proof that the covariance function for a fractional Brownian motion / fractional Gaussian free field is well defined Given $0 < t_1 < \dots < t_n$, we can show that the matrix $\Omega$ whose entries are defined by $M_{i,j} = min(t_i,t_j)$ is symmetric definite positive.
The proof is immediate once one recognizes that this is the covariance matrix of $(W_{t_i})$ where $W$ is a Brownian motion. One would write that:
$$M = t_1 E_1 E_1^T + (t_2-t_1) E_2 E_2^T + \dots + (t_n-t_{n-1}) E_n E_n^T$$
where $E_i$ is a vector of 0s except for the elements starting at position $i$.
One can note that we can also write $M_{i,j} = min(t_i, t_j) = t_i + t_j -|t_i-t_j|$.
I would like to prove that the matrix whose entries are $M_{i,j} = t_i^{2h}+t_j^{2h}-|t_i-t_j|^{2h}$ is also positive. This is the covariance matrix of the fractional brownian motion, but I would like to prove the positivity without relying on this fact. I am merely mentioning it to give context and intuition.
Extra points if the proof applies for a (potentially fractional) Gaussian free field.
 A: Note that the covariance function of fBm with Hurst parameter $H \in (0,1)$ can be written as an integral\begin{align*}
C(s,t) &= \frac{1}{2} \left( t^{2 H} + s^{2 H} - |t - s|^{2 H} \right) \\
&= \frac{1}{c_H^2} \int_{\mathbb{R}} \left[ ((t-u)^+)^{H-1/2} - ((-u)^+)^{H-1/2} \right] \left[ ((s-u)^+)^{H-1/2} - ((-u)^+)^{H-1/2} \right] du
\end{align*} where $a^+ = \max(a,0)$ and $
c_H = \sqrt{ \frac{1}{2H} + \int_0^{\infty} ((1+u)^{H-1/2} - u ^{H-1/2})^2 du } < \infty \;.
$ See, e.g., Proposition 2.3 of

Nourdin, Ivan, Selected aspects of
fractional Brownian motion., Bocconi & Springer Series 4. Milano:
Springer (ISBN 978-88-470-2822-7/hbk). x, 122 p. (2012).
ZBL1274.60006.

Applying this integral form of the covariance function \begin{align*}
\frac{1}{2} &\sum_{j=1}^n \sum_{i=1}^n x_i x_j M_{i,j} = \sum_{j=1}^n \sum_{i=1}^n  x_i x_j C(t_i, t_j) \\
&= \frac{1}{c_H^2} \int_{\mathbb{R}} \sum_{j=1}^n \sum_{i=1}^n x_i x_j  \left[ ((t_i-u)^+)^{H-1/2} - ((-u)^+)^{H-1/2} \right] \left[ ((t_j-u)^+)^{H-1/2} - ((-u)^+)^{H-1/2} \right] du \\
&= \frac{1}{c_H^2} \int_{\mathbb{R}} \left[ \sum_{i=1}^n x_i \left[ ((t_i-u)^+)^{H-1/2} - ((-u)^+)^{H-1/2} \right] \right]^2 du \ge 0 \quad \text{as required.}
\end{align*}
This proves that the $n \times n$ matrix $M$ is positive semidefinite, which implies that $M$ is the covariance matrix of some Gaussian random $n$-vector.
