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

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the Ornstein-Uhlenbeck rates $\theta_{k}$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §rates and the sequence $\theta=(\theta_{k})_{k\in\mathbb{N}}$; the set $\mathcal{F}C_{\mathrm{pol}}(X)$ of continuous cylindrical functions of polynomial growth with their representations, the Mehler maps $M_{t}$ and the Mehler semigroup $P_{t}$ of The Mehler Maps and the Mehler Semigroup with Noise Weights on Continuous Cylindrical Functions of Polynomial Growth; the Ornstein-Uhlenbeck operator $L^{a}$; the cylindrical Hermite polynomials $H_{\alpha}$ ($\alpha\in\mathcal{A}$) and the weighted orders $\theta\cdot\alpha$. $\exp$ is the exponential function. Let $s,t\in\mathbb{R}$ with $0\le s$ and $0\le t$, and let $F,G\in\mathcal{F}C_{\mathrm{pol}}(X)$.

1. (The class)¶ $\mathcal{F}C_{\mathrm{pol}}(X)$ is closed under pointwise sums, real multiples and products, and contains the constant functions, $\mathcal{F}C^{1}_{b}(X)$, $\mathcal{F}C^{2}_{b}(X)$ and every $H_{\alpha}$ ($\alpha\in\mathcal{A}$). Every $F\in\mathcal{F}C_{\mathrm{pol}}(X)$ is continuous, hence Borel, and $\int_{X}|F|^{p}\,d\gamma_{c}<\infty$ for every real $p\ge1$, so that $F\in L^{p}(\gamma_{c})$.

2. (Cylindrical form)¶ Let $(n,\psi,B,q)$ be a representation of $F$, and let $\eta_{k}(t)$, $\xi_{k}(t)$ be the numbers of The Mehler Maps and the Mehler Semigroup with Noise Weights on Continuous Cylindrical Functions of Polynomial Growth §map. For $u\in\mathbb{R}^{n}$ put

$$\psi_{t}(u)=\int_{\mathbb{R}^{n}}\psi\bigl(\eta_{1}(t)u_{1}+\xi_{1}(t)z_{1},\dots,\eta_{n}(t)u_{n}+\xi_{n}(t)z_{n}\bigr)\,\gamma_{c^{(n)}}(dz),$$

with $\gamma_{c^{(n)}}$ the diagonal Gaussian measure on $\mathbb{R}^{n}$ of Variance Sequences and Their Truncations §truncations. Then the integrand is integrable, $\psi_{t}$ is continuous, $P_{t}F=\psi_{t}\circ p_{n}$, and there is a real $B'\ge0$, depending only on $B$, $q$ and $c$, with $|\psi_{r}(u)|\le B'(1+\lVert u\rVert^{q})$ for every real $r\ge0$ and every $u\in\mathbb{R}^{n}$; in particular $(n,\psi_{t},B',q)$ is a representation of $P_{t}F\in\mathcal{F}C_{\mathrm{pol}}(X)$. Moreover $P_{0}F=F$, $P_{t}$ is linear on $\mathcal{F}C_{\mathrm{pol}}(X)$, and $P_{t}$ maps each constant function to itself.

3. (Semigroup law)¶ $P_{s}(P_{t}F)=P_{s+t}F$.

4. (Invariance)¶ $\int_{X}P_{t}F\,d\gamma_{c}=\int_{X}F\,d\gamma_{c}$.

5. (Positivity)¶ If $F(x)\ge0$ for every $x\in X$, then $P_{t}F(x)\ge0$ for every $x\in X$.

6. (Contraction)¶ For every real $p\ge1$, $\lVert P_{t}F\rVert_{p}\le\lVert F\rVert_{p}$.

7. (Symmetry)¶ $G\,P_{t}F$ and $F\,P_{t}G$ are integrable with respect to $\gamma_{c}$, and $\int_{X}G\,P_{t}F\,d\gamma_{c}=\int_{X}F\,P_{t}G\,d\gamma_{c}$.

8. (Hermite eigenfunctions)¶ For every $\alpha\in\mathcal{A}$, $P_{t}H_{\alpha}=\exp(-t\,\theta\cdot\alpha)\,H_{\alpha}$.

9. (Commutation with partial derivatives)¶ If $F\in\mathcal{F}C^{1}_{b}(X)$, then $P_{t}F\in\mathcal{F}C^{1}_{b}(X)$, $\partial_{k}F\in\mathcal{F}C_{\mathrm{pol}}(X)$ and $\partial_{k}(P_{t}F)=\exp(-\theta_{k}t)\,P_{t}(\partial_{k}F)$ for every $k\in\mathbb{N}$.

10. (Smoothing)¶ If $t>0$ and $F$ is bounded, then $P_{t}F\in\mathcal{F}C^{2}_{b}(X)$.

11. (Continuity in time)¶ For every $x\in X$, the function $r\mapsto P_{r}F(x)$ is continuous on $[0,\infty)$, and $\lim_{r\to t}\lVert P_{r}F-P_{t}F\rVert_{p}=0$ for every real $p\ge1$, the limits being taken over real $r\ge0$.

12. (Generator)¶ If $F\in\mathcal{F}C^{2}_{b}(X)$, then $P_{t}F\in\mathcal{F}C^{2}_{b}(X)$, $L^{a}F\in\mathcal{F}C_{\mathrm{pol}}(X)$, $L^{a}(P_{t}F)=P_{t}(L^{a}F)$, and for every $x\in X$

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

the limit being taken over real $h\ne0$ with $t+h\ge0$.