Proof of Well-Posedness of the Renormalised Wick-Square Problem: Existence, Comparison and Uniqueness of Renormalised Viscosity Solutions Differing from the Free Solution by a Bounded Lipschitz Function

Each result cited is universally quantified over the data in its own statement.

Conventions. By The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §triple, $(H,V,A)$ is a Hilbert triple in the setting of Hilbert Triples: Standing Notation and Background, with $H=H^{-3}$, $V=H^{-2}$, $D(A)=H^{-1}\subseteq V\subseteq H$, and $|x|_{H}\le|x|_{V}$ for $x\in V$ (Hilbert Triples: Standing Notation and Background §triple). Elementary arithmetic in $\mathbb{R}$ (the triangle inequality, maxima of two numbers, the limit laws and the order of limits) is used under The Real Numbers: Standing Notation and Background §background. Continuity of real functions on subsets of $(H,d_{H})$ is equivalent to sequential continuity (Real Hilbert Spaces: Standing Notation and Background §topology). Every function that is Lipschitz with a constant $\Lambda\ge0$ (Lipschitz Map Between Metric Spaces) on a subset $S$ of a metric space is uniformly continuous on $S$: given $\varepsilon>0$, the number $\varepsilon/(\Lambda+1)$ serves as $\delta$. Every function uniformly continuous on $S$ is continuous at every point of $S$ relative to $S$ (Continuous Map Between Metric Spaces): the $\delta$ of uniform continuity serves at each point.

Step 0 (Extension from the state space). We show: (E) if $M,\Lambda\ge0$ and $f:H^{-1}\to\mathbb{R}$ satisfies $|f(x)|\le M$ and $|f(x)-f(y)|\le\Lambda|x-y|_{H}$ for $x,y\in H^{-1}$, then there is $\hat{f}:H\to\mathbb{R}$, continuous on $H$, with $\hat{f}=f$ on $H^{-1}$, $|\hat{f}(x)|\le M$ and $|\hat{f}(x)-\hat{f}(y)|\le\Lambda|x-y|_{H}$ for all $x,y\in H$. Indeed, $H^{-1}=D(A)$ is nonempty (it contains $0_{H}$) and dense in $H$ by The Resolvent of the Form Operator of a Hilbert Triple: Minimisation, Contraction, and Density of the Domain §dense-in-h, that is dense in the topological space of the metric space $(H,d_{H})$ (Real Hilbert Space §topology); $f$ is Lipschitz, hence uniformly continuous, on $H^{-1}$. By Extension of a Uniformly Continuous Real Function from a Dense Subset §existence with $(X,d)=(H,d_{H})$ and $S=H^{-1}$ there is $\hat{f}:H\to\mathbb{R}$, continuous on $H$, with $\hat{f}=f$ on $H^{-1}$, and by Extension of a Uniformly Continuous Real Function from a Dense Subset §bounds, $|\hat{f}(x)|\le M$ for $x\in H$. Let $x,y\in H$. By The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds §dense (with $U=H$, open in $H$ by Hilbert Triples: Standing Notation and Background §open-sets), choose for each $j\in\mathbb{N}$ points $x_{j},y_{j}\in H^{-1}$ with $|x_{j}-x|_{H}<1/j$ and $|y_{j}-y|_{H}<1/j$. Then $|\hat{f}(x_{j})-\hat{f}(y_{j})|=|f(x_{j})-f(y_{j})|\le\Lambda|x_{j}-y_{j}|_{H}\le\Lambda(|x-y|_{H}+2/j)$; as $x_{j}\to x$ and $y_{j}\to y$ in $H$ and $\hat{f}$ is continuous, letting $j\to\infty$ gives $|\hat{f}(x)-\hat{f}(y)|\le\Lambda|x-y|_{H}$.

