An upper bound of the lower tail of the mass of balls under the critical $2d$ stochastic heat flow
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908946581684224 |
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| author | Nakashima, Makoto |
| author_facet | Nakashima, Makoto |
| contents | We study the critical two-dimensional stochastic heat flow $\mathscr{Z}_t^{\vartheta}$, recently constructed as the scaling limit of directed polymers in a random environment and as the weak limit of the solution to a mollified stochastic heat equation. Focusing on the mass of balls $\mathscr{Z}_t^{\vartheta}(B_r(0),B_r(a))$
($a\in \mathbb{R}^2$, $r>0$), we establish an upper bound on its lower tail. As a consequence, we prove the integrability of the logarithm of $\mathscr{Z}_t^{\vartheta}(B_r(0),B_r(a))$
and its strict positivity. These results provide partial answers to open questions concerning the local behavior of $\mathscr{Z}_t^\vartheta$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_18080 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An upper bound of the lower tail of the mass of balls under the critical $2d$ stochastic heat flow Nakashima, Makoto Probability 60H17, 65C35 We study the critical two-dimensional stochastic heat flow $\mathscr{Z}_t^{\vartheta}$, recently constructed as the scaling limit of directed polymers in a random environment and as the weak limit of the solution to a mollified stochastic heat equation. Focusing on the mass of balls $\mathscr{Z}_t^{\vartheta}(B_r(0),B_r(a))$ ($a\in \mathbb{R}^2$, $r>0$), we establish an upper bound on its lower tail. As a consequence, we prove the integrability of the logarithm of $\mathscr{Z}_t^{\vartheta}(B_r(0),B_r(a))$ and its strict positivity. These results provide partial answers to open questions concerning the local behavior of $\mathscr{Z}_t^\vartheta$. |
| title | An upper bound of the lower tail of the mass of balls under the critical $2d$ stochastic heat flow |
| topic | Probability 60H17, 65C35 |
| url | https://arxiv.org/abs/2507.18080 |