Proof of The Data of the Hamilton-Jacobi Equation of the Heat Equation with Square-Integrable White Noise on the Torus: the Noise, the Nonlinearities and the Drift

Each result cited is universally quantified over the data in its own statement, and is applied here to the Sobolev triple $(H,V,A)$ of order $s$ and the enumeration $\kappa$ fixed in the statement. The clauses of The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale are applied with its $m$ equal to $s$ or to $s+1$, as indicated. By The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian (i), $(\zeta_{s+1,\kappa(j)})_{j\in\mathbb{N}}$ is an orthonormal basis of $H$, which is in particular an orthonormal sequence in $H$.

Claim 1. Let $j\in\mathbb{N}$ and put $k=\kappa(j)$. By The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian (iii), $f_{j}=\hat{E}_{k}=\rho_{k}^{s+1}\zeta_{s+1,k}$, and by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (iii) with $m=s$, $\hat{E}_{k}\in H^{-s}(\mathbb{T}^{n})=V$ with $|\hat{E}_{k}|_{H^{-s}}=\rho_{k}^{s}$; so $|f_{j}|_{V}^{2}=(\rho_{k}^{s})^{2}=\tfrac{1}{\mu_{k}^{s}}$ by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (i) with $m=s$.

Since $\kappa$ is a bijection, by Bijection of Sets each element of $\mathbb{Z}^{n}$ is the value of $\kappa$ at exactly one natural number; in particular $\kappa(j)=\kappa(j')$ implies $j=j'$. As $n\le s$, Summability of the Negative Powers of the Fourier Weights of the Torus (ii) applies to $\kappa$ and gives that the series $\sum_{j=1}^{\infty}\tfrac{1}{\mu_{\kappa(j)}^{s}}$ converges with sum at most $(1+\tfrac{1}{\pi^{2}})^{n}$. Its terms are the numbers $|f_{j}|_{V}^{2}$, so by Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them (i) the sequence $f$ is square-summable in $V$ with $\sigma(f)$ equal to that sum.

Claim 2. Since $H$ contains the orthonormal sequence $(\zeta_{s+1,\kappa(j)})_{j\in\mathbb{N}}$, A Real Hilbert Space Containing an Orthonormal Sequence Is Not Finite-Dimensional (i) gives the assertion.

Claim 3. We verify the three conditions of Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple (iv).

Monotonicity. Let $x,y\in V$. By claims 1 and 2 of Elementary Identities in a Vector Space the additive inverse of $0_{H}$ is $0_{H}$, so $B(x)-B(y)=0_{H}-0_{H}=0_{H}$; and $\langle 0_{H},x-y\rangle_{H}=0$ by Elementary Identities in a Real Inner Product Space (ii). Since $0\le 0$, the condition of Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple (i) holds.

$A$-monotonicity. Let $x\in D(A)$; then $Ax\in H$ by Hilbert Triples: Standing Notation and Background (iii), and $\langle B(x),Ax\rangle_{H}=\langle 0_{H},Ax\rangle_{H}=0$ by Elementary Identities in a Real Inner Product Space (ii). Since $0\le 0$, the condition of Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple (ii) holds.

Boundedness on $V$-bounded sets. Let $R\in\mathbb{R}$ be positive and take $\beta=0$. Every $x\in V$ satisfies $|B(x)|_{H}=|0_{H}|_{H}=0$ by Elementary Identities in a Real Inner Product Space (ii), hence $|B(x)|_{H}\le\beta$; in particular this holds for every $x\in V$ with $|x|_{V}\le R$. So the condition of Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple (iii) holds, and $B$ is a monotone nonlinearity for $(H,V,A)$.

