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 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 blowup 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.

McKean SDEs with singular coefficients

“…and L has been defined in (14). We say that the martingale problem with distributional drift B admits uniqueness if, whenever we have two solutions (X 1 , P 1 ) and (X 2 , P 2 ), then the law of X 1 under P 1 equals the law of X 2 under P 2 .…”

“…We sometimes omit the brackets and write F f in place of F (f ). The result below on F is taken from [14], Proposition 3.1 and equation (32). Lemma 3.1 (Issoglio [14]).…”

“…The result below on F is taken from [14], Proposition 3.1 and equation (32). Lemma 3.1 (Issoglio [14]). Under Assumption 4 and if α ∈ (0, 1) then…”

“…Remark A.2. In [14], the end of the proof of Proposition 4.1 incorrectly uses Gronwall's lemma. The proper argument should instead cite a generalised Gronwall's inequality, like the one stated above.…”

Blow‐up regions for a class of fractional evolution equations with smoothed quadratic nonlinearities

“…In all these papers, the coefficient 𝔟 is multiplied rather than convoluted, and it is interpreted as one realisation of some random noise (hence viewing the whole equation as one realisation of a stochastic PDE). For example, in [6] the author studied a non-linear equation of the form…”