Ordered parameters and their prior distributions

I have two parameters in my Stan model mo and m1 and we know that m1 > m0, so I placed an ordering on them. I have then identified 2 prior distributions for the parameters. However, because they are ordered, do I have to define the relationship between m0 and m1 using a bivariate distribution?

parameters {
      real<lower=0> m0;
      real<lower=m0> m1;

model {
   // Priors
      m0 ~ student_t(N_obs0 - 1, lmean0, sqrt(s0sq/(N_obs0 - 1)));
      m1 ~ student_t(N_obs1 - 1, lmean1, sqrt(s1sq/(N_obs1 - 1)));

Stan has the ordered parameter type - 1.8 Ordered logistic and probit regression | Stan User’s Guide

Thanks! But how would I use the ordered_logistic() while keeping the student_t() distributions for m0 and m1?

If you specify a prior that does not respect the ordering constraint (or any constraint in Stan), then functionally what you get is a prior that is truncated by the constraint. So if you apply student t priors, you will get a joint prior corresponding to a bivariate t distribution truncated at the m0 = m1 line. The margins of this thing will not be student t. If you want student t margins, then you would need to work out for yourself the log probability density for some bivariate prior that has student t margins and respects the ordering constraint.