PomeLemma 11.23.14.6C85B622ADCF The averages of $|S|$ and $P|S|$ are
\[ \int _ X|S|\mu =\frac8{\pi ^2}, \qquad \int _ XP|S|\mu =\frac8{3\pi ^2}. \]
11.23.13 Total variation between two evolving probabilitiesF9C293B132B8
The entropies of 11.23.6 compare one evolving probability with the fixed target. Two solutions of the flow with different initial potentials can nevertheless move apart. On a flat torus, even with a closed background, their total-variation distance can increase at the initial time, while no convex entropy relative to the target increases. We show this by computing the initial derivatives explicitly. By a logarithmic Monge–Ampère flow with target $\nu $ we mean a smooth solution of the equation in (11.23.7), with an arbitrary smooth initial potential.
For this example, let $E=\mathbb C/(\mathbb Z+\mathrm i\mathbb Z)$ and $X=E^2$, with covering coordinates $z_ j=x_ j+\mathrm i y_ j$. Write
\[ \eta _ j=\frac{\mathrm i}2\mathrm dz_ j\wedge \mathrm d\bar z_ j, \qquad \kappa =\eta _1+\eta _2, \qquad \mu =\eta _1\wedge \eta _2=\frac{\kappa ^2}2. \]
Thus $\mu $ is a probability volume form and $\int _ X\kappa ^2=2$. Put
\[ c_ j=\cos (2\pi x_ j),\qquad S=c_1+c_2,\qquad P=c_1c_2. \]
Proof. The integrands do not depend on $y_1,y_2$. The degree-two map of real tori $(a,b)\mapsto (a+b,a-b)$ preserves normalized Haar averages. With the angles $A=2\pi a$ and $B=2\pi b$, it gives
\[ S=2\cos A\cos B,\qquad P=\frac{\cos (2A)+\cos (2B)}2. \]
The one-variable averages of $|\cos A|$ and $|\cos A|^3$ are $2/\pi $ and $4/(3\pi )$, respectively, so the average of $\cos (2A)|\cos A|$ is $2/(3\pi )$. Multiplying the averages over the two independent angles gives both formulas. $\square$
For a fixed real number $K>3$, define the smooth target probability $\nu _ K$ by
\[ Z_ K=\int _ X\mathrm e^{KP}\mu , \qquad f_ K=Z_ K^{-1}\mathrm e^{KP}, \qquad \nu _ K=f_ K\mu . \]
For $0<\varepsilon <1/\pi $, the initial potentials $\varepsilon S$ and zero give positive forms. Set
\[ A_\varepsilon =1-\pi \varepsilon c_1, \qquad B_\varepsilon =1-\pi \varepsilon c_2, \qquad r_\varepsilon =A_\varepsilon B_\varepsilon , \]
so that $\kappa +\mathrm{dd}^{\mathrm{c}}(\varepsilon S)=A_\varepsilon \eta _1+B_\varepsilon \eta _2$ and its normalized Monge–Ampère probability is $r_\varepsilon \mu $.
Consider the short smooth paths
7732A51ABE6E
11.23.15
\begin{equation} U_\varepsilon (t)=\varepsilon S+t\log \frac{r_\varepsilon }{f_ K}, \qquad V(t)=-t\log f_ K. \end{equation}
They remain positive for sufficiently small $|t|$, and at $t=0$ their velocities are exactly those prescribed by the logarithmic Monge–Ampère flow with target $\nu _ K$. Their normalized probabilities are
\[ \mu _\varepsilon (t)=\frac{(\kappa +\mathrm{dd}^{\mathrm{c}}U_\varepsilon (t))^2}2, \qquad \mu _0(t)=\frac{(\kappa +\mathrm{dd}^{\mathrm{c}}V(t))^2}2, \]
which have mass one because the background $\kappa $ is closed. Define their total-variation distance, with the convention for probabilities, by
\[ D_\varepsilon (t) =\frac12\int _ X|\mu _\varepsilon (t)-\mu _0(t)|. \]
Proposition 11.23.16.621E75C5A9F4 For the paths (11.23.15),
\[ D'_\varepsilon (0) =4\varepsilon \left(\frac K3-1\right)+o(\varepsilon ) \quad \text{as }\varepsilon \downarrow 0. \]
In particular, this derivative is positive for sufficiently small $\varepsilon >0$. Any two logarithmic Monge–Ampère flows with target $\nu _ K$ and these initial potentials have the same positive initial derivative of their total-variation distance.
