An upper bound of the lower tail of the mass of balls under the critical $2d$ stochastic heat flow

We study the critical two-dimensional stochastic heat flow Zₜϑ, recently constructed as the scaling limit of directed polymers in a random environment and as the limit of the solution to a mollified stochastic heat equation. Focusing on the mass of balls Zₜϑ(Bᵣ(0),Bᵣ(a)) (a∈ R², $r>0$), we establish an upper bound on its lower tail. As a consequence, we prove the integrability of the logarithm of Zₜϑ(Bᵣ(0),Bᵣ(a)) and its strict positivity. These results provide partial answers to open questions concerning the local behavior of Zₜ^ϑ.