With a friend, we are trying to understand the article of Igor Wigman : "On the distribution of the nodal sets of random spherical Harmonics" :
https://arxiv.org/abs/0805.2768
We study the eigenfunctions of the Laplacian on the 2-dimensional sphere $\mathbb{S}^2$. It is known that the eigenvalues of the Laplacian :
$ \Delta f + E f = 0 $
are $E = E_n = n(n+1)$
and that the eigenspace $\mathscr{E}_n$ corresponding to $E_n$ has a dimension of $2n+1$ and we have found a $\mathbb{L}^2(\mathbb{S}^2)$ - orthonormal basis of $\mathscr{E}_n$ which is $(\eta_{1},...,\eta_{2n+1})$ (page 2 of the article).
We consider a random eigenfunction
$f(x) = \sqrt \frac{4\pi}{2n+1} \cdot \sum_{k=1}^{2n+1}a_k . \eta_k(x)$
Where $a_k$ are Gaussian $N(0,1)$, this allowes us to induce a Gaussian probability measure on $\mathscr{E}_n \cong \mathbb{R}^{2n+1}$ (page 3 of the article).
Therefore, $f(x).f(y)$ is a random variable and the "Spherical Harmonics Addition Theorem" gives us :
$\mathbb{E}[f(x)f(y)] = \frac{4\pi}{2n+1} \cdot \sum_{k=1}^{2n+1}\eta_k(x)\eta_k(y) = P_l(cos(d(x,y))) = u(x,y)$
with $P_l$ the Legendre function (I think that for our problem, the writing of this function is not necessary) and $d(x,y)$ the distance between $x$ and $y$ on the sphere. In this expression, we see that $u$ is rotationaly invariant :
$u(Rx,Ry) = u(x,y)$ where $R$ is any rotation of $\mathbb{S}^2$
As we are in $\mathbb{S}^2$, we will use the spherical coordinates $x=(\theta_x , \varphi_x)$
And our problem is that we do not understand the affirmation page 13 :
"By the rotational symmetry on $\mathbb{S}^2$ " :
$\mathbb{E} [ ( \frac{\partial }{\partial \theta}f(x))^2 ] \\ =\mathbb{E}[\frac{(\frac{\partial }{\partial \theta}f(x))^2 + (\frac{1}{sin \theta} \frac{\partial }{\partial \varphi}f(x))^2}{2}]\\ =\frac{1}{2}\mathbb{E} \left(\nabla f(x) \cdot \nabla f(x) \right)\\ =\frac{1}{2 \cdot |\mathbb{S}^2|} \int_{\mathbb{S}^2} \mathbb{E} \left(\nabla f(x) \cdot \nabla f(x) \right) dx \\$
What we've already done : Using the definition of the derivation as the limit of the rate of change along a curve and the fact that $u(.,.)$ is rotationaly invariant, that the expression $\mathbb{E} [ ( \frac{\partial }{\partial \theta}f(x))^2 ]$ equals $\partial_{\theta_x} \partial_{\theta_y} u |_{x=y}$ alongside a suitable rotation "around" one of the variables we have shown the first and second equality.
Question : That said, we would like to know where the last equality comes from.
Thank you for your help !
The idea, for $\mathbb{S}^m$, would be (following Wigman's notation): $$ \mathbb{E}\left(\partial_i f(x)^2\right)=\mathbb{E}\left(\langle \nabla f(x), e_i^x\rangle^2\right)=\frac1m m\mathbb{E}\left(\langle \nabla f(x), e_i^x\rangle^2\right)=\frac1m \sum_{i=1}^m\mathbb{E}\left(\langle \nabla f(x), e_i^x\rangle^2\right), $$ thus, $$ \mathbb{E}\left(\partial_i f(x)^2\right)=\frac1m\sum_{i=1}^m\mathbb{E}\left(\langle \nabla f(x), \nabla f(x)\rangle\right). $$ By rotational symmetry, $$ \mathbb{E}\left(\langle \nabla f(x), \nabla f(x)\rangle\right)=\mathbb{E}\left(\langle \nabla f(Rx), \nabla f(Rx)\rangle\right). $$ Therefore, $$ \mathbb{E}\left(\partial_i f(x)^2\right)=\frac{1}{m|\mathbb{S}^m|}\int_{\mathbb{S}^m} \mathbb{E}\left(\langle \nabla f(y), \nabla f(y)\rangle\right)d\sigma_{\mathbb{S}^m}(y). $$