Theorem 14.18.2.F90961841CA0 For a smooth family $F_ t$ and its solutions,
\begin{align*} \dot b_\gamma (F_ t)& =-\int _ X\dot F_ t\, \mathrm{d}\mu _{u_ t}^\gamma ,\\ \ddot b_\gamma (F_ t)& =-\int _ X\ddot F_ t\, \mathrm{d}\mu _{u_ t}^\gamma -\frac1{n-1}\int _ X \operatorname {tr}\bigl ((Q_ t^{-1}\dot Q_ t)^2\bigr ) \, \mathrm{d}\mu _{u_ t}^\gamma . \end{align*}
In particular, $b_\gamma $ is concave, and along $F_ t=F_0+tG$ its second derivative is strictly negative unless $G$ is constant. Moreover, for smooth real functions $F,G$ on $X$ and every real constant $a$,
\[ b_\gamma (F+a)=b_\gamma (F)-a, \qquad |b_\gamma (G)-b_\gamma (F)|\leq \| G-F\| _\infty . \]
Proof.
By (14.13.1),
\[ \mathcal G(u)=\frac1{n-1}\log \det _\gamma Q_ u. \]
Differentiating $\mathcal G(u_ t)=F_ t+b_\gamma (F_ t)$ once gives $\mathcal L_{u_ t}\dot u_ t=\dot F_ t+\dot b_\gamma (F_ t)$, and integration against the adjoint probability measure gives the first formula.
Because $\Gamma _ u$ is affine in $u$, its second derivative along the family is $\mathcal D_\gamma \ddot u_ t$. The formula for the second derivative of $\log \det $ used in the proof of theorem 14.8.1 therefore gives
\[ \mathcal L_{u_ t}\ddot u_ t -\frac1{n-1}\operatorname {tr}\bigl ((Q_ t^{-1}\dot Q_ t)^2\bigr ) =\ddot F_ t+\ddot b_\gamma (F_ t). \]
Integrating against $\mu _{u_ t}^\gamma $ at this fixed parameter gives the second formula; no derivative of the measure is needed.
For affine $F_ t$, we have $\ddot F_ t=0$, and positivity of the density of $\mu _{u_ t}^\gamma $ proves concavity. If the second derivative vanishes, then $\dot Q_ t=0$, so $\dot\Gamma _{u_ t}=0$ and $\mathcal L_{u_ t}\dot u_ t=0$. By proposition 14.12.1, $\dot u_ t$ is constant, and the first differentiated equation forces $G$ to be constant. Conversely, adding a constant to $F$ changes only $b$, by the opposite constant. Finally, integrate the first derivative along the segment from $F$ to $G$: since $\mu _{u_ t}^\gamma $ is a probability measure, the derivative lies between $-\sup _ X(G-F)$ and $-\inf _ X(G-F)$, which proves the Lipschitz bound.
$\square$
Pome
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