Well-Posedness of 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 $1\le n$, and 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:

$$H=H^{-(s+1)}(\mathbb{T}^{n}),\qquad V=H^{-s}(\mathbb{T}^{n}),$$

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 $\langle\,\cdot\,,\cdot\,\rangle_{H}$, $\langle\,\cdot\,,\cdot\,\rangle_{V}$, norms $|\cdot|_{H}$, $|\cdot|_{V}$ and distance $d_{H}$; and $A$ the form operator of the triple, with domain $D(A)$, which is 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 the trace of $V$ on it is $V$; and $\mathrm{Sym}(V)$ is as 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 $2=1+1$, and for $p\in H$ write $|p|_{H}^{2}=|p|_{H}|p|_{H}$.

Let $\kappa$ be an enumeration of the lattice, a bijection from $\mathbb{N}$ onto $\mathbb{Z}^{n}$, and let $f=(f_{j})_{j\in\mathbb{N}}$ be the trigonometric white noise along $\kappa$: $f_{j}=\hat{E}_{\kappa(j)}$ is the Fourier coefficient family of the $\kappa(j)$th class $E_{\kappa(j)}$ 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 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 $\gamma,C_{g}\in\mathbb{R}$ satisfy $0<\gamma$ and $0\le C_{g}$, let $\omega_{g}$ be a modulus of continuity, and let $g:V\to\mathbb{R}$ satisfy

$$|g(x)|\le C_{g}\quad\text{for every }x\in V, \qquad |g(x)-g(y)|\le\omega_{g}\bigl(|x-y|_{V}\bigr)\quad\text{for all }x,y\in V .$$

Let $\theta,\nu\in\mathbb{R}$ satisfy $0\le\theta\le1$ and $0\le\nu$, and let $F$ be the operator of The Hamilton-Jacobi Equation of the Controlled Heat Equation with Square-Integrable White Noise on the Torus (i) for these data, so that the Hamilton-Jacobi equation of the controlled heat equation with square-integrable white noise is $F=0$ on $H$. Then the following hold.

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

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

is a second-order equation operator on $H$ relative to $(H,V,A)$ that is degenerate elliptic, and the hypotheses of Well-Posedness for a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple are satisfied by the present data, with $\gamma$ in the role of $\lambda_{0}$ there, with the zero map of $V$ into $H$ as $B$, with $L:H\to H$ the map $L(x)=-x$, and with $\ell=1$.

2. (Well-posedness)(ii) There is a function $u:H\to\mathbb{R}$ that is a viscosity solution of $F$ on $H$, satisfies

$$|u(x)|\le\frac{C_{g}}{\gamma}\qquad\text{for every }x\in H,$$

and is uniformly continuous on $H$ with respect to $d_{H}$ and the metric of Real Hilbert Spaces: Standing Notation and Background (i); the quotient above is the quotient of $C_{g}$ by the nonzero $\gamma$, and is nonnegative. Moreover, if $u'$ is a viscosity solution of $F$ on $H$ that is continuous on $H$ and for which some $C''\in\mathbb{R}$ satisfies $|u'(x)|\le C''$ for every $x\in H$, then $u'(x)=u(x)$ for every $x\in H$.