Fractional stochastic wave equation driven by a Gaussian noise rough in space

Spatial averages for the parabolic Anderson model driven by rough noise

“…. > s σ(n) > 0; we refer readers to Hu et al (2015); Huang et al (2017a); Song et al (2020) for the rigorous derivation of (2.3).…”

“…Previously the random field solution theory for the stochastic heat equation driven by a Gaussian noise white in time and rough in space, was restricted to the range H 1 ∈ (1/4, 1/2) (see, for instance, Balan et al, 2015;Hu et al, 2017Hu et al, , 2018. Condition H 0 + H 1 > 3/4 in case (H2) breaks the barrier 1/4, which is required under (H1) if ϕ(x) = |x| 1−2H1 , and it has been observed in Song et al (2020) that the random field solution exists for the PAM when H 0 ∈ (1/2, 1), H 1 ∈ (0, 1/2) and H 0 + H 1 > 3/4.…”

Solving the hyperbolic Anderson model 1: Skorohod setting

“…(iii) In the special case d = 1, a study of the fractional space-time noise was carried out in [15]. More specifically [15] handled the case of a fractional noise in time with index α 0 ∈ (0, 1), while γ was rougher than in [1,3].…”

“…(iii) In the special case d = 1, a study of the fractional space-time noise was carried out in [15]. More specifically [15] handled the case of a fractional noise in time with index α 0 ∈ (0, 1), while γ was rougher than in [1,3]. Namely in [15] the spatial component of the noise is assumed to be the distributional second derivative of the function x → |x| 2H with H < 1/2.…”

Decompositions of stochastic convolution driven by a white-fractional Gaussian noise

“…The problem of stochastic partial differential equations (SPDEs) driven by this Gaussian process has been widely studied during the past two decades. Here, we only list some recent results [7][8][9]15,16,22].…”