A Sobolev space theory for the stochastic partial differential equations with space-time non-local operators
- Authors
- Kim, Kyeong-Hun; Park, Daehan; Ryu, Junhee
- Issue Date
- 9월-2022
- Publisher
- SPRINGER BASEL AG
- Keywords
- Stochastic partial differential equations; Sobolev space theory; Space-time non-local operators; Maximal L-p-regularity; Space-time white noise
- Citation
- JOURNAL OF EVOLUTION EQUATIONS, v.22, no.3
- Indexed
- SCIE
SCOPUS
- Journal Title
- JOURNAL OF EVOLUTION EQUATIONS
- Volume
- 22
- Number
- 3
- URI
- https://scholar.korea.ac.kr/handle/2021.sw.korea/143744
- DOI
- 10.1007/s00028-022-00813-7
- ISSN
- 1424-3199
- Abstract
- We deal with the Sobolev space theory for the stochastic partial differential equation (SPDE) driven by Wiener processes partial derivative(alpha)(t)u = (phi(Delta)u + f (u)) + delta(beta)(t) Sigma(infinity)(k=1) integral(t)(0) g(k) (u)dw(s)(k), t > 0, x is an element of R-d as well as the SPDE driven by space-time white noise partial derivative(alpha)(t)u = (phi(Delta)u + f (u)) + partial derivative(beta-1)(t), t > 0, x. R-d. Here, alpha is an element of (0, 1), ss < alpha + 1/2, {w(t)(k) : k = 1, 2,...} is a family of independent one-dimensional Wiener processes and. (W) over dot is a space-timewhite noise defined on [0,infinity)xR(d). The time non-local operator partial derivative(alpha)(t) denotes the Caputo fractional derivative of order alpha, the function phi is a Bernstein function, and the spatial non-local operator phi(Delta) is the integro-differential operator whose symbol is -phi(vertical bar xi vertical bar(2)). In other words, phi(Delta) is the infinitesimal generator of the d-dimensional subordinate Brownian motion. We prove the uniqueness and existence results in Sobolev spaces and obtain the maximal regularity results of solutions.
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