Proof of Existence and Uniqueness of a Continuous Periodic Density for the Torus Heat Semigroup Started at the Origin

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

Conventions. The number $s$ with $0<s\le\tfrac12$ is fixed. Densities are those of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities, taken on the measure space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\lambda_{d})$; so a density is a nonnegative Borel function $h$ on $\mathbb{R}^{d}$ with $\nu(A)=\int\mathbf{1}_{A}h\,d\lambda_{d}$ for every Borel $A$, where $\nu$ is the measure in question. As recorded in the statement, $\delta_{0}\in\mathcal{P}(\mathbb{T}^{d})$, so $S_{s}\delta_{0}$ is defined and belongs to $\mathcal{P}(\mathbb{T}^{d})$ by The Heat Semigroup on the Probability Measures on the Torus §heat; in particular it is a finite measure on $\mathcal{B}(\mathbb{R}^{d})$. By Optimal Transport on the Flat Torus: Standing Notation §conventions and The Flat Torus: Standing Notation §periodic, $C_{\mathrm{per}}$ is the class of Lattice-Periodic Functions and the Periodic Function Classes §classes: the $\mathbb{Z}^{d}$-periodic maps $\mathbb{R}^{d}\to\mathbb{R}$ that are continuous on $(\mathbb{R}^{d},d_{E})$ as maps into $\mathbb{R}$ with the metric $d_{\mathbb{R}}(a,b)=|a-b|$ of The Absolute Value Metric on the Real Line.

Step 1 (Existence). By Heat Smoothing on the Torus: Distance to the Identity, Contraction, a Smooth Positive Periodic Density, Finite Entropy, and Finite Fisher Information §density, applied with $\delta_{0}$ in place of $\mu$, there is a $\mathbb{Z}^{d}$-periodic function $p:\mathbb{R}^{d}\to\mathbb{R}$ of class $C^{2}$ on $\mathbb{R}^{d}$ such that $\mathbf{1}_{Q}p$ is a density of $S_{s}\delta_{0}$ with respect to $\lambda_{d}$. The set $\mathbb{R}^{d}$ is open in itself by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and claim 3 of that lemma, with $U=\mathbb{R}^{d}$, $m=1$ and $k=2$, shows that $p$ is continuous relative to $\mathbb{R}^{d}$ at every point of $\mathbb{R}^{d}$ as a map into $(\mathbb{R},d_{\mathbb{R}})$. Hence $p\in C_{\mathrm{per}}$, and $\Theta=p$ has the required property.

Step 2 (Uniqueness). Let $\Theta_{1},\Theta_{2}\in C_{\mathrm{per}}$ be such that $\mathbf{1}_{Q}\Theta_{1}$ and $\mathbf{1}_{Q}\Theta_{2}$ are both densities of $S_{s}\delta_{0}$ with respect to $\lambda_{d}$. By The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities §uniqueness, applied to the measure space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\lambda_{d})$ and the finite measure $S_{s}\delta_{0}$, the set

$$N=\{x\in\mathbb{R}^{d}:\mathbf{1}_{Q}(x)\Theta_{1}(x)\ne\mathbf{1}_{Q}(x)\Theta_{2}(x)\}$$

is Borel with $\lambda_{d}(N)=0$. Put $D=\Theta_{1}-\Theta_{2}=\Theta_{1}+(-1)\Theta_{2}$, which lies in $C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions §algebra. Every $x\in Q$ with $D(x)\ne0$ lies in $N$, since $\mathbf{1}_{Q}(x)=1$.

Suppose, for a contradiction, that $D(v)\ne0$ for some $v\in\mathbb{R}^{d}$. Put $u=\pi(v)$. By The Half-Open Unit Cell Tiles Euclidean Space §wrap, $u\in Q$ and $u=v-m$ for the integer vector $m\in\mathbb{Z}^{d}$ provided by The Half-Open Unit Cell Tiles Euclidean Space §tiling; so $D(u)=D(u+m)=D(v)$ by periodicity (applied at the point $u$ with the lattice vector $m$), and $\varepsilon=|D(u)|$ is positive. We now choose, in this order, the numbers $\delta$, $t$ and $r$.

Choice of $\delta$. As $D$ is continuous at $u$, there is a real $\delta>0$ such that $|D(x)-D(u)|<\varepsilon$ for every $x\in\mathbb{R}^{d}$ with $d_{E}(u,x)<\delta$ (Continuous Map Between Metric Spaces), and $d_{E}(u,x)=\lVert u-x\rVert$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$. For such $x$ we have $D(x)\ne0$: otherwise $|D(u)|<\varepsilon=|D(u)|$.

Choice of $t$ and of a point of the open cell. Since $0\le u_{i}<1$ for every $i\in[d]$, the number $\tau=\min\{1-u_{i}:i\in[d]\}$ is positive. Choose a real $t$ with $0<t<\tau$ and $t\le\delta/(4d)$, and let $w\in\mathbb{R}^{d}$ be the point with coordinates $w_{i}=u_{i}+t$. Then $0<w_{i}<1$ for every $i$, so $w\in\mathring{Q}$, the open cell of The Half-Open Unit Cell Tiles Euclidean Space. By claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $\lVert w-u\rVert^{2}=\sum_{i=1}^{d}t^{2}=dt^{2}\le d\,\delta^{2}/(16d^{2})\le\delta^{2}/16$, as $1\le d$; hence $\lVert w-u\rVert\le\delta/4$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, both numbers being nonnegative.

Choice of $r$. By The Half-Open Unit Cell Tiles Euclidean Space §cell the set $\mathring{Q}$ is open, that is, open in $(\mathbb{R}^{d},d_{E})$ (Euclidean Space and Lebesgue Measure: Standing Notation §space); so by Open Subset of a Metric Space there is a real $r_{1}>0$ with $B(w,r_{1})\subseteq\mathring{Q}$. Put $r=\min\{r_{1},\delta/2\}>0$.

Contradiction. Let $x\in B(w,r)$. Then $x\in\mathring{Q}\subseteq Q$ (The Half-Open Unit Cell Tiles Euclidean Space §cell), and by claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$,

$$\lVert x-u\rVert\le\lVert x-w\rVert+\lVert w-u\rVert<\frac{\delta}{2}+\frac{\delta}{4}<\delta,$$

where $\lVert x-w\rVert=d_{E}(x,w)<r\le\delta/2$ by claim 2 of the same lemma. So $D(x)\ne0$ by the choice of $\delta$, and $x\in N$ because $x\in Q$. Thus $B(w,r)\subseteq N$. By claim 1 of Balls Have Positive Lebesgue Measure and Bounded Sets Have Finite Lebesgue Measure, $B(w,r)$ is a Borel set with $\lambda_{d}(B(w,r))>0$, while monotonicity of measures (claim 2 of Basic Properties of a Measure) gives $\lambda_{d}(B(w,r))\le\lambda_{d}(N)=0$. This contradiction shows that $D(v)=0$ for every $v\in\mathbb{R}^{d}$, that is, $\Theta_{1}=\Theta_{2}$.

Steps 1 and 2 together show that there is exactly one $\Theta\in C_{\mathrm{per}}$ such that $\mathbf{1}_{Q}\Theta$ is a density of $S_{s}\delta_{0}$ with respect to $\lambda_{d}$. $\blacksquare$