Proof of Well-Posedness of the Hamilton-Jacobi Equation of the Controlled Heat Equation with Square-Integrable White Noise on the Torus

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 $s$, the enumeration $\kappa$ and the data named in the present statement. Let $B:V\to H$ be the zero map, $B(x)=0_{H}$, and $L:H\to H$ the map $L(x)=-x$, 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, whose noise sequence $f$ and whose triple are those of the present statement.

We apply Well-Posedness for a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple with its Hilbert triple taken to be $(H,V,A)$, whose standing separability hypothesis Hilbert Triples: Standing Notation and Background §separable holds by The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian §triple. Its hypotheses are met, one by one, as follows.

1. $H$, as a vector space over $\mathbb{R}$, is not finite-dimensional, 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 §not-finite-dimensional.

2. The number $\lambda_{0}$ of that theorem is taken to be $\gamma$, which satisfies $0<\gamma$, and $C_{g}$ satisfies $0\le C_{g}$; $\omega_{g}$ is a modulus of continuity; and $g:V\to\mathbb{R}$ is bounded in absolute value by $C_{g}$ and satisfies $|g(x)-g(y)|\le\omega_{g}(|x-y|_{V})$ for all $x,y\in V$. These are hypotheses of the present statement.

3. $B:V\to H$ is a monotone nonlinearity for $(H,V,A)$, 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 §zero-nonlinearity.

4. The number $\ell$ of that theorem is taken to be $1$, which is nonnegative, and $L:H\to H$ is Lipschitz with constant $1$ from $(H,d_{H})$ to itself, 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 §lipschitz.

5. The numbers $\theta$ and $\nu$ satisfy $0\le\theta\le1$ and $0\le\nu$, by hypothesis.

6. The sequence $f$ 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 §noise, and $\mathrm{Tr}_{f}$ is the trace along $f$, as fixed in the present statement.

Claim 1. By items 1 to 6 every hypothesis of Well-Posedness for a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple is satisfied by the present data, with $\gamma$ in the role of $\lambda_{0}$, with $B$ the zero map, and with $L(x)=-x$ and $\ell=1$. The function displayed in that theorem for these data has value $\gamma r-\tfrac{\nu}{2}\mathrm{Tr}_{f}X+\tfrac{\theta}{2}|p|_{H}^{2}+\langle Ax+B(x)+L(x),p\rangle_{H}-g(x)$ at $(x,r,p,X)$, and $Ax+B(x)+L(x)=Ax+0_{H}+(-x)=Ax-x$ by claim 1 of Elementary Identities in a Vector Space and claim 2 of that lemma; so it is the function $F$ of The Hamilton-Jacobi Equation of the Controlled Heat Equation with Square-Integrable White Noise on the Torus §operator, displayed in the present statement. That theorem records that $F$ is a second-order equation operator on $H$ relative to $(H,V,A)$ that is degenerate elliptic, by A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses §operator. This proves claim 1.

Claim 2. By Well-Posedness for a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple §existence there is a function $u:H\to\mathbb{R}$ that is a viscosity solution of $F$ on $H$, satisfies $|u(x)|\le C$ for every $x\in H$ with $C=\tfrac{C_{g}}{\gamma}$ the quotient of $C_{g}$ by the nonzero $\gamma$, which that theorem records as nonnegative, and is uniformly continuous on $H$ with respect to $d_{H}$ and the metric of Real Hilbert Spaces: Standing Notation and Background §numbers. By Well-Posedness for a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple §well-posedness this $u$ is continuous on $H$, and any viscosity solution $u'$ of $F$ on $H$ that is continuous on $H$ and satisfies $|u'(x)|\le C''$ for every $x\in H$, for some $C''\in\mathbb{R}$, agrees with $u$ at every point of $H$. These are the assertions of claim 2.