Abstract
We consider the narrow wedge solution to the Kardar–Parisi–Zhang stochastic PDE under the characteristic scaling of time, space and fluctuations. We study the correlation of fluctuations at two different times. We show that, when the times are close to each other, the correlation approaches one at a power-law rate with exponent , while, when the two times are remote from each other, the correlation tends to zero at a power-law rate with exponent . We also prove exponential-type tail bounds for differences of the solution at two space-time points.
Three main tools are pivotal to proving these results: (1) a representation for the two-time distribution in terms of two independent narrow wedge solutions, (2) the Brownian Gibbs property of the KPZ line ensemble and (3) recently proved one-point tail bounds on the narrow wedge solution.
Acknowledgments
Aspects of this project were prompted by the question that we have mentioned and that were asked by Amir Dembo and Jean-Dominique Deuschel during a conference at MSRI in fall 2010. In an earlier version of this work, Xuan Wu and Yier Lin pointed out a missing detail in the proof of Proposition ?? regarding controlling the probability of the event therein. The authors thank Xuan and Yier for their close reading and comments. The authors also wish to acknowledge useful comments from Patrik Ferrari, Pierre Le Doussal and Kazumasa Takeuchi. Ivan Corwin was partially supported by the Packard Fellowship for Science and Engineering and by the NSF through grants DMS-1811143 and DMS-1664650. Alan Hammond was partially supported by the NSF through grants DMS-1512908 and DMS-1855550 and by a Miller Professorship at U.C. Berkeley.
Citation
Ivan Corwin. Promit Ghosal. Alan Hammond. "KPZ equation correlations in time." Ann. Probab. 49 (2) 832 - 876, March 2021. https://doi.org/10.1214/20-AOP1461
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