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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 the delta(-like) initial 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 [75] to prove an n-point, upper-tail LDP for the KPZ equation and to characterize the corresponding spacetime limit shape.
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Jointly invariant measures for the Kardar–Parisi–Zhang equation
We give an explicit description of a family of jointly invariant measures of the KPZ equation singled out by asymptotic slope conditions. These are couplings of Brownian motions with drift, and can be extended to a cadlag process indexed by all real drift parameters. We name this process the KPZ horizon (KPZH). As a corollary, we resolve a recent conjecture by showing the existence of a random, countably infinite dense set of drift values at which the Busemann process of the KPZ equation is discontinuous. This signals instability, and shows the failure of the one force–one solution principle and the existence of at least two extremal semi-infinite polymer measures in the exceptional directions. The low-temperature limit of the KPZH is the stationary horizon (SH), the unique jointly invariant measure of the KPZ fixed point under the same slope conditions. The high-temperature limit of the KPZH is a coupling of Brownian motions that differ by linear shifts, which is jointly invariant under the Edwards–Wilkinson fixed point.
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- PAR ID:
- 10651177
- Publisher / Repository:
- Springer-Verlag
- Date Published:
- Journal Name:
- Probability Theory and Related Fields
- Volume:
- 192
- Issue:
- 1-2
- ISSN:
- 0178-8051
- Page Range / eLocation ID:
- 303 to 372
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00440-025-01362-z
