Open Access
2023 Phase analysis for a family of stochastic reaction-diffusion equations
Davar Khoshnevisan, Kunwoo Kim, Carl Mueller, Shang-Yuan Shiu
Author Affiliations +
Electron. J. Probab. 28: 1-66 (2023). DOI: 10.1214/23-EJP983


We consider a reaction-diffusion equation of the type


subject to a “nice” initial value and periodic boundary, where T=[1,1] and W˙ denotes space-time white noise. The reaction term V:RR belongs to a large family of functions that includes Fisher–KPP nonlinearities [V(x)=x(1x)] as well as Allen-Cahn potentials [V(x)=x(1x)(1+x)], the multiplicative nonlinearity σ:RR is non random and Lipschitz continuous, and λ>0 is a non-random number that measures the strength of the effect of the noise W˙.

The principal finding of this paper is that: (i) When λ is sufficiently large, the above equation has a unique invariant measure; and (ii) When λ is sufficiently small, the collection of all invariant measures is a non-trivial line segment, in particular infinite. This proves an earlier prediction of Zimmerman et al. (2000). Our methods also say a great deal about the structure of these invariant measures.

Funding Statement

Research supported in part by the NSF grant DMS-1855439, DMS-1608575, and DMS-1307470 [D.K.], the NRF grants 2019R1A5A1028324, 2021R1A6A1A10042944, and RS-2023-00244382 [K.K], a Simons grant [C.M.], and MOST grant MOST109-2115-M-008-006-MY2 [S.-Y. S.]. Parts of this research were funded by the NSF grant DMS-1440140 while D.K., K.K., and C.M. were in residence at the Mathematical Sciences Research Institute at UC Berkeley in Fall of 2015. D.K., K.K., and C.M. wish to thank the Banff International Research Center for their support under the Research in Teams Program.


Download Citation

Davar Khoshnevisan. Kunwoo Kim. Carl Mueller. Shang-Yuan Shiu. "Phase analysis for a family of stochastic reaction-diffusion equations." Electron. J. Probab. 28 1 - 66, 2023.


Received: 26 December 2021; Accepted: 25 June 2023; Published: 2023
First available in Project Euclid: 27 July 2023

MathSciNet: MR4620551
zbMATH: 07733582
arXiv: 2012.12512
MathSciNet: MR4529085
Digital Object Identifier: 10.1214/23-EJP983

Primary: 60H15
Secondary: 35R60

Keywords: Invariant measures , phase transition , Stochastic partial differential equations

Vol.28 • 2023
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