THE COMPACT SUPPORT PROPERTY FOR SOLUTIONS TO THE STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH COLORED NOISE
SCIE
SCOPUS
- Title
- THE COMPACT SUPPORT PROPERTY FOR SOLUTIONS TO THE STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH COLORED NOISE
- Authors
- Han, Beom-Seok; Kim, Kunwoo; Yi, Jaeyun
- Date Issued
- 2024-01
- Publisher
- Society for Industrial and Applied Mathematics Publications
- Abstract
- We study the compact support property for nonnegative solutions of the following stochastic partial differential equations (SPDEs): ∂ tu = aijuxixj (t,x) + biuxi (t,x) + cu + h(t, x,u(t,x)) F (t,x), (t,x) ∈ (0,∞ )× Rd, where F is a spatially homogeneous Gaussian noise that is white in time and colored in space, and h(t, x,u) satisfies K 1uλ ≤ h(t, x,u) ≤ K(1+u) for λ ∈ (0, 1) and K ≥ 1. We show that if the initial data u0 ≥ 0 has a compact support, then, under the reinforced Dalang's condition on F (which guarantees the existence and the Hölder continuity of a weak solution), all nonnegative weak solutions u(t, ) have the compact support for all t > 0 with probability 1. Our results extend the works by Mueller and Perkins [Probab. Theory Related Fields, 93 (1992), pp. 325-358] and Krylov [Probab. Theory Related Fields, 108 (1997), pp. 543-557], in which they show the compact support property only for the one-dimensional SPDEs driven by space-time white noise on (0,∞ ) × R.
- URI
- https://oasis.postech.ac.kr/handle/2014.oak/123652
- DOI
- 10.1137/23M1557283
- ISSN
- 0036-1410
- Article Type
- Article
- Citation
- SIAM Journal on Mathematical Analysis, vol. 55, no. 6, page. 7665 - 7703, 2024-01
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