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Event Type:

Analysis Seminar

Date/Time:

Monday, November 2, 2015 - 12:00 to 13:00

Location:

Kidder 356

Guest Speaker:

Mathew Titus

Institution:

Graduate Student

Abstract:

Let \((B_t)\) be a d-dimensional Brownian motion, and consider the stochastic

differential equation \(dX_t = x + B_t + \int_0^t b(X_s) ds\) where \(b(x)\) is a

gradient; that is \(b(x) = grad (V(x))\) for some potential function \(V\). The

Markov process so generated provides a semigroup operator, \(E[f(X_t) | X_0]\),

on distributions on euclidean space. There exist criteria to discern

whether there is a nontrivial invariant distribution which the system tends

to over time, i.e. tests for ergodicity. Under some mild conditions, the

common class of operators above does exhibit ergodic behavior. In this talk

we follow the work of Ganidis and discuss the question of how quickly the

distributions approach their invariant distribution. We find that the

growth rate of \(V\) will imply bounds on the rate of convergence.