Heat equation driven by a space-time fractional noise

The extension of standard stochastic models (SDEs, SPDEs) to general fractional noises is known to be a tricky issue, which cannot be studied within the classical martingale setting. We will see how the recently-introduced theory of regularity structures allows us to overcome these difficulties, in the case of a heat equation model with non-linear perturbation driven by a space-time fractional Brownian motion.

The analysis relies in particular on the exhibition of an explicit process at the core of the dynamics, the so-called K-rough path, the definition of which shows strong similarities with that of a classical rough path.