A pathwise approach to a stochastic porous medium equation with nonlinear, conservative noise
- Benjamin Fehrman (MPI MiS, Leipzig)
This talk will introduce a pathwise well-posedness theory for a stochastic porous medium equation with nonlinear, conservative noise. Since the noise will be described by a rough path, taking for instance a generic sample path of fractional Brownian motion, such equations are not classically well-posed. Furthermore, owing to the nonlinear structure, it is not possible to make sense of the equation through a simple transformation. Therefore, we will instead pass to the kinetic formulation which, for smooth driving paths, yields an equivalent linear equation. On this level, the noise can be understood in terms of the underlying characteristics, which are described by a well-posed and stable system of rough differential equations. A meaningful interpretation of the original equation can then be obtained by propagating test functions along the corresponding inverse characteristics. Time permitting, applications of these methods to fast diffusion and porous medium equations with multiplicative noise will also be discussed.