Are there simple conditions on an $d \times d$ matrix B under which $$ f(t, s) = \exp(—B |t — s|^\alpha), \qquad \alpha \in (0, 2), \qquad t, \, s \in \mathbb{R}$$ is positive definite as a matrix-valued function (see def below)?
Is it true in general that if $r(t, s)$ is a positive definite matrix-valued function, then so is $\exp(r(t, s))$? This is obviously true if $r$ is scalar valued.
The reason for this particular question is that I am interested in the corresponding class of multivariate fractional Ornstein-Uhlenbeck processes.
P.S. By positive definiteness I mean $$ \sum_{i, j} \mathbf{x}_i^\top f (t_i, t_j) \, \mathbf{x}_j \geq 0 \qquad \forall \, t_i \in \mathbb{R}, \quad \mathbf{x}_i \in \mathbb{R}^d. $$