Dense blowup for parabolic SPDEs
- Dense blowup for parabolic SPDEs
- Le Chen; Jingyu Huang; Davar Khoshnevisan; KIM, KUNWOO
- Date Issued
- Institute of Mathematical Statistics
- The main result of this paper is that there are examples of stochastic partial differential equations [hereforth, SPDEs] of the type
such that the solution exists and is unique as a random field in the sense of Dalang  and Walsh , yet the solution has unbounded oscillations in every open neighborhood of every space-time point. We are not aware of the existence of such a construction in spatial dimensions below 33.
En route, it will be proved that when σ(u)=uσ(u)=u there exist a large family of parabolic SPDEs whose moment Lyapunov exponents grow at least sub exponentially in its order parameter in the sense that there exist A1,β∈(0,1)A1,β∈(0,1) such that
γ––(k):=lim inft→∞t−1infx∈R3logE(|u(t,x)|k)⩾A1exp(A1kβ)for all k⩾2.
γ_(k):=lim inft→∞t−1infx∈R3logE(|u(t,x)|k)⩾A1exp(A1kβ)for all k⩾2.
This sort of “super intermittency” is combined with a local linearization of the solution, and with techniques from Gaussian analysis in order to establish the unbounded oscillations of the sample functions of the solution to our SPDE.
- Article Type
- Electronic Journal of Probability, vol. 24, no. 118, page. 1 - 33, 2019-10
- Files in This Item:
- There are no files associated with this item.
Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.