Proposition 17.3.4.9086ABF36774 Let $u,v\in \mathcal E^\infty (X,\theta ,\phi )$ and put $u_ t=(1-t)u+tv$ for $0\leq t\leq 1$. Then
46190A7C1B2F
17.3.5
\begin{equation} \frac{\mathrm d}{\mathrm dt}E_\theta ^\phi (u_ t) =\int _ X(v-u)\, \mu _{u_ t}, \end{equation}
with one-sided derivatives at the endpoints, and
BC6BC1B194C2
17.3.6
\begin{equation} E_\theta ^\phi (v)-E_\theta ^\phi (u) =\frac1{n+1}\sum _{j=0}^ n\int _ X(v-u) \langle \theta _ v^ j\wedge \theta _ u^{n-j}\rangle . \end{equation}
In particular, the energy is increasing, and $E_\theta ^\phi (u+C)=E_\theta ^\phi (u)+Cm$ for every constant $C$.
Proof.
Let $w\in \mathcal E^\infty (X,\theta ,\phi )$. For $0\leq r\leq 1$ the potential $\phi +r(w-\phi )=(1-r)\phi +rw$ again belongs to $\mathcal E^\infty (X,\theta ,\phi )$, and multilinearity gives
\[ \mu _{\phi +r(w-\phi )}=\sum _{j=0}^ n\binom nj r^ j(1-r)^{n-j}\langle \theta _ w^ j\wedge \theta _\phi ^{n-j}\rangle . \]
Since $\binom nj\int _0^1r^ j(1-r)^{n-j}\, \mathrm dr=1/(n+1)$, as one sees by integrating by parts $j$ times, we obtain
73519BE96F0E
17.3.7
\begin{equation} E_\theta ^\phi (w)=\int _0^1\int _ X(w-\phi )\, \mu _{\phi +r(w-\phi )}\, \mathrm dr. \end{equation}
Now put $h=v-u$ and $q_{r,t}=\phi +r(u_ t-\phi )$. Since $\theta _{q_{r,t}}=(1-r)\theta _\phi +r(1-t)\theta _ u+rt\theta _ v$, the measures $\mu _{q_{r,t}}$ are polynomials in $(r,t)$ whose coefficients are mixed measures of total mass $m$, and the weights $u_ t-\phi =(u-\phi )+th$ are bounded. All expressions below are therefore polynomials in $(r,t)$ with finite coefficients, and the derivatives are derivatives of polynomials. Identities between such polynomials of measures follow from the corresponding identities between polynomials in commuting variables $x,y,z$, by substituting the mixed measure $\langle \theta _\phi ^ a\wedge \theta _ u^ b\wedge \theta _ v^ c\rangle $ for the monomial $x^ ay^ bz^ c$; multilinearity shows that this substitution is compatible with the products occurring here. For instance, the identities $\partial _ t\ell ^ n=nr(z-y)\ell ^{n-1}$ and $\partial _ r\ell ^ n=n\bigl ((1-t)y+tz-x\bigr )\ell ^{n-1}$ for $\ell =(1-r)x+r(1-t)y+rtz$ give
25D7DD3C6946
17.3.8
\begin{equation} \partial _ t\mu _{q_{r,t}}=nr\, \mathrm{dd}^{\mathrm{c}}h\wedge \langle \theta _{q_{r,t}}^{n-1}\rangle ,\qquad \partial _ r\mu _{q_{r,t}}=n\, \mathrm{dd}^{\mathrm{c}}(u_ t-\phi )\wedge \langle \theta _{q_{r,t}}^{n-1}\rangle , \end{equation}
in the sense of (17.2.1). Differentiating (17.3.7) with $w=u_ t$ and using the first of these identities, we obtain
\[ \frac{\mathrm d}{\mathrm dt}E_\theta ^\phi (u_ t) =\int _0^1\left[\int _ Xh\, \mu _{q_{r,t}} +nr\int _ X(u_ t-\phi )\, \mathrm{dd}^{\mathrm{c}}h\wedge \langle \theta _{q_{r,t}}^{n-1}\rangle \right]\mathrm dr. \]
By theorem 17.2.5, applied to the bounded quasi-psh differences $u_ t-\phi $ and $h$, the second integral equals $\int _ Xh\, \mathrm{dd}^{\mathrm{c}}(u_ t-\phi )\wedge \langle \theta _{q_{r,t}}^{n-1}\rangle $, which is $\frac1n\partial _ r\int _ Xh\, \mu _{q_{r,t}}$ by the second identity. The integrand in $r$ is therefore $\partial _ r\bigl [r\int _ Xh\, \mu _{q_{r,t}}\bigr ]$, and since $q_{1,t}=u_ t$,
\[ \frac{\mathrm d}{\mathrm dt}E_\theta ^\phi (u_ t)=\int _ Xh\, \mu _{u_ t}. \]
This is (17.3.5). Integrating it from $t=0$ to $t=1$, and expanding $\mu _{u_ t}=\sum _ j\binom nj t^ j(1-t)^{n-j}\langle \theta _ v^ j\wedge \theta _ u^{n-j}\rangle $ with the same beta integrals, gives (17.3.6). If $u\leq v$, every term of the cocycle formula is nonnegative, so the energy is increasing. Every mixed measure has total mass $m$, so taking $v=u+C$ in the cocycle formula gives the last assertion.
$\square$
Pome
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