Question: Calculate the expected time of first visit to state 2 given we start in state 2.
Is the answer to this the mean recurrence time of 2 or simply zero?
I at first thought that the answer was zero, since we start in state 2 and therefore take zero time to get there. But then I thought of the mean recurrence time, and was confused as to what to do.
The answer depends exactly on how the first visit to a given state has been defined. In my experience convention says that the answer is zero. If the question asked for the expected return time to state 2 then you would indeed be after the mean recurrence time. Hope this clears things up.
To further clarify the matter:
We define the hitting time on i to be $$ H_i = \inf(n \ge 0:X_n=i), $$ and the first passage time to i to be $$ T_i = \inf(n \ge 1 : X_n =i). $$ Note that we can have a hitting time of zero, as in your case, but the first passage time cannot be zero. The mean recurrence time is then defined as $$ m_i = E_i (T_i) = E(T_i |X_0=i), $$ which is strictly greater than zero.