Consider the Gaussian stochastic process (in $\mathbb R^2$ without loss of generality) $Z_t=\left( X_t, Y_t\right)$. What is the conditional law of $X_t | Y_t$ ?
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If $Z_t = (X_t,Y_t)$ is a multivariate Gaussian, then $X_t|Y_t$ will also be Gaussian. This can be shown by computing the pdf for $X_t|Y_t$, using the pdf for a multivariate Gaussian, and the formula for conditional density functions.