Path integral approach to analytic continuation of Liouville theory: the pencil region

We study the problem of analytic continuation of Liouville Conformal Field Theory using the probabilistic approach of David, Kupiainen, Rhodes and Vargas [DKRV16] based on the theory of Gaussian Multiplicative Chaos. The key idea is to apply stochastic calculus techniques to some Brownian motions associated to the Gaussian Free Field. We strengthen the results in [KRV17] and are able to provide rigorous justification of analytic continuation of Liouville Theory in an infinite trianglar region, which we call the pencil region. This is in accordance with the physics literature [HMW11]. Along the lines, we develop new tools and estimates of Gaussian measures and Liouville correlation functions using this probabilistic approach.