Apply (E) to $g$ with $M=C_{g}$, $\Lambda=\ell_{g}$, and let $\bar{g}:V\to\mathbb{R}$ be the restriction of the resulting extension to $V$. Then $|\bar{g}(x)|\le C_{g}$ and $|\bar{g}(x)-\bar{g}(y)|\le\ell_{g}|x-y|_{H}$ for $x,y\in V$, and $\bar{g}|_{H^{-1}}=g$. Hence The Riccati Shift: the Renormalised Operator at the Free Solution plus a Regular Function is the Shifted Operator, Which Satisfies the Hilbert-Triple Comparison Hypotheses, and Its Touching Points Lie in the State Space and Renormalised Viscosity Solutions versus Viscosity Solutions of the Shifted Equation on the Sobolev Triple of Order Two apply to $C_{g}$, $\ell_{g}$ and $\bar{g}$; we let $F^{\sharp}$ be the shifted operator for $\bar{g}$, and the renormalised notions there, taken for the running cost $\bar{g}|_{H^{-1}}=g$, are those of the theorem.

Step 1 (Existence). Put $C=C_{g}/\gamma$, nonnegative since $\gamma>0$ (The Wick-Square Problem on the Torus: Standing Notation §parameters). Let $x\in D(A)$, and let $0_{\mathrm{Sym}}$ be the zero form of $\mathrm{Sym}(V)$. By The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §noise, $\mathrm{Tr}_{f}0_{\mathrm{Sym}}$ is the lattice sum of the zero family, all of whose cube sums are $0$, so it is $0$ (Cube Sums of Families on the Integer Lattice §lattice-sum); $G(0_{H},0_{H})=G(0\cdot0_{H},0_{H})=0$ by bilinearity of $G\in\mathrm{Sym}(H)$ (The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §gradient-form); and $\langle Ax+B_{q}x,0_{H}\rangle_{H}=0$ by Elementary Identities in a Real Inner Product Space §zero. Hence by The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem on the Sobolev Triple of Order Two §operator

$$F^{\sharp}(x,-C,0_{H},0_{\mathrm{Sym}})=-C_{g}-\bar{g}(x)\le0,\qquad F^{\sharp}(x,C,0_{H},0_{\mathrm{Sym}})=C_{g}-\bar{g}(x)\ge0 .$$

$F^{\sharp}$ is a second-order equation operator on $H$ relative to $(H,V,A)$ (The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem on the Sobolev Triple of Order Two §operator) satisfying all the structural hypotheses of Existence of a Bounded, Uniformly Continuous Viscosity Solution on a Hilbert Triple by The Riccati Shift: the Renormalised Operator at the Free Solution plus a Regular Function is the Shifted Operator, Which Satisfies the Hilbert-Triple Comparison Hypotheses, and Its Touching Points Lie in the State Space §hypotheses. That theorem gives $w:H\to\mathbb{R}$ that is a viscosity solution of $F^{\sharp}$ on $H$ (Existence of a Bounded, Uniformly Continuous Viscosity Solution on a Hilbert Triple §solution), satisfies $|w(x)|\le C$ for $x\in H$ (Existence of a Bounded, Uniformly Continuous Viscosity Solution on a Hilbert Triple §bounded), and is uniformly continuous on $H$ (Existence of a Bounded, Uniformly Continuous Viscosity Solution on a Hilbert Triple §uniformly-continuous), hence continuous on $H$ (Conventions).

Next apply Lipschitz Regularity of Bounded Continuous Viscosity Solutions of a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple, whose standing hypothesis that $H$ is not finite-dimensional holds by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §infinite, with $\lambda_{0}=\gamma$, the constants $C_{g},\ell_{g}$, the cost $\bar{g}$, $B=B_{q}$ (a monotone nonlinearity by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §drift), the given $\nu>0$, the sequence $f$ (square-summable in $V$ by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §noise), $\Gamma=G$ (with $0_{\mathrm{Sym}}\preceq G\preceq I_{H}$ by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §gradient-form), $C'=C$ and $u=w$. The operator of that theorem is then, term by term, $F^{\sharp}$, of which $w$ is a bounded continuous viscosity solution. By Lipschitz Regularity of Bounded Continuous Viscosity Solutions of a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple §lipschitz, with $\Lambda_{0}=\ell_{g}/\gamma+2C_{g}/\gamma+1$,

