Abstract
This paper introduces new classes of fractional and multifractional random fields arising from elliptic, parabolic and hyperbolic equations with random innovations derived from fractional Brownian motion. The case of stationary random initial conditions is also considered for parabolic and hyperbolic equations.
Highlights
There has been some recent interest in studying stochastic partial differential equations driven by a fractional noise
We provide a structure for developing mean-square weak-sense and strong-sense solutions to stochastic elliptic, hyperbolic and parabolic equations driven by fractional Gaussian noise, whose integral is fractional Brownian motion
We provide an overview of the mean-square solution of stochastic elliptic, hyperbolic and parabolic problems driven by fractional Gaussian random fields
Summary
There has been some recent interest in studying stochastic partial differential equations driven by a fractional noise (see Duncan et al, 2002; Tindel et al, 2003; Muller and Tribe 2004; Hu et al, 2004; Maslowski and Nualart, 2005; Hu and Nualart, 2009a, 2009b; Sanz-Solé and Torrecilla, 2009; Sanz-Solé and Vailermot, 2010, among others). We provide a structure for developing mean-square weak-sense (generalized) and strong-sense (pointwise definition) solutions to stochastic elliptic, hyperbolic and parabolic equations driven by fractional Gaussian noise, whose integral is fractional Brownian motion. Mild solutions for a class of fractional SPDEs have been developed for elliptic and parabolic problems by Sanz-Solé and Torrecilla (2009), and Sanz-Solé and Vuillermot (2010) They defined the stochastic convolution integrals of the Green function with fractional noise as Wiener integrals. We interpret the corresponding stochastic integrals of non-random Green functions with respect to fractional noise as Wiener integrals in the spectral domain This approach gives us an opportunity, for relatively simple situations, to obtain an explicit parabolic, hyperbolic and elliptic parametric family of models involving fractional Gaussian random fields. New Green functions for the case of the heat equation with quadratic potential were constructed in Leonenko and Ruiz-Medina (2006, 2008)
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