A non-linear parabolic PDE with a distributional coefficient and its applications to stochastic analysis

Abstract

We consider a non-linear parabolic partial differential equation (PDE) on ${\mathbb{R}}^d$ with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity is of quadratic type in the gradient of the unknown. Under suitable conditions on the parameters we prove local existence and uniqueness of a mild solution to the PDE, and investigate properties like continuity with respect to the initial condition and blow-up times. We prove a global existence and uniqueness result assuming further properties on the non-linearity. To conclude we consider an application of the PDE to stochastic analysis, in particular to a class of non-linear backward stochastic differential equations with distributional drivers.

26 pages - new Section 4 added, with results on global existence and uniqueness. Final published article available online from Journal of Differential Equations

References in corpus (4)

• A solution theory for quasilinear singular SPDEs
• Controlling Rough Paths
• Parabolic equations with rough coefficients and singular forcing
• Quadratic BSDEs with rough drivers and $L^2$--terminal condition

Cited by in corpus (2)

• Blow-up for a nonlinear PDE with fractional Laplacian and singular quadratic nonlinearity
• A Feynman-Kac result via Markov BSDEs with generalized driver