The Wick-Ordered $Phi^4$ Potential on the Torus of Dimension at Most Two: Limit of the Galerkin Potentials Almost Everywhere and in Every $L^p$, Conditional Expectations, Cameron-Martin Derivatives, and the Lower Bound at Each Cutoff

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, with the Lebesgue spaces $L^{p}(\gamma_{c})$ and norms $\lVert\cdot\rVert_{p}$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §lebesgue, suppose $n\le2$, and let $\varkappa,\mathfrak{m}\in\mathbb{R}$ with $0<\varkappa$. Let $V$ be the Wick-ordered $\phi^{4}$ potential with coupling $\varkappa$ and mass $\mathfrak{m}$, and let $V_{N}$, $d_{N}$, $\partial_{j}V_{N}$, $\mathcal{F}_{N}$, conditional expectations, $\mathcal{H}_{\le4}$ and the constant $C$ be as in 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, the constant being that of 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 §increment; $C_{N}$ is the Wick constant at cutoff $N$. $\mathbb{D}^{1,2}_{c}$ and the partial derivatives $\partial_{j}F$ of its elements are those of Cameron-Martin Partial Derivatives on the Gaussian Sobolev Space with the Variances as Weights §partial. Powers with real exponents are those of Real Power of a Positive Real Number, and $|s|$ is the absolute value of $s\in\mathbb{R}$.

1. (Measurability)¶ $V$ is Borel.

2. (Limit)¶ For $\gamma_{c}$-almost every $x\in X$ the sequence $(V_{2^{j}}(x))_{j\in\mathbb{N}}$ converges to $V(x)$. The class of $V$ lies in $\mathcal{H}_{\le4}$, and for every real $p\ge2$ it lies in $L^{p}(\gamma_{c})$, with

$$\lVert V-V_{N}\rVert_{p}\le(p-1)^{2}\,C^{1/2}\,N^{-1/8}\qquad\text{for every }N\in\mathbb{N}.$$

3. (Conditional expectations)¶ For every $N\in\mathbb{N}$, $V_{N}$ is a conditional expectation of $V$ given $\mathcal{F}_{N}$.

4. (Cameron-Martin derivatives)¶ For every $N\in\mathbb{N}$ the class of $V_{N}$ lies in $\mathbb{D}^{1,2}_{c}$, with partial derivatives the classes of the functions $\partial_{j}V_{N}$. The class of $V$ lies in $\mathbb{D}^{1,2}_{c}$, and for every $j\in\mathbb{N}$ and every real $p\ge2$, $\partial_{j}V\in L^{p}(\gamma_{c})$ and $\lVert\partial_{j}V-\partial_{j}V_{N}\rVert_{p}\to0$ as $N\to\infty$.

5. (Lower bound at each cutoff)¶ For every $N\in\mathbb{N}$ and $x\in X$,

$$V_{N}(x)\ \ge\ -B_{N},\qquad B_{N}=\frac{(3\varkappa C_{N}+|\mathfrak{m}|)^{2}}{4\varkappa}+\frac{|\mathfrak{m}|\,C_{N}}{2}.$$

6. (Uniform lower bound in dimension one)¶ If $n=1$, the lattice sum $C_{\infty}=\sum_{k\in\mathbb{Z}}\mu_{k}^{-1}$ is defined by Lattice Sums of Powers of the Fourier Weights in Dimension at Most Two: Power Sums, the Growth of the Wick Constants, and the Quartic Convolution Sums §powers, $C_{N}\le C_{\infty}$ for every $N$, and $V(x)\ge-B_{\infty}$ for every $x\in X$, where $B_{\infty}$ is the number $B_{N}$ with $C_{\infty}$ in place of $C_{N}$.