Let $H \in (0, 1)$, $D \in \mathbb{R}$ and assume that the following function $$ r ( t, s ) = \frac{1}{2} \, \Big[ t^{2H} + s^{2H} - | t - s |^{2H} \Big] + D \, t^H s^H, \quad t, \, s \geq 0 $$ is positive definite. Then, clearly $D \geq -1$, since otherwise $r(t, t)$ is negative. It is also clear that if $D \geq 0$, then this is a covariance of $B_H (t) + \sqrt{D} \, t^H \, \xi$, where $\xi$ is independent of $B_H$ standard normal rv. Hence, $r$ is again positive definite.

Question: can $D \in (-1,0)$?

May be related to A simple decomposition for fractional Brownian motion with parameter $H<1/2$, but I didn't find the connection.