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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 $Ω$, assuming as a background hypothesis only that the essential boundary of $Ω$ 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 &#34;lateral&#34; data in $L^p$, for some $p<\infty$, in this setting.