We consider two Ito equations that evolve on different time scales. The equations are fully coupled in the sense that all of the coefficients may depend on both the ``slow'' and the ``fast'' variables and the diffusion terms may be correlated. The diffusion term in the slow process is small. A large deviation principle is obtained for the joint distribution of the slow process and of the empirical process of the fast variable. By projecting on the slow and fast variables, we arrive at new results on large deviations in the averaging framework and on large deviations of the empirical measures of ergodic diffusions, respectively. The proof relies on the property that an exponentially tight sequence of probability measures on a metric space is large deviation relatively compact. The identification of the large deviation rate function is accomplished by analyzing the large deviation limit of an exponential martingale.
On large deviations of coupled diffusions with time scale separation
Mar. 12, 2015 3pm (Webber 20…
Algebraic Geometry
Jason Cantarella (University of Georgia)
X
Knutson-Hausmann and Howard-Manon-Millson have given descriptions of the space of polygons of fixed edge lengths in R^3 as symplectic reductions of complex Grassmannians or products of spheres. These structures turn out to be genuinely useful in analyzing the geometric probability theory of random closed walks, and we give some example theorems along these lines. We compare these to the picture of polygon space as a GIT quotient. The applications of this picture are currently more mysterious. The talk describes joint work with Clayton Shonkwiler (CSU), Tetsuo Deguchi (Ochanomizu University), Rob Kusner (U Mass), and Alexander Grosberg (NYU).