Abstract
In this article, we consider the stochastic wave and heat equations on $\mathbb{R}$ with non-vanishing initial conditions, driven by a Gaussian noise which is white in time and behaves in space like a fractional Brownian motion of index $H$, with $1/4 < H < 1/2$. We assume that the diffusion coefficient is given by an affine function $\sigma(x)=ax+b$, and the initial value functions are bounded and Holder continuous of order $H$. We prove the existence and uniqueness of the mild solution for both equations. We show that the solution is $L^{2}(\Omega)$-continuous and its $p$-th moments are uniformly bounded, for any $p \geq 2$.
Highlights
1 2 where σ(x) = ax + b is an affine function and Xdenotes the formal derivative of a spatially homogeneous Gaussian noise X, which is white in time and behaves in space like a fractional
To obtain uniqueness we need to bound a spatial increment of the process σ(u) − σ(v) in terms of the same increment for the process u − v, assuming that u and v are two solutions
Since its covariance is invariant under translations, the noise can be viewed as a stationary random distribution, in the sense introduced by Itô in [25]
Summary
This section is divided in three parts. In this case, the process {X(t, x)}x∈R (defined in Remark 2.1 below) is a Gaussian process with stationary increments and spectral measure tμ. We obtain a new criterion for integrability with respect to X, using tools from the theory of fractional Sobolev spaces, borrowed from [18]
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