Let $(X_t)$ be a real sample continuous stochastic process with density function $f_t$. Let $$Y_t= X_t - (1-t) X_0 - tX_1, \, t \in [0,1].$$
Suppose that $(Y_t)$ is Markov with regard to its own filtration. Does it mean it is forcibly a Gaussian process? Or can we find a real sample continuous stochastic process $(X_t)$, that has a density function, such that $(Y_t)$ is Markov and not Gaussian?
Bonus: if $(Y_t)$ is Markov with regard to the filtration of $(X_t)$, does it change the answer to the above question?