Upper Bounds for Exponential Moments of Positive Brownian Functionals

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.