A Weak Reverse Holder Inequality for Caloric Measure

Following a result of Bennewitz-Lewis for non-doubling harmonic measure, we prove a criterion for non-doubling caloric measure to satisfy a weak reverse Holder inequality on an open set $\Omega$, assuming as a background hypothesis only that the essential boundary of $\Omega$ satisfies an appropriate parabolic version of Ahlfors-David regularity (which entails some backwards in time thickness). We also show that the weak reverse Holder estimate is equivalent to solvability of the initial Dirichlet problem with "lateral" data in $L^p$, for some $p<\infty$, in this setting.