Stochastic Partial Differential Equations Driven by a ZgfBm Noise

Stochastic partial differential equations (SPDEs) provide a powerful framework for modeling physical phenomena subject to random fluctuations. This chapter analyses a new SPDE featuring a Laplacian operator, driven by a white-colored noise. This noise process behaves as a Wiener process in the spatial variable and as a generalized fractional Brownian motion, ZgfBm in time with Hurst parameter H ∈ (1/2, 1). By considering the more general ZgfBm, the chapter provides a unified framework for the analysis of SPDEs. To properly define and work with stochastic integrals with respect to the noise W a,b,H , the chapter first introduces the associated canonical Hilbert space, and then details the construction of the integral. A powerful consequence of the construction of the noise W a,b,H and its associated Wiener integral, and a key tool in our analysis, is the transfer formula.