Claim 4. Let $x,y\in H$. The vector $(-x)+y$ satisfies $(x-y)+\bigl((-x)+y\bigr)=0_{H}$, by the commutativity and associativity of addition in the vector space $H$, so it is the additive inverse of $x-y$ by the uniqueness in claim 2 of Elementary Identities in a Vector Space; by that same uniqueness $-(-y)=y$, so $L(x)-L(y)=(-x)-(-y)=(-x)+y=-(x-y)$. Therefore

$$d_{H}\bigl(L(x),L(y)\bigr)=\bigl|L(x)-L(y)\bigr|_{H}=\bigl|-(x-y)\bigr|_{H}=|x-y|_{H}=d_{H}(x,y),$$

the outer equalities by Real Inner Product Spaces (iii) and the middle one by Elementary Identities in a Real Inner Product Space (iv). Since $1\,d_{H}(x,y)=d_{H}(x,y)$ and $0\le1$ by claim 1 of Elementary Arithmetic in an Ordered Field, $L$ is Lipschitz with constant $1$ in the sense of Lipschitz Map Between Metric Spaces.

Claim 5. Let $u\in C^{2}_{\mathrm{per}}$. By The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian (iv), $u|_{Q}$ and $(\Delta u)|_{Q}$ lie in $\mathcal{L}^{2}(\mathbb{T}^{n})$, so $x=\widehat{[\,u|_{Q}\,]}$ and $\hat{D}$ with $D=[\,(\Delta u)|_{Q}\,]$ are defined, and by the same clause $x\in D(A)$ and $Ax=x-\hat{D}$. Since $B(x)=0_{H}$ and $L(x)=-x$, claim 1 of Elementary Identities in a Vector Space and the vector space conditions of Vector Space over a Field give

$$Ax+B(x)+L(x)=Ax+0_{H}+(-x)=Ax-x=(x-\hat{D})-x=-\hat{D},$$

the last step because $(x-\hat{D})-x=(x+(-x))+(-\hat{D})=0_{H}+(-\hat{D})=-\hat{D}$ by commutativity and associativity of addition.

This proves claim 5.

Claim 6. Let $w\in D(A)$ and write $w_{j}=\langle w,\zeta_{s+1,\kappa(j)}\rangle_{H}$. By The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian (ii) the series $\sum_{j=1}^{\infty}\mu_{\kappa(j)}w_{j}\zeta_{s+1,\kappa(j)}$ converges in $H$ with sum $Aw$. By Orthonormal Expansions in a Real Hilbert Space (iii), applied to the orthonormal basis $(\zeta_{s+1,\kappa(j)})_{j\in\mathbb{N}}$ of $H$ and to the vector $w$, the series $\sum_{j=1}^{\infty}w_{j}\zeta_{s+1,\kappa(j)}$ converges in $H$ with sum $w$. By Elementary Properties of Series in a Real Inner Product Space (i), applied in $H$ with the scalar $-1$, the series $\sum_{j=1}^{\infty}(-1)w_{j}\zeta_{s+1,\kappa(j)}$ converges with sum $(-1)w$, which is $-w$ by claim 5 of Elementary Identities in a Vector Space. Applying Elementary Properties of Series in a Real Inner Product Space (i) again, now to the two convergent series $\sum_{j=1}^{\infty}\mu_{\kappa(j)}w_{j}\zeta_{s+1,\kappa(j)}$ and $\sum_{j=1}^{\infty}(-1)w_{j}\zeta_{s+1,\kappa(j)}$, the series whose $j$th term is $\mu_{\kappa(j)}w_{j}\zeta_{s+1,\kappa(j)}+(-1)w_{j}\zeta_{s+1,\kappa(j)}$ converges in $H$ with sum $Aw+(-w)=Aw-w$.

For each $j\in\mathbb{N}$ the vector space conditions of Vector Space over a Field give

$$\mu_{\kappa(j)}w_{j}\zeta_{s+1,\kappa(j)}+(-1)w_{j}\zeta_{s+1,\kappa(j)}=\bigl((\mu_{\kappa(j)}-1)w_{j}\bigr)\zeta_{s+1,\kappa(j)},$$

and $\mu_{\kappa(j)}-1=4\pi^{2}\lVert\kappa(j)\rVert^{2}$ by the definition of the Fourier weights. The convergent series just obtained is therefore $\sum_{j=1}^{\infty}4\pi^{2}\lVert\kappa(j)\rVert^{2}w_{j}\zeta_{s+1,\kappa(j)}$, with sum $Aw-w$; and $Aw+B(w)+L(w)=Aw+0_{H}+(-w)=Aw-w$ by claim 1 of Elementary Identities in a Vector Space and claim 2 of that lemma. This proves claim 6.