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: ∂ 2 u - P ( a ( u ) ∂ 1 2 u + 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 C α - 2 for α > 2 3 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 x 2 and only mildly coloured in the space variable x 1 ; any spatial covariance operator ( 1 + | ∂ 1 | ) - λ 1 with λ 1 > 1 3 is admissible. On the deterministic side we obtain a C α -estimate for u , assuming that we control products of the form v ∂ 1 2 v and vf with v solving the constant-coefficient equation ∂ 2 v - a 0 ∂ 1 2 v = f . As a consequence, we obtain existence, uniqueness and stability with respect to ( f , v f , v ∂ 1 2 v ) 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 ) ∂ 1 2 u , 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 ∂ t u - P ( a ∂ 1 2 u + σ f ) = 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.