Let $X_t$ be a continuous stochastic process which is Hölder continuous for every $a<1/2$.
Let's also assume that we know the probability distribution of the process, i.e. if we fix $t_1>0$, then the random variable $X_{t_1}$ has a known probability density function $f_{t_1}$. We assume that $f_{t_1}$ is a smooth function for all $t_1>0$ and the mapping $t_1 \mapsto f_{t_1}$ is continuous.
If we fix $t_2>t_1>0$, what can we say about the probability distribution of $Y_{t_2,t_1} = X_{t_2} - X_{t_1}$? Is there a simple, closed form formula that gives its probability density function? Do we need any more assumptions to answer this question?