Coming up from $-\infty$ for KPZ via stochastic control

We derive a lower bound, independent of the initial condition, for the solution of the KPZ equation on the torus through its representation as the value function of a (conditional) stochastic control problem. With the same techniques, we also prove a bound for its oscillation, again independent of initial conditions, from which a Harnack type inequality for the rough heat equation (on the torus) can be obtained.