An $L^p$-Variational Formula on Wiener Space

We establish a stochastic control representation for $L^p$-norms on Wiener space. For every $p\ge1$ and every non-negative universally measurable functional $φ$, we show that $$\|φ(W)\|_p = \sup_a \mathbb{E}\Big[ e^{-\frac12\int_0^T\|a_t\|^2\,dt} φ\Big( W+\sqrt{p-1}\int_0^\cdot a_t\,dt \Big) \Big],$