The Galerkin Wick-Ordered $Phi^4$ Potentials on the Torus of Dimension at Most Two under the Free Field: Chaos Order, Conditional Expectations, and Increments with Their Cameron-Martin Derivatives

In the settings of The Real Numbers: Standing Notation and Background, The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation and Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, the last read with the space $X=H^{-m}(\mathbb{T}^{n})$, the orthonormal basis $(e_{j})_{j\in\mathbb{N}}$, the noise weights $a$ and the variance sequence $c$ of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §gaussian, so that $\gamma_{c}$ is the free field, suppose $n\le2$, and let $\varkappa,\mathfrak{m}\in\mathbb{R}$ with $0<\varkappa$. For $N\in\mathbb{N}$, $V_{N}$ is the Galerkin Wick-ordered $\phi^{4}$ potential with cutoff $N$, coupling $\varkappa$ and mass $\mathfrak{m}$, $d_{N}$ and $J_{N}$ are the Galerkin head dimension and the active indices at cutoff $N$, and $\partial_{j}V_{N}$ ($j\in\mathbb{N}$) are the functions of Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient, $V_{N}$ being an admissible cylindrical potential with head dimension $d_{N}$ by The Galerkin Wick-Ordered Phi^4 Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass §admissible; thus $\partial_{j}V_{N}=0$ for $d_{N}<j$. $\mathcal{F}_{N}$ is the $\sigma$-algebra on $X$ generated by the coordinate maps $x\mapsto x_{j}$ with $j\in J_{N}$. Conditional expectations are those of Conditional Expectation of a Square-Integrable Random Variable on the probability space $(X,\mathcal{B}(X),\gamma_{c})$, and $\mathcal{H}_{\le r}$ ($r\in\mathbb{N}_{0}$) is the chaos up to order $r$.

1. (Chaos order)¶ For every $N\in\mathbb{N}$ and $j\in\mathbb{N}$, $V_{N}$ and $\partial_{j}V_{N}$ are square-integrable with respect to $\gamma_{c}$, and their classes lie in $\mathcal{H}_{\le4}$ and $\mathcal{H}_{\le3}$ respectively.

2. (Conditional expectations)¶ For $N,M\in\mathbb{N}$ with $N\le M$, $V_{N}$ is a conditional expectation of $V_{M}$ given $\mathcal{F}_{N}$.

3. (Increments)¶ There is a real number $C\ge0$, depending only on $n$, $\varkappa$ and $\mathfrak{m}$, such that for all $N,M\in\mathbb{N}$ with $N\le M$, with $N^{-1/4}$ as in Real Power of a Positive Real Number,

$$\int_{X}(V_{M}-V_{N})^{2}\,d\gamma_{c}\le C\,N^{-1/4}\qquad\text{and}\qquad\sum_{j=1}^{d_{M}}c_{j}\int_{X}(\partial_{j}V_{M}-\partial_{j}V_{N})^{2}\,d\gamma_{c}\le C\,N^{-1/4}.$$