Question
Skorokhod embedding allows any mean-zero, finite variance random variable to be viewed as one of these processes at a suitably chosen stopping time. For 10 points each:
[10e] Name these continuous-time processes that have independent, Gaussian increments and continuous sample paths. These Wiener processes are used to model particles in a medium.
ANSWER: Brownian motion
[10m] Letting tau be the exit time from a region, the expected value of any function of a Brownian motion at tau has this convenient property. The mean-value property applies to these functions, whose Hessians have zero trace.
ANSWER: harmonic functions
[10h] By choosing a suitable harmonic function to analyze exit times of annuli, one can show that standard Brownian motion returns to any neighborhood of zero in precisely this situation. Likewise, simple random walk is recurrent in precisely this situation, inspiring a joke about a drunkard.
ANSWER: dimensions one and two [accept equivalents like recurrent for dimensions less than three; accept a line and a plane] (The joke is that “a drunk man will find his way home, but a drunk bird may get lost forever.”)
<Morrison, Other Science>
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