Claude Le Bris & Pierre-Louis Lions 
Parabolic Equations with Irregular Data and Related Issues [PDF ebook] 
Applications to Stochastic Differential Equations

This book studies the existence and uniqueness of solutions to parabolic-type equations with irregular coefficients and/or initial conditions. It elaborates on the Di Perna-Lions theory of renormalized solutions to linear transport equations and related equations, and also examines the connection between the results on the partial differential equation and the well-posedness of the underlying stochastic/ordinary differential equation.