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

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; the integer lattice $\mathbb{Z}^{n}$, Euclidean space $\mathbb{R}^{n}$ with its norm $\lVert\,\cdot\,\rVert$, the cell $Q$, the set $\mathcal{L}^{2}(\mathbb{T}^{n})$, the space $L^{2}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, the periodic class $C^{2}_{\mathrm{per}}$, the restriction $u|_{Q}$ and the Laplacian $\Delta$ are the ones fixed there. Let $\pi$ be the real number of The Number Pi §pi (the wrapping map of The Flat Torus: Standing Notation §cell is not used here), let $2=1+1$ and $4=2+2$, and let $t^{l}$ denote the natural power of a real number $t$ with exponent $l\in\mathbb{N}$, with $t^{2}=tt$. Let $s\in\mathbb{N}$ satisfy $n\le s$; the lemma Summability of the Negative Powers of the Fourier Weights of the Torus is used below with its $s$ taken to be this $s$.

We work also in the setting of Hilbert Triples: Standing Notation and Background, used here with the Hilbert triple $(H,V,A)$ taken to be the Sobolev triple of order $s$ of the torus, that is,

the Sobolev spaces of orders $-(s+1)$ and $-s$, whose elements are coefficient families on $\mathbb{Z}^{n}$, with their inner products written $\langle\,\cdot\,,\cdot\,\rangle_{H}$ and $\langle\,\cdot\,,\cdot\,\rangle_{V}$, and with $A$ the form operator of this triple, which acts on the coefficient families of twice continuously differentiable periodic functions as one minus the Laplacian by The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian §laplacian; the standing hypothesis Hilbert Triples: Standing Notation and Background §separable holds for this triple by The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian §triple, and $D(A)$, $0_{H}$, $d_{H}$, $|\cdot|_{H}$ and $|\cdot|_{V}$ are as fixed there. The Fourier weights $\mu_{k}=1+4\pi^{2}\lVert k\rVert^{2}$, the positive numbers $\rho_{k}$ with $\rho_{k}^{2}=\tfrac{1}{\mu_{k}}$, the Fourier coefficient family $\hat{U}$ of a class $U\in L^{2}(\mathbb{T}^{n})$, the classes $E_{k}$ of the trigonometric system, the rescaled trigonometric basis $\zeta_{s+1,k}$ of $H$ and the enumerations of the lattice are as in The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian. Let $\kappa$ be an enumeration, that is, a bijection from $\mathbb{N}$ onto $\mathbb{Z}^{n}$.

Let $f=(f_{j})_{j\in\mathbb{N}}$ be the sequence in $H$ given by

the Fourier coefficient families of the trigonometric classes taken along $\kappa$, which lie in $H$ 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 §embedding; let $B:V\to H$ be the map with $B(x)=0_{H}$ for every $x\in V$, and let $L:H\to H$ be the map with $L(x)=-x$ for every $x\in H$. Square-summability of a sequence in $V$ and its sum $\sigma$ are as defined there. Then the following hold.

1. (The noise)¶ For every $j\in\mathbb{N}$, $f_{j}\in V$, $f_{j}=\rho_{\kappa(j)}^{s+1}\,\zeta_{s+1,\kappa(j)}$, and

The sequence $f$ is square-summable in $V$, and

2. (The ambient space is infinite-dimensional)¶ $H$, which is a vector space over $\mathbb{R}$, is not finite-dimensional.

3. (The zero map is a monotone nonlinearity)¶ $B$ is a monotone nonlinearity for $(H,V,A)$.

4. (Minus the identity is a contraction)¶ $L$ is Lipschitz with constant $1$ from $(H,d_{H})$ to itself, the number $1$ being nonnegative.

5. (The drift is minus the Laplacian)¶ Let $u\in C^{2}_{\mathrm{per}}$. Then $u|_{Q}$ and $(\Delta u)|_{Q}$ lie in $\mathcal{L}^{2}(\mathbb{T}^{n})$ by The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian §laplacian, so that $x=\widehat{[\,u|_{Q}\,]}$ is defined; moreover $x\in D(A)$ and

6. (The drift in coefficients)¶ Let $w\in D(A)$ and write $w_{j}=\langle w,\zeta_{s+1,\kappa(j)}\rangle_{H}$ for $j\in\mathbb{N}$. Then the series $\sum_{j=1}^{\infty}4\pi^{2}\lVert\kappa(j)\rVert^{2}w_{j}\,\zeta_{s+1,\kappa(j)}$ converges in $H$ and