Proof of The $Phi^4$ Measure on the Torus of Dimension at Most Two as the Gibbs Measure of a Gibbs-Sobolev Potential, and Its Galerkin Approximation in Total Variation

Each result cited is universally quantified over the data in its own statement.

Step 1 (claim 1). Let $q>1$ and $q'=q/(q-1)$, so $q'>1$. $V$ is Borel by 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 §borel. (a) For condition Gibbs-Sobolev Potentials: Exponentially Integrable Potentials with Cameron-Martin Derivatives of Finite Moments §exponential: since $-qV/\beta=-(q/\beta)V$ and $q/\beta>0$, the function $\exp(-qV/\beta)$ is integrable by Nelson's Exponential Integrability for the Wick-Ordered Phi^4 Potential on the Torus of Dimension at Most Two §integrable with $t=q/\beta$. (b) For condition Gibbs-Sobolev Potentials: Exponentially Integrable Potentials with Cameron-Martin Derivatives of Finite Moments §sobolev: by 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 §limit with $p=2$, the class of $V$ lies in $L^{2}(\gamma_{c})$, that is, $\int_{X}V^{2}\,d\gamma_{c}<\infty$; and it lies in $\mathbb{D}^{1,2}_{c}$ by 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 §sobolev. (c) For condition Gibbs-Sobolev Potentials: Exponentially Integrable Potentials with Cameron-Martin Derivatives of Finite Moments §derivatives: for $k\in\mathbb{N}$, 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 §sobolev with $p=2q'\ge2$ gives $\partial_{k}V\in L^{2q'}(\gamma_{c})$, $\partial_{k}V$ being the Cameron-Martin partial derivative of Cameron-Martin Partial Derivatives on the Gaussian Sobolev Space with the Variances as Weights §partial in both items. Hence (a), (b) and (c) of Gibbs-Sobolev Potentials: Exponentially Integrable Potentials with Cameron-Martin Derivatives of Finite Moments §potential hold, and $V$ is a Gibbs-Sobolev potential at temperature $\beta$ with exponent $q$.

Step 2 (the Gibbs measures are defined). Each $V_{N}$ is continuous, hence Borel, by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity, being an admissible cylindrical potential 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; and $V$ is Borel by Step 1. Write $t=1/\beta>0$, $w=\exp(-V/\beta)=\exp(-tV)$, $w_{N}=\exp(-V_{N}/\beta)=\exp(-tV_{N})$, $Z=Z_{V,\beta}$ and $Z_{N}=Z_{V_{N},\beta}$. By Nelson's Exponential Integrability for the Wick-Ordered Phi^4 Potential on the Torus of Dimension at Most Two §convergence with $p=1$, $w$ and every $w_{N}$ are integrable, so the weights, normalisers and Gibbs measures of The Gibbs Measure of a Borel Potential Relative to a Diagonal Gaussian Measure §weight, The Gibbs Measure of a Borel Potential Relative to a Diagonal Gaussian Measure §normaliser and The Gibbs Measure of a Borel Potential Relative to a Diagonal Gaussian Measure §gibbs are defined, with $0<Z$ and $0<Z_{N}$; and, writing $\eta_{N}=\int_{X}|w_{N}-w|\,d\gamma_{c}=\lVert w_{N}-w\rVert_{1}$ (Elementary Properties of the p-Seminorm §power with $p=1$), the same claim gives $\eta_{N}\to0$ as $N\to\infty$.

Step 3 (normalisers). By linearity and the bound $|\int_{X}f\,d\gamma_{c}|\le\int_{X}|f|\,d\gamma_{c}$ of Linearity and Monotonicity of the Lebesgue Integral §integrable,

$$|Z_{N}-Z|=\Bigl|\int_{X}(w_{N}-w)\,d\gamma_{c}\Bigr|\le\eta_{N},$$

so $(Z_{N})_{N\in\mathbb{N}}$ converges to $Z$ by claim 3(c) of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities, applied with $b_{N}=\eta_{N}\to0$ (Step 2) and $L=Z$.

Step 4 (a bound uniform in $A$). Let $N\in\mathbb{N}$ and $A\in\mathcal{B}(X)$. The functions $\mathbf{1}_{A}w_{N}$ and $\mathbf{1}_{A}w$ are both integrable, being Borel, nonnegative and at most $w_{N}$ and $w$ (monotonicity in Linearity and Monotonicity of the Lebesgue Integral §nonnegative). By The Gibbs Measure of a Borel Potential Relative to a Diagonal Gaussian Measure §gibbs,

$$\gamma^{V_{N}}_{\beta}(A)-\gamma^{V}_{\beta}(A)=\frac{1}{Z_{N}}\int_{X}\mathbf{1}_{A}(w_{N}-w)\,d\gamma_{c}+\Bigl(\frac{1}{Z_{N}}-\frac{1}{Z}\Bigr)\int_{X}\mathbf{1}_{A}w\,d\gamma_{c}.$$

Since $|\mathbf{1}_{A}(w_{N}-w)|\le|w_{N}-w|$, the first term has absolute value at most $\eta_{N}/Z_{N}$, by Linearity and Monotonicity of the Lebesgue Integral §integrable. Since $0\le\mathbf{1}_{A}w\le w$, $0\le\int_{X}\mathbf{1}_{A}w\,d\gamma_{c}\le Z$, and $|1/Z_{N}-1/Z|=|Z-Z_{N}|/(Z_{N}Z)$, the second term has absolute value at most $|Z-Z_{N}|/Z_{N}\le\eta_{N}/Z_{N}$ by Step 3. Hence

$$\bigl|\gamma^{V_{N}}_{\beta}(A)-\gamma^{V}_{\beta}(A)\bigr|\le\frac{2\eta_{N}}{Z_{N}}\qquad\text{for every }A\in\mathcal{B}(X).$$

Step 5 (choice of $N_{0}$; claim 2). Let $\varepsilon>0$ be given; then choose (using that $\eta_{N}\to0$) $N_{0}\in\mathbb{N}$ with $\eta_{N}\le\min\{Z/2,\,\varepsilon Z/4\}$ for every $N\ge N_{0}$, which exists because $\eta_{N}\to0$ and $\min\{Z/2,\varepsilon Z/4\}>0$. For $N\ge N_{0}$, Step 3 gives $Z_{N}\ge Z-|Z-Z_{N}|\ge Z-\eta_{N}\ge Z/2$, so by Step 4

$$\bigl|\gamma^{V_{N}}_{\beta}(A)-\gamma^{V}_{\beta}(A)\bigr|\le\frac{2\,(\varepsilon Z/4)}{Z/2}=\varepsilon\qquad\text{for every }A\in\mathcal{B}(X).$$

Together with Steps 2 and 3 this proves claim 2.