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

Abstract

Let u = {u(t, x); (t, x) ∈ R + × R} be the solution to a linear stochastic heat equation driven by a Gaussian noise, which is a Brownian motion in time and a fractional Brownian motion in space with Hurst parameter H ∈ (0, 1). For any given x ∈ R (resp., t ∈ R + ), we show a decomposition of the stochastic process t → u(t, x) (resp., x → u(t, x)) as the sum of a fractional Brownian motion with Hurst parameter H/2 (resp., H) and a stochastic process with C ∞ -continuous trajectories. Some applications of those decompositions are discussed.