Math Colloquium | Brownian Motion on Aff(R) and Its Approximation by Random Walk, Feb. 16

Dr. Stanislav Molchanov

The Mathematics Colloquium series presents Dr. Stanislav Molchanov on “Brownian Motion on Aff(R) and its approximation by the random walk” on Friday, Feb. 16, at 4:10 p.m. in Morton 219.

Molchanov is Professor of Mathematics at the University of North Carolina-Charlotte.

Abstract: For any Lee group G, one can define (in terms of the corresponding Lee algebra) the Laplacian, parabolic heat semigroup and the Brownian motion.  The natural simulation of this Brownian motion can be based on the symmetric random walk on the subgroup G’ ⊂ G, generated by finitely many elements of G. In some situations (the nilpotent matrix groups , similar to Heisenberg group H3), G’ is the discrete subgroup of G with arbitrary small-scale lattice, then the local limit theorems indicate that random walk on G’ gives the “good” approximation for the Brownian motion.

But typically finite generated subgroups G’ ⊂ G are not the lattices but chaotically distributed in G. In this case, one can use the new class of “quasilocal”limit theorems. This important topic will be illustrated by the Brownian motion on the particular solvable Lee group Aff(R). This Brownian motion is related (due to M. Yor) to the financial mathematics (Asian options etc).