High moments of the SHE in the clustering regimes
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866915031168319488 |
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| author | Tsai, Li-Cheng |
| author_facet | Tsai, Li-Cheng |
| contents | We analyze the high moments of the Stochastic Heat Equation (SHE) via a transformation to the attractive Brownian Particles (BPs), which are Brownian motions interacting via pairwise attractive drift. In those scaling regimes where the particles tend to cluster, we prove a Large Deviation Principle (LDP) for the empirical measure of the attractive BPs. Under what we call the 1-to-$n$ initial-terminal condition, we characterize the unique minimizer of the rate function and relate the minimizer to the spacetime limit shapes of the Kardar--Parisi--Zhang (KPZ) equation in the upper tails. The results of this paper are used in the companion paper Lin and Tsai (2023) to prove an $n$-point, upper-tail LDP for the KPZ equation and to characterize the corresponding spacetime limit shape. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_14375 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | High moments of the SHE in the clustering regimes Tsai, Li-Cheng Probability Mathematical Physics We analyze the high moments of the Stochastic Heat Equation (SHE) via a transformation to the attractive Brownian Particles (BPs), which are Brownian motions interacting via pairwise attractive drift. In those scaling regimes where the particles tend to cluster, we prove a Large Deviation Principle (LDP) for the empirical measure of the attractive BPs. Under what we call the 1-to-$n$ initial-terminal condition, we characterize the unique minimizer of the rate function and relate the minimizer to the spacetime limit shapes of the Kardar--Parisi--Zhang (KPZ) equation in the upper tails. The results of this paper are used in the companion paper Lin and Tsai (2023) to prove an $n$-point, upper-tail LDP for the KPZ equation and to characterize the corresponding spacetime limit shape. |
| title | High moments of the SHE in the clustering regimes |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2304.14375 |