# homogeneous processes and homogeneous-increment processes

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.

1. What are some homogeneous-increment process that are not homogeneous?

2. Is it possible to have some necessary and sufficient condition for a homogeneous-increment process to be homogeneous?

Thanks and regards!

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$$X_t=t$$