\(L_p\)-theory for the fractional time stochastic heat equation with an infinite-dimensional fractional Brownian motion. (English) Zbl 1475.60118
Summary: In this paper, we study the \(L_p\)-theory of the fractional time stochastic heat equation
\[
\partial_t^{\alpha} u(t,x)=\triangle u(t,x)+f(t,x)+\sum_{k=1}^{\infty} \partial_t^{\gamma} \int_0^t g^k (s,x)\delta \beta_s^k,
\]
where \(\alpha, \gamma\in (0,1), \gamma <\alpha +\frac{1}{2}, \partial_t^{\alpha}\) denotes the Caputo derivative of order \(\alpha\), and \(\{ \beta^k, k=1,2,\ldots\}\) is a sequence of i.i.d. fractional Brownian motions with a same Hurst index \(H\in (\frac{1}{2},1)\). The integral with respect to fractional Brownian motion is the Skorohod integral. By using the Malliavin calculus techniques and fractional calculus, we obtain a generalized Littlewood-Paley inequality, and prove the existence and uniqueness of \(L_p\)-solution to such equation.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60H05 | Stochastic integrals |
| 60G22 | Fractional processes, including fractional Brownian motion |
Keywords:
stochastic heat equation; fractional Brownian motion; fractional calculus; Littlewood-Paley’s inequality; \(L_p\)-solutionReferences:
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