The Hamilton-Jacobi Equation of the Controlled Heat Equation with Square-Integrable White Noise on the Torus

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; the periodic class $C^{2}_{\mathrm{per}}$, the cell $Q$ and the restriction $u_{0}|_{Q}$ of a map $u_{0}$ to it are the ones fixed there. Let $s\in\mathbb{N}$ satisfy $n\le s$. We work also in the setting of Hilbert Triples: Standing Notation and Background, used with the Hilbert triple $(H,V,A)$ taken to be the Sobolev triple of order $s$ of the torus: $H=H^{-(s+1)}(\mathbb{T}^{n})$ and $V=H^{-s}(\mathbb{T}^{n})$ are the Sobolev spaces of orders $-(s+1)$ and $-s$, whose elements are coefficient families on the integer lattice $\mathbb{Z}^{n}$, with their inner products $\langle\,\cdot\,,\cdot\,\rangle_{H}$ and $\langle\,\cdot\,,\cdot\,\rangle_{V}$ and norm $|\cdot|_{H}$; $A$ is the form operator of the triple, with domain $D(A)$, which acts as one minus the Laplacian on the Fourier coefficient families of twice continuously differentiable periodic functions by The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian (iv); the standing hypothesis Hilbert Triples: Standing Notation and Background (ii) holds by The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian (i); $H$ is open in $H$ by Hilbert Triples: Standing Notation and Background (v), so that the trace of $D(A)$ on it is $D(A)$; and $\mathrm{Sym}(H)$ and $\mathrm{Sym}(V)$ are the sets of bounded symmetric bilinear forms on $H$ and on $V$ fixed in Hilbert Triples: Standing Notation and Background (vi). The exponent $p$ of The Flat Torus: Standing Notation (i) is not used here: below, $p$ denotes the gradient slot of $F$. Let $2=1+1$, for $p\in H$ write $|p|_{H}^{2}=|p|_{H}|p|_{H}$, and let $\Delta$ be the Laplacian of a map of class $C^{2}$ on $\mathbb{R}^{n}$.

Let $\kappa$ be an enumeration of the lattice, that is, a bijection from $\mathbb{N}$ onto $\mathbb{Z}^{n}$, and let $f=(f_{j})_{j\in\mathbb{N}}$ be the sequence with $f_{j}=\hat{E}_{\kappa(j)}$, the Fourier coefficient family of the $\kappa(j)$th class of the trigonometric system, as in 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; it is called the trigonometric white noise along $\kappa$, being the orthonormal basis $(E_{\kappa(j)})_{j\in\mathbb{N}}$ of the square-integrable classes carried into $V$ by the Fourier coefficient map. It lies in $V$ and is square-summable in $V$ by 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 (i), and $\mathrm{Tr}_{f}$ is the trace along $f$. Let $\gamma,\theta,\nu\in\mathbb{R}$ satisfy $0<\gamma$, $0\le\theta\le1$ and $0\le\nu$, and let $g:V\to\mathbb{R}$ be a function.

1. (The operator)(i) $F$ denotes the function on $D(A)\times\mathbb{R}\times H\times\mathrm{Sym}(V)$ whose value at $(x,r,p,X)$ is

$$F(x,r,p,X)=\gamma\,r-\tfrac{\nu}{2}\,\mathrm{Tr}_{f}X+\tfrac{\theta}{2}\,|p|_{H}^{2}+\bigl\langle Ax-x,\,p\bigr\rangle_{H}-g(x),$$

each term being a real number: $\mathrm{Tr}_{f}X$ by Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them (ii), $Ax\in H$ by Hilbert Triples: Standing Notation and Background (iii), and $g(x)$ because $x\in D(A)\subseteq V$. Thus $F$ is a second-order equation operator on $H$ relative to $(H,V,A)$, its domain being $W\times\mathbb{R}\times H\times\mathrm{Sym}(V)$ with $W=D(A)$.

2. (The equation)(ii) The Hamilton-Jacobi equation of the controlled heat equation with square-integrable white noise on $\mathbb{T}^{n}$, with discount $\gamma$, control weight $\theta$, noise intensity $\nu$ and running cost $g$, is the equation $F=0$ on $H$: for a function $u:H\to\mathbb{R}$ it reads

$$\gamma\,u(x)-\tfrac{\nu}{2}\,\mathrm{Tr}_{f}\bigl(D^{2}u(x)|_{V}\bigr)+\tfrac{\theta}{2}\,|Du(x)|_{H}^{2}+\bigl\langle Ax-x,\,Du(x)\bigr\rangle_{H}=g(x)\qquad(x\in D(A)),$$

where for $u$ of class $C^{2}$ on $H$ the gradient $Du(x)\in H$, the Hessian $D^{2}u(x)\in\mathrm{Sym}(H)$ and its restriction $D^{2}u(x)|_{V}\in\mathrm{Sym}(V)$ are those of Hilbert Triples: Standing Notation and Background (v) and Hilbert Triples: Standing Notation and Background (vi); for a general $u:H\to\mathbb{R}$ the equation is understood in the viscosity sense, a solution being a viscosity solution of $F$ on $H$. The drift $Ax-x$ is minus the Laplacian read in Fourier coefficients: for $x$ the Fourier coefficient family of the restriction to the cell of a function $u_{0}\in C^{2}_{\mathrm{per}}$, the vector $Ax-x$ is minus the Fourier coefficient family of the restriction of $\Delta u_{0}$, by 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 (v) together with 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 (vi), which identifies $Ax+B(x)+L(x)$ there with $Ax-x$.