The Mehler Law Flow of a Probability Measure on a Hilbert Space

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the Mehler maps $M_{t}:X\times X\to X$ ($t\ge0$) of The Mehler Maps and the Mehler Semigroup with Noise Weights on Continuous Cylindrical Functions of Polynomial Growth, which are Borel by that clause. $X\times X$ carries the norm and distance of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs; these have the same open sets as the product metric by Properties of the Product of Two Real Inner Product Spaces §metric, so its Borel $\sigma$-algebra is the product $\sigma$-algebra $\mathcal{B}(X)\otimes\mathcal{B}(X)$ by The Borel Sigma-Algebra of a Product of Two Separable Metric Spaces is the Product Sigma-Algebra §product, $X$ being separable by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space. For $\mu\in\mathcal{P}(X)$, $\mu\otimes\gamma_{c}$ is the product probability measure on $\mathcal{B}(X)\otimes\mathcal{B}(X)$ of Existence and Uniqueness of the Product Measure.

(The Mehler law flow)¶ For $\mu\in\mathcal{P}(X)$ and a real $t\ge0$, $\mu P_{t}\in\mathcal{P}(X)$ is the push-forward $(M_{t})_{\#}(\mu\otimes\gamma_{c})$ of $\mu\otimes\gamma_{c}$ by the Borel map $M_{t}$, as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward; the family $(\mu P_{t})_{t\ge0}$ is the Mehler law flow from $\mu$ with noise weights $a$.