Let $(X_t)_{t\in[0,1]}$ be a centered Gaussian Process, the mapping $(t,\omega)\rightarrow X_t(\omega)$ is measurable, and $K(s,t)=Cov(X_s,X_t)$ the covariance matrix. Assume further that the mapping $t\rightarrow X_t$ from $[0,1] \, \text{into} \, L^2$ is continuous. (This is equivalent with the continuity of of $K(s,t)$ into $[0,1]^2$).
Question: Let K be twice continuously differentiable. Show that for every $t \in[0,1]$ the limit $$\dot X_t:= \lim_{s \rightarrow t} \frac{X_s-X_t}{s-t}$$ exists in $L^2(\Omega)$.Further show that $\dot X_t$ is a centered Gaussian Process and compute its Covariance function.
My understanding, is that the covariance will be $K_{st}(s,t)$ the mixed partial derivative, but I don't see how to deal with technicalities. Any help? Good reference?