Abstract
This paper is concerned with the following stochastically fractional heat equation on(t,x)∈[0,T]×Rddriven by fractional noise:∂u(t,x)/∂t=Dδαu(t,x)+WH(t,x)⋄u(t,x), where the Hurst parameterH=(h0,h1,…,hd)and⋄denotes the Skorokhod integral. A unique solution of that equation in an appropriate Hilbert space is constructed. Moreover, the Lyapunov exponent of the solution is estimated, and the Hölder continuity of the solution on both space and time parameters is discussed. On the other hand, the absolute continuity of the solution is also obtained.
Highlights
IntroductionThere has been intense interest (see, e.g., Podlubny [1], Samko et al [2], Heydari et al [3, 4], Cattani et al [5, 6], Liao [7], and Hu [8]) in fractional calculus and its applications
There has been intense interest in fractional calculus and its applications
Many mathematical problems in physics and engineering with respect to systems and processes are represented by a kind of equations, more precisely fractional order differential equations driven by fractional order noise
Summary
There has been intense interest (see, e.g., Podlubny [1], Samko et al [2], Heydari et al [3, 4], Cattani et al [5, 6], Liao [7], and Hu [8]) in fractional calculus and its applications. Boulanba et al [20] studied the existence, uniqueness, Holder regularity, and absolute continuity of the solution for a class of fractional stochastic partial differential equations driven by spatially correlated noise. Nualart and Ouknine [31] explored the existence and uniqueness of mild solution to a class of second-order heat equations with additive fractional noise (fractional in time and white in space) when the Hurst parameter H > 1/2. Tindel et al [32] studied a linear stochastic evolution equation driven by an infinite-dimensional fBm in the cases of the Hurst parameter above and below 1/2, respectively. Hu and Nualart [33] studied the d-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of an fBm with H ∈ (0, 1) in time.
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