Proof. For a real function $h$ depending only on $x_1,x_2$, the coefficient matrix of $\mathrm{dd}^{\mathrm{c}}h$ with respect to the forms $(\mathrm i/2)\mathrm dz_ j\wedge \mathrm d\bar z_ k$ is $(4\pi )^{-1}(h_{x_ jx_ k})$. Put $R_\varepsilon (t)=\mathrm d\mu _\varepsilon (t)/\mathrm d\mu $ and $R_0(t)=\mathrm d\mu _0(t)/\mathrm d\mu $. Differentiating the determinant at the diagonal initial matrices gives
\[ \begin{split} \left.\partial _ t(R_\varepsilon -R_0)\right|_{t=0} =\frac1{4\pi }\bigl (& B_\varepsilon \partial _{x_1}^2\log A_\varepsilon +A_\varepsilon \partial _{x_2}^2\log B_\varepsilon \\ & -K(B_\varepsilon -1)\partial _{x_1}^2P -K(A_\varepsilon -1)\partial _{x_2}^2P\bigr ). \end{split} \]
Since $\partial _{x_ j}^2P=-4\pi ^2P$, Taylor expansion in $\varepsilon $ yields the uniform estimate
\[ \left.\partial _ t(R_\varepsilon -R_0)\right|_{t=0} =\pi ^2\varepsilon S(1-KP)+O(\varepsilon ^2). \]
At $t=0$, the difference of the densities is
\[ q_\varepsilon =r_\varepsilon -1 =-\pi \varepsilon S+\pi ^2\varepsilon ^2P. \]
For each fixed $x_2$, its coefficient of $c_1$ is $-\pi \varepsilon (1-\pi \varepsilon c_2)\ne 0$, so as a function of $x_1$ it has at most two zeros in a period, and its zero set has measure zero. Differentiation under the integral sign, justified by the uniform bound on the derivatives of the smooth densities, gives
\[ D'_\varepsilon (0)=\frac12\int _ X\operatorname {sgn}(q_\varepsilon ) \left.\partial _ t(R_\varepsilon -R_0)\right|_{t=0}\mu . \]
The set $\{ S=0\} $ also has measure zero, and $\operatorname {sgn}(q_\varepsilon )\to -\operatorname {sgn}(S)$ off this set. Dominated convergence and lemma 11.23.14 give
\[ \lim _{\varepsilon \downarrow 0}\frac{D'_\varepsilon (0)}\varepsilon =\frac{\pi ^2}2\int _ X|S|(KP-1)\mu =4\left(\frac K3-1\right). \]
Every flow with the stated initial data has the velocities (11.23.15) at $t=0$, so its initial density derivatives, and hence the derivative of the distance, coincide with this computation. $\square$
Corollary 11.23.17.21368BD09EBB For these paths, the initial derivative of every $C^2$ convex entropy of each probability relative to $\nu _ K$ is nonpositive, while the total-variation distance between the two probabilities has positive initial derivative for sufficiently small $\varepsilon >0$.
Proof. The initial velocities are those of the logarithmic flow, and both initial forms are closed. The chain-rule calculation in the proof of proposition 11.23.8 therefore applies at $t=0$: its background term vanishes and its gradient term is nonpositive. The preceding proposition gives the assertion about the pair. $\square$
Since only initial velocities enter, the example shows, without any existence or convergence theorem for the flow, that the logarithmic Monge–Ampère flow need not contract the total-variation distance infinitesimally, even though no convex entropy relative to the target increases at the initial time.
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