An old result of F.W.~Gehring asserts that if $f$ is an a.e.\ finite measurable function on the unit circle in the complex plane, then there is a harmonic function $h$ in the unit disk with non-tangential limits equal to $f$ at a.e.\ point of the circle. It's clear that $h$ is far from unique --- think of the Poisson integral $u$ of a singular probability measure on the circle, and consider $h+c\cdot u$, for any constant $c\in\RRR$.  

A conceptually related result, due to R.M.~Dudley (but anticipated by C.W.~Lamb), concerns a real-valued Brownian motion $\{B_t: 0\le t\le 1\}$, and asserts that if $F$ is a random variable measurable over $\sigma\{B_t: 0\le t\le 1\}$then there is a predictable process $H=(H_t)_{0\le t\le 1}$ with $\int_0^1 H_t^2\,dt<\infty$ a.s.\such that $F=\int_0^1 H_t\,dB_t$a.s. Once again, $H$ is not unique.

But if $F$ is integrable then we have the martingale $M_t:=\EEE[F\mid\FF_t]$ which admits a stochastic integral representation $M_t=\int_0^t K_s\,dB_s$, $0\le t\le 1$, a.s., and of course $F=M_1=\int_0^1K_s\,dB_s$, a.s.Under certain additional conditions we have $\EEE\int_0^1 K_s^2\,ds\le\EEE\int_0^1 H^2_s\,ds$, for any process $H$ as inDudley's stochastic integral representation of $F$.

My goal in this work is to gain a probabilistic understanding of Gehring's result, and to explore side conditions on $H$ ensuring some sort of uniqueness.

学术海报.pdf