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Figure 7.13. The set up for the study of ''escape from a domain of attraction.'' The phase line for the Ornstein-Uhlenbeck process has a single, stable steady state at the origin. We surround the origin by an interval [A, B], where A < 0 and B > 0, and assume that Y(0) is in this interval. Because the long time limit of q(x,t,y,s), q(x), is positive outside of [A, B] (Eq. (7.49)), escape from the interval is guaranteed.

Perhaps the most interesting insight from Eq. 7.49 pertains to ''escapes from domains of attraction'' (which we will revisit in the next chapter). The phase line for the deterministic system the underlies the Ornstein-Uhlenbeck process has a single steady state at the origin (Figure 7.13). Suppose that we start the process at some point A < y < B. Equations (7.48) and (7.49) tell us that there is always positive probability that Y(t) will be outside of the interval [A, B]. In other words, the Ornstein-Uhlenbeck process will, with probability equal to 1, escape from [A, B]. As we will see in Chapter 8, how it does this becomes very important to our understanding of evolution and conservation.

When Ornstein and Uhlenbeck did this work, they envisioned that Y(t) was the velocity of a Brownian particle, experiencing friction (hence the relaxation proportional to velocity) and random fluctuations due to the smaller molecules surrounding it. We need to integrate velocity in order to find position, so if X(t) denotes the position of this particle

and now we have another stochastic integral to deal with. But that is the subject for a more advanced book (see Connections).

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