In other words, the time to escape from [— L, L] when starting at x > 0 grows like T (x) ~ exp(L2/e). Thus, on average it takes a very long time to escape from a domain of attraction.
This conclusion - of a very long average time to escape - accounts for the reality that most trajectories starting at x > 0 will be drawn back towards the origin and spend a long time there before ultimately escaping. However, now let us focus on a special subset of trajectories: those which start at x > 0 and escape (through L) without ever having returned to the origin. We can thus define i(x, t) = Prfexit (0, L] by time t without ever having returned to
Now, since w(x, t) is the probability of exiting from (0, L] without having crossed 0 by time t, wt(x, t) is the probability density for the time of exit. That is ut(x, t)dt = Prfexit from (0,L] in the interval t, t + dt without having crossed 0|X(0) = x}
and consequently the mean time for trajectories that start at x and exit without having crossed 0 is h(x) =
so that w(x) is the probability of ever escaping from (0, L] without first revisiting L. We recognize that m(x, t) Prfexiting by time t and never hitting 0|X(0) = x > 0}
Was this article helpful?