The Mehler Law Flow: Duality, Finite Second Moments, Invariance of the Gaussian Measure, Generation by the Noise Ornstein-Uhlenbeck Functional, and Decrease of Relative Entropy

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the set $\mathcal{F}C_{\mathrm{pol}}(X)$ of continuous cylindrical functions of polynomial growth and their representations, the Mehler semigroup $P_{t}$ and the Mehler law flow $(\mu P_{t})_{t\ge0}$ of The Mehler Law Flow of a Probability Measure on a Hilbert Space; the Ornstein-Uhlenbeck operator $L^{a}$; and, for $\nu\in\mathcal{P}_{2}(X)$, $n\in\mathbb{N}$ and $g\in C^{2}_{b}(\mathbb{R}^{n})$, the set of Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded with $q=n$, the noise Ornstein-Uhlenbeck functional $L^{a}_{\nu}(g)$. Let $\mu\in\mathcal{P}_{2}(X)$ and let $s,t\in\mathbb{R}$ with $0\le s$ and $0\le t$.

1. (Flow)¶ $\mu P_{t}\in\mathcal{P}_{2}(X)$, $\mu P_{0}=\mu$, $(\mu P_{s})P_{t}=\mu P_{s+t}$, and $\gamma_{c}P_{t}=\gamma_{c}$.

2. (Duality)¶ Let $F\in\mathcal{F}C_{\mathrm{pol}}(X)$ have a representation $(n,\psi,B,q)$ with $q\le2$. Then $F$ is integrable with respect to $\mu P_{t}$, $P_{t}F$ is integrable with respect to $\mu$, and

$$\int_{X}F\,d(\mu P_{t})=\int_{X}P_{t}F\,d\mu .$$

3. (The operator and the functional)¶ For every $\nu\in\mathcal{P}_{2}(X)$, $n\in\mathbb{N}$ and $g\in C^{2}_{b}(\mathbb{R}^{n})$, the function $L^{a}(g\circ p_{n})$ is integrable with respect to $\nu$, and

$$L^{a}_{\nu}(g)=-\int_{X}L^{a}(g\circ p_{n})\,d\nu .$$

4. (Generator of the law flow)¶ Let $n\in\mathbb{N}$ and $g\in C^{2}_{b}(\mathbb{R}^{n})$, and let $\Phi(r)=\int_{X}g\circ p_{n}\,d(\mu P_{r})$ for real $r\ge0$. Then

$$\lim_{h\to0}\frac{\Phi(t+h)-\Phi(t)}{h}=-L^{a}_{\mu P_{t}}(g),$$

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

5. (Relative entropy)¶ If $\mu$ has finite relative entropy with respect to $\gamma_{c}$, then so has $\mu P_{t}$, and $H(\mu P_{t}\,|\,\gamma_{c})\le H(\mu\,|\,\gamma_{c})$.