Parabolic partial differential equations with irregular data. Related issues. Application to stochastic differential equations : Notes de C.

Summary "We study the existence and the uniqueness of the solution to parabolic type equations with irregular coefficients and/or initial conditions. The coefficients considered in the equation typically belong to Lebesgue or Sobolev spaces, the initial condition may be only Lebesgue integrable, the second order term in the equation may be degenerate. The arguments elaborate on the DiPerna-Lions theory of renormalized solutions to linear transport equations and related equations. The connection between the results on the partial differential equation and the well-posedness of the underlying stochastic/ordinary differential equation is examined. We in particular follow up on two previous articles. These notes, written up jointly by the two authors, lay out the background on the various issues and present the recent results obtained by the second author. They are an expanded version of the lectures delivered at Collège de France during the academic year 2012-13.