According to Rick Durrett's book "Stochastic Calculus" (p. 305) rescaled Ehrenfest chains converge to an Ornstein-Uhlenbeck process:
We have two urns containing a total of $2n$ balls. At each time step a randomly chosen ball switches its urn.
Durrett says
We expect the number of balls in the left urn to be about $n+C\sqrt{n}$.
Question 1: How is this claim justified?
(I believe that the expected number of balls should depend on the time step and initial distribution of balls too.)
Now let $Z^n_m$ be the number of balls in the left urn minus $n$ and let $Y^{1/n}_{m/n}=\frac{1}{\sqrt{n}}Z^n_m$ (so we expect it to be around $C$). The process $(Y^{1/n}_{m/n})_{m\in\mathbb{N}_0}$ has state space $S_n=\{k/\sqrt{n}:-n\leq k\leq n\}$. Durrett says
The transition probability is $$\Pi_{1/n}(x,x+n^{-1/2})=\frac{n-x\sqrt{n}}{2n}\quad \Pi_{1/n}(x,x-n^{-1/2})=\frac{n+x\sqrt{n}}{2n}$$
Question 2: Why are the transition probabilities given by these expressions?
(From the model description at each time step the number of balls should change, by $\pm 1$. So the sum of those two probabilities should be 1, right?)
Now we can check the conditions of theorem 7.1 on p. 297 (I think there are some typos in the book, but it's okay) to conclude that the sequence $(X^{1/n})_{n\in\mathbb{N}}$ of time-continuous processes $(X^{1/n}_t)_{t\geq 0}$ given by $X^{1/n}_t=Y^{1/n}_{[nt]/n}$ converges weakly to the solution $(X_t)_{t\geq 0}$ of the Ornstein-Uhlenbeck SDE (as $n\rightarrow \infty$) $$d X_t=-X_t dt + d B_t$$
Except for Donsker's theorem, I don't know much about finding proper rescalings of stochastic processes and determining their scaling limits. If there are other introductory texts you find helpful, please let me know!
Any help or comment is greatly appreciated!