ carries the norm and distance of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs.

1. (The Mehler maps)¶ For a real $t\ge0$ and $k\in\mathbb{N}$ put $\eta_{k}(t)=\exp(-\theta_{k}t)$ and $\xi_{k}(t)=\sqrt{1-\exp(-2\theta_{k}t)}$; since $\theta_{k}t\ge0$, one has $0<\eta_{k}(t)\le1$ and $0\le1-\exp(-2\theta_{k}t)<1$, so $0\le\xi_{k}(t)\le1$, and $\eta_{k}(t)^{2}+\xi_{k}(t)^{2}=1$. For $(x,y)\in X\times X$, $(\eta_{k}(t)x_{k}+\xi_{k}(t)y_{k})^{2}\le2x_{k}^{2}+2y_{k}^{2}$, the series $\sum_{k}x_{k}^{2}$ and $\sum_{k}y_{k}^{2}$ converge by Orthonormal Expansions in a Real Hilbert Space §parseval, hence so does $\sum_{k}(2x_{k}^{2}+2y_{k}^{2})$ by Elementary Properties of Series of Real Numbers §linearity; so by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison and Orthonormal Expansions in a Real Hilbert Space §riesz-fischer the series below converges in $X$, and the Mehler map $M_{t}:X\times X\to X$ is

$$M_{t}(x,y)=\sum_{k=1}^{\infty}\bigl(\eta_{k}(t)\,x_{k}+\xi_{k}(t)\,y_{k}\bigr)e_{k}.$$

By Orthonormal Expansions in a Real Hilbert Space §riesz-fischer its $k$-th coordinate is $\eta_{k}(t)x_{k}+\xi_{k}(t)y_{k}$ and $|M_{t}(x,y)|^{2}=\sum_{k}(\eta_{k}(t)x_{k}+\xi_{k}(t)y_{k})^{2}$. Every vector of $X$ is determined by its coordinates, being the sum of its expansion by Orthonormal Expansions in a Real Hilbert Space §expansion; since the $k$-th coordinate of $M_{t}(x,y)$ depends linearly on $(x,y)$, for real $\lambda,\lambda'$ and $(x,y),(x',y')\in X\times X$ the vectors $M_{t}(\lambda x+\lambda'x',\lambda y+\lambda'y')$ and $\lambda M_{t}(x,y)+\lambda'M_{t}(x',y')$ have the same coordinates, hence are equal, and $M_{t}$ is linear. Further $|M_{t}(x,0)|\le|x|$ and $|M_{t}(0,y)|\le|y|$ by the same identity and Orthonormal Expansions in a Real Hilbert Space §parseval, since $\eta_{k}(t)^{2}\le1$ and $\xi_{k}(t)^{2}\le1$; since $M_{t}(x,y)=M_{t}(x,0)+M_{t}(0,y)$ by linearity, the triangle inequality gives $|M_{t}(x,y)|\le|x|+|y|$. As $2|x||y|\le|x|^{2}+|y|^{2}$, one has $(|x|+|y|)^{2}\le2(|x|^{2}+|y|^{2})$, and $|x|^{2}+|y|^{2}=|(x,y)|^{2}$ for the norm of $X\times X$ by the product pairing of The Product of Two Real Inner Product Spaces §product; so $|M_{t}(x,y)|\le\sqrt{2}\,|(x,y)|$, and by linearity $|M_{t}(x,y)-M_{t}(x',y')|=|M_{t}(x-x',y-y')|\le\sqrt{2}\,|(x,y)-(x',y')|$. Consequently $M_{t}$ is Lipschitz with constant $\sqrt{2}$, hence continuous (given $\varepsilon>0$, take $\delta=\varepsilon/\sqrt{2}$), hence Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space.

2. (The Mehler semigroup)¶ Let $F\in\mathcal{F}C_{\mathrm{pol}}(X)$ with representation $(n,\psi,B,q)$, let $t\ge0$ and $x\in X$. The function $y\mapsto F(M_{t}(x,y))$ is continuous, hence Borel, and integrable with respect to $\gamma_{c}$. Indeed, $\lVert p_{n}(z)\rVert\le|z|$ for $z\in X$ by Orthonormal Expansions in a Real Hilbert Space §bessel, so $|F(M_{t}(x,y))|\le B\bigl(1+(|x|+|y|)^{q}\bigr)\le B\bigl(1+(1+|x|)^{q}(1+|y|^{2})^{q}\bigr)$, where the second inequality holds because $|y|\le1+|y|^{2}$ (if $|y|\le1$ this is clear, and otherwise $|y|\le|y|^{2}$), so that $|x|+|y|\le|x|+1+|y|^{2}\le(1+|x|)(1+|y|^{2})$, the last step since $|x|\,|y|^{2}\ge0$; raising both sides to the power $q$, by the monotonicity of $u\mapsto u^{q}$ on $u\ge0$ and $(uv)^{q}=u^{q}v^{q}$, gives $(|x|+|y|)^{q}\le(1+|x|)^{q}(1+|y|^{2})^{q}$. Put $\bar{c}=\sum_{k=1}^{\infty}c_{k}$, positive with $c_{k}\le\bar{c}$ for every $k$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, and $\alpha=1/(4\bar{c})$. The function $u\mapsto(1+u/\alpha)^{q}$ is a polynomial function, nonnegative for $u\ge0$, so claim 1 of The Exponential Function Dominates Every Polynomial Function, applied with $\varepsilon=1$, gives $u_{0}\ge1$ with $(1+u/\alpha)^{q}\exp(-u)<1$ for $u\ge u_{0}$; multiplying by $\exp(u)>0$ and using $\exp(-u)\exp(u)=\exp(0)=1$ (claims 1 and 2 of Basic Properties of the Exponential Function) gives $(1+u/\alpha)^{q}\le\exp(u)$ for $u\ge u_{0}$, while $(1+u/\alpha)^{q}\le(1+u_{0}/\alpha)^{q}$ for $0\le u\le u_{0}$; with $u=\alpha|y|^{2}$ this gives $(1+|y|^{2})^{q}\le(1+u_{0}/\alpha)^{q}+\exp(\alpha|y|^{2})$, and $y\mapsto\exp(\alpha|y|^{2})$ is integrable with respect to $\gamma_{c}$ 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 $2\alpha c_{k}\le1/2$. The Mehler semigroup with noise weights $a$ acts on $F$ by

$$P_{t}F(x)=\int_{X}F\bigl(M_{t}(x,y)\bigr)\,\gamma_{c}(dy)\qquad(t\ge0,\ x\in X).$$