$$|w(x)-w(y)|\le\Lambda_{0}\,|x-y|_{H}\qquad\text{for all }x,y\in H .$$

By Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §solution, $w$ is both a viscosity subsolution and a viscosity supersolution of $F^{\sharp}$ on $H$; as $w$ is bounded and continuous on $H$, Renormalised Viscosity Solutions versus Viscosity Solutions of the Shifted Equation on the Sobolev Triple of Order Two §from-shifted shows that $u=u_{0}+w|_{H^{-1}}$ is a renormalised viscosity subsolution and supersolution, i.e. a renormalised viscosity solution (Renormalised Viscosity Subsolutions, Supersolutions and Solutions of the Wick-Square Problem, Tested by the Gaussian Penalty plus a Regular Function §solution). Since $u-u_{0}=w|_{H^{-1}}$, one has $|u(x)-u_{0}(x)|\le C_{g}/\gamma$ and $|(u-u_{0})(x)-(u-u_{0})(y)|\le\Lambda_{0}|x-y|_{H}$ for $x,y\in H^{-1}$; in particular $u-u_{0}$ is bounded and Lipschitz from $(H^{-1},d_{H})$ to $\mathbb{R}$, so $u\in\mathcal{C}$. This proves clause 1.

Step 2 (Comparison). Let $u_{1},u_{2}\in\mathcal{C}$, $u_{1}$ a renormalised viscosity subsolution and $u_{2}$ a renormalised viscosity supersolution. For $i\in\{1,2\}$, choose $M_{i},\Lambda_{i}\ge0$ with $|(u_{i}-u_{0})(x)|\le M_{i}$ and $|(u_{i}-u_{0})(x)-(u_{i}-u_{0})(y)|\le\Lambda_{i}|x-y|_{H}$ for $x,y\in H^{-1}$, and let $w_{i}:H\to\mathbb{R}$ be the extension of $u_{i}-u_{0}$ given by (E): continuous on $H$, $|w_{i}|\le M_{i}$ on $H$, Lipschitz with constant $\Lambda_{i}$ from $(H,d_{H})$ to $\mathbb{R}$, and $u_{i}=u_{0}+w_{i}|_{H^{-1}}$. By Renormalised Viscosity Solutions versus Viscosity Solutions of the Shifted Equation on the Sobolev Triple of Order Two §to-shifted, $w_{1}$ is a viscosity subsolution and $w_{2}$ a viscosity supersolution of $F^{\sharp}$ on $H$. With $C_{12}=\max\{M_{1},M_{2}\}$ one has $w_{1}\le C_{12}$ and $-C_{12}\le w_{2}$ on $H$, so The Riccati Shift: the Renormalised Operator at the Free Solution plus a Regular Function is the Shifted Operator, Which Satisfies the Hilbert-Triple Comparison Hypotheses, and Its Touching Points Lie in the State Space §comparison gives $w_{1}(x)\le w_{2}(x)$ for every $x\in V$. For $x\in H^{-1}\subseteq V$, therefore, $u_{1}(x)=u_{0}(x)+w_{1}(x)\le u_{0}(x)+w_{2}(x)=u_{2}(x)$. This proves clause 2.

Step 3 (Uniqueness). By Step 1 there is a renormalised viscosity solution in $\mathcal{C}$. If $u,u'\in\mathcal{C}$ are renormalised viscosity solutions, each is both a subsolution and a supersolution (Renormalised Viscosity Subsolutions, Supersolutions and Solutions of the Wick-Square Problem, Tested by the Gaussian Penalty plus a Regular Function §solution), so Step 2 applied to $(u,u')$ and to $(u',u)$ gives $u\le u'$ and $u'\le u$ on $H^{-1}$; hence $u=u'$. This proves clause 3.

Step 4 (The free solution). Let $g$ be the zero function. Then the hypotheses of the theorem hold with the constants $0$ and $0$ in place of $C_{g}$ and $\ell_{g}$, and the set $\mathcal{C}$ and the renormalised notions do not involve these constants. Steps 0 and 1, carried out with these constants, give a renormalised viscosity solution $u\in\mathcal{C}$ with $|u(x)-u_{0}(x)|\le0/\gamma=0$ for every $x\in H^{-1}$, so $u=u_{0}$. Thus $u_{0}$ is a renormalised viscosity solution in $\mathcal{C}$, and by Step 3 it is the only one. This proves clause 4.