Gonzalo Bley

(University of Virginia)

(University of Virginia)

Analysis Seminar

Thursday, 19 May, 2016, at 16:15-17:15, in Aud. D3 (1531-215)

Description:

A positive Brownian functional $\mathcal{A}_T$ on the finite-time horizon $[0, T]$ is a non-negative $L^2$ random variable on the canonical Wiener space of $\mathbb{R}^d$-valued, continuous paths defined on $[0, T]$ with respect to Wiener measure starting at some initial distribution. In this work we show how the Clark-Ocone formula, well-known from Malliavin Calculus, plus a simple supermartingale estimate can be used to provide an upper bound for the exponential moment of a given Brownian functional. One application of the method replaces the initial functional by another one, at the price of an upper bound and some constants and exponents. The effectiveness of the method depends on the functional at hand, although it can be applied in principle to any Brownian functional. We give concrete examples, arising from non-relativistic quantum mechanics and field theory, where the method delivers a ``less singular'' Brownian functional -- a finite number of applications then yield an explicit log-linear upper bound on the original exponential moment, of the form $e^{CT}$, for some non-negative constant $C$. Special types of Brownian functionals arise in many-body quantum mechanics, defined on a high-dimensional Brownian motion and involving the sum of several terms. By carefully constructing a combinatorial block design for the particles involved, we show how one can produce highly refined upper bounds, that in some cases have the correct dependence on the number of particles of the system involved.

The central applications of this work are in non-relativistic quantum field theory, and those interested in them should attend the first talk in the mathematical physics seminar. This talk is partly based on joint work with L. Thomas from the University of Virginia.

The central applications of this work are in non-relativistic quantum field theory, and those interested in them should attend the first talk in the mathematical physics seminar. This talk is partly based on joint work with L. Thomas from the University of Virginia.

Contact person: Jacob Schach Møller