August 2023 Dissipation in parabolic SPDEs II: Oscillation and decay of the solution
Davar Khoshnevisan, Kunwoo Kim, Carl Mueller
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 59(3): 1610-1641 (August 2023). DOI: 10.1214/22-AIHP1289


We consider a stochastic heat equation of the type, tu=x2u+σ(u)W˙ on (0,)×[1,1] with periodic boundary conditions and non-degenerate positive initial data, where σ:RR is a non-random Lipschitz continuous function and W˙ denotes space-time white noise. If additionally σ(0)=0 then the solution is known to be strictly positive; see Mueller (Stoch. Stoch. Rep. 37 (1991) 225–245). In that case, we prove that the oscillation of the logarithm of the solution decays sublinearly as time tends to infinity. Among other things, it follows that, with probability one, all limit points of t1supx[1,1]logu(t,x) and t1infx[1,1]logu(t,x) must coincide. As a consequence of this fact, we prove that, when σ is linear, there is a.s. only one such limit point and hence the entire path decays almost surely at an exponential rate.

On considère une équation de la chaleur stochastique du type, tu=x2u+σ(u)W˙ sur (0,)×[1,1] avec des conditions aux limites périodiques et des données initiales positives non dégénérées, où σ:RR est une fonction continue lipschitzienne non aléatoire et W˙ désigne un bruit blanc spatio-temporel. Si en plus σ(0)=0, alors la solution est strictement positive, voir Mueller (Stoch. Stoch. Rep. 37 (1991) 225–245). Dans ce cas, nous prouvons que l’oscillation du logarithme de la solution décroît de manière souslinéaire lorsque le temps tend vers l’infini. Entre autres, il s’ensuit que, avec probabilité un, tous les points limites de t1supx[1,1]logu(t,x) et t1infx[1,1]logu(t,x) doivent coïncider. En conséquence de ce fait, nous prouvons que, quand σ est linéaire, il n’y a presque sûrement qu’un seul point limite. Par conséquent, toute la trajectoire décroît exponentiellement presque sûrement.

Funding Statement

Research supported in part by the NSF grant DMS-1855439 [D.K.], NRF grants 2019R1A5A1028324 and 2020R1A2C 4002077 [K.K.], and Simons Foundation grant 513424 [C.M.].


We thank Vlad Bally and Francesco Russo for help with the French abstract.


Download Citation

Davar Khoshnevisan. Kunwoo Kim. Carl Mueller. "Dissipation in parabolic SPDEs II: Oscillation and decay of the solution." Ann. Inst. H. Poincaré Probab. Statist. 59 (3) 1610 - 1641, August 2023.


Received: 26 January 2022; Revised: 24 May 2022; Accepted: 31 May 2022; Published: August 2023
First available in Project Euclid: 31 August 2023

MathSciNet: MR4635721
Digital Object Identifier: 10.1214/22-AIHP1289

Primary: 60H15
Secondary: 35R60

Keywords: Almost sure Lyapunov exponents , decay , ‎oscillation‎ , Stochastic heat equation

Rights: Copyright © 2023 Association des Publications de l’Institut Henri Poincaré


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Vol.59 • No. 3 • August 2023
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