Quasilinear SPDEs via Rough Paths

We are interested in (uniformly) parabolic PDEs with a nonlinear dependence of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time periodic setting with a single spatial variable: ∂2u−P(a(u)∂21u+σ(u)f)=0, where P is the projection on mean-zero functions, and f is a distribution which is only controlled in the low regularity norm of Cα−2 for α>23 on the parabolic Hölder scale. The example we have in mind is a random forcing f and our assumptions allow, for example, for an f which is white in the time variable x2 and only mildly coloured in the space variable x1; any spatial covariance operator (1+|∂1|)−λ1 with λ1>13 is admissible. On the deterministic side we obtain a Cα-estimate for u, assuming that we control products of the form v∂21v and vf with v solving the constant-coefficient equation ∂2v−a0∂21v=f. As a consequence, we obtain existence, uniqueness and stability with respect to (f,vf,v∂21v) of small space-time periodic solutions for small data. We then demonstrate how the required products can be bounded in the case of a random forcing f using stochastic arguments. For this we extend the treatment of the singular product σ(u)f via a space-time version of Gubinelli’s notion of controlled rough paths to the product a(u)∂21u, which has the same degree of singularity but is more nonlinear since the solution u appears in both factors. In fact, we develop a theory for the linear equation ∂tu−P(a∂21u+σf)=0 with rough but given coefficient fields a and σ and then apply a fixed point argument. The PDE ingredient mimics the (kernel-free) Safonov approach to ordinary Schauder theory.