On Complex Hypercontractivity

We give a new proof of a hypercontractivity theorem for the Mehler transform with a complex parameter, earlier proved by Weissler (1979,J. Funct. Anal.32, 102–121) and Epperson (1989,J. Funct. Anal.87, 1–30). The proof uses stochastic integrals and Itô calculus. The method also yields new proofs of some related results.