If I understand correctly, a homogeneous process must be homogeneous-increment, because for example, $L(X_{t_2} - X_{t_1}) = L(X_{t_2 + \tau} - X_{t_1 + \tau})$ can be proven by $L(X_{t_2} - X_{t_1}, X_{t_1}) = L(X_{t_2}, X_{t_1}) = L(X_{t_2 + \tau}, X_{t_1 + \tau}) = L(X_{t_2 + \tau} - X_{t_1 + \tau}, X_{t_1 + \tau})$ and then marginalize $L(X_{t_2} - X_{t_1}, X_{t_1})$ wrt $X_{t_1}$, and $L(X_{t_2 + \tau} - X_{t_1 + \tau}, X_{t_1 + \tau})$ wrt $ X_{t_1 + \tau}$ and the equality still holds.
I am now wondering about the reverse.
What are some homogeneous-increment process that are not homogeneous?
Is it possible to have some necessary and sufficient condition for a homogeneous-increment process to be homogeneous?
Thanks and regards!