What the Hamilton-Jacobi Equation of the Heat Equation with Square-Integrable White Noise Models, and What Is Not Proved

This remark records what the equation The Hamilton-Jacobi Equation of the Controlled Heat Equation with Square-Integrable White Noise on the Torus §equation, solved in Well-Posedness of the Hamilton-Jacobi Equation of the Controlled Heat Equation with Square-Integrable White Noise on the Torus §well-posed, is meant to model, and which parts of that reading the corpus establishes. Nothing here is used elsewhere; the notation is that of the corollary, with $n\le s$, $H=H^{-(s+1)}(\mathbb{T}^{n})$, $V=H^{-s}(\mathbb{T}^{n})$, $A$ the form operator of the Sobolev triple, and $f_{j}=\hat{E}_{\kappa(j)}$ the trigonometric white noise along an enumeration $\kappa$. Here $\pi$ is the real number of The Number Pi §pi (the wrapping map of The Flat Torus: Standing Notation §cell is not used), $\Delta$ is the Laplacian of a map of class $C^{2}$ on $\mathbb{R}^{n}$, and $\mu_{k}=1+4\pi^{2}\lVert k\rVert^{2}$ are the Fourier weights.

The spaces and the operator. For $m\in\mathbb{N}$, the elements of $H^{-m}(\mathbb{T}^{n})$ are coefficient families $c$ on $\mathbb{Z}^{n}$ for which the series $\sum_{j}\mu_{\kappa(j)}^{-m}c(\kappa(j))^{2}$ converges, along any enumeration $\kappa$, 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 §series and 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 §weights; this is the classical Fourier description of the Sobolev space of order $-m$, realised here on the coefficient side rather than as a space of distributions. The square-integrable classes embed through $U\mapsto\hat{U}$, injectively because the trigonometric system is complete (The Trigonometric System is an Orthonormal Basis of the Square-Integrable Space of the Torus §complete); the form operator acts as one minus the Laplacian, in that on the coefficient family of the restriction of a function $u_{0}\in C^{2}_{\mathrm{per}}$ it subtracts the coefficient family of the restriction of $\Delta u_{0}$, by The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian §laplacian; and the coefficient families $\hat{E}_{k}$ of the trigonometric classes are eigenvectors of it, with eigenvalues $\mu_{k}$, by The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian §eigen. So the map $x\mapsto Ax-x$ agrees, on those coefficient families, with minus the Laplacian read in Fourier coefficients, 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 §drift, and the equation is the Hamilton-Jacobi equation of the heat equation.

Why the noise is square-integrable white noise. Read heuristically: a Gaussian noise on a Hilbert space is described by its covariance. The sequence $(E_{\kappa(j)})_{j\in\mathbb{N}}$ is an orthonormal basis of $L^{2}(\mathbb{T}^{n})$ by The Trigonometric System is an Orthonormal Basis of the Square-Integrable Space of the Torus §basis, so, formally, the series $\sum_{j}E_{\kappa(j)}\,d\beta_{j}$, with independent standard Brownian motions $\beta_{j}$, has covariance the identity of $L^{2}(\mathbb{T}^{n})$: it is space-time white noise on the torus. The noise of the equation is exactly this basis carried into $V$ by the embedding $U\mapsto\hat{U}$, and what the corpus proves about it is the deterministic statement 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: $|f_{j}|_{V}^{2}=\tfrac{1}{\mu_{\kappa(j)}^{s}}$, whose series over $j$ converges when $n\le s$, with sum at most $(1+\tfrac{1}{\pi^{2}})^{n}$. That square-summability is the hypothesis the trace $\mathrm{Tr}_{f}$ of Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them §trace needs, and it is why the equation is posed on the negative-order scale: white noise is not square-summable in $L^{2}(\mathbb{T}^{n})$ itself, since $\lVert E_{k}\rVert_{L^{2}}=1$ for every $k$.

The control problem (heuristic). Read purely formally, Well-Posedness of the Hamilton-Jacobi Equation of the Controlled Heat Equation with Square-Integrable White Noise on the Torus §well-posed solves the dynamic programming equation of the following problem: a state $X_{t}$ in $H$ obeys the controlled stochastic heat equation

and the controller pays

Here $g$ is applied to $X_{t}\in H$ although it is defined only on $V$; this too is part of the heuristic. Minimising the expected cost over controls $a$ gives, formally, the equation $F=0$ of The Hamilton-Jacobi Equation of the Controlled Heat Equation with Square-Integrable White Noise on the Torus §equation: the discount $\gamma$ is the zeroth-order coefficient, the drift $\Delta X=X-AX$ enters through $\langle AX-X,Du\rangle_{H}$ with the sign reversed because it is moved to the left-hand side, the Hamiltonian $\tfrac{\theta}{2}|Du|_{H}^{2}$ is what the pointwise minimisation over $a_{t}$ leaves behind, and $\tfrac{\nu}{2}\mathrm{Tr}_{f}D^{2}u$ is the second-order term of the noise.

What is not proved. Four caveats.

First, no stochastic object above is constructed. The displayed dynamics and cost are heuristic: the corpus carries real-valued and vector-valued Brownian motion but no Hilbert-space-valued Wiener process, no stochastic integral against one, and no verification theorem tying a viscosity solution to a value function. The corollary is a theorem about a partial differential equation, and it is complete as such.

Second, the hypothesis of Well-Posedness of the Hamilton-Jacobi Equation of the Controlled Heat Equation with Square-Integrable White Noise on the Torus is $n\le s$, whereas square-summability of the weights holds under the weaker $n<2s$; that sharpening is not in the corpus. What Summability of the Negative Powers of the Fourier Weights of the Torus §summable supplies is $n\le s$, and it costs only a larger $s$, that is, a weaker space.

Third, the trace $\mathrm{Tr}_{f}$ and the sum $\sigma(f)$ are attached to the sequence $f$, hence to the enumeration $\kappa$. That they do not depend on $\kappa$, nor on the choice of orthonormal basis of $L^{2}(\mathbb{T}^{n})$ carried into $V$ — which is what would make "white noise on $L^{2}(\mathbb{T}^{n})$" a property of the space alone — rests on rearrangement of series of nonnegative terms and on basis-independence of the trace, neither of which the corpus carries.

Fourth, the nonlinearity is absent: the corollary takes the zero map as its monotone nonlinearity. The cubic term of the Allen-Cahn model of Well-Posedness of the Viscous Allen-Cahn Hamilton-Jacobi Equation on the Torus is not available on a space of negative order, where the cube of a coefficient family is not defined; supplying it requires renormalisation and is not attempted here.