Weak Solutions of Semilinear PDEs in Sobolev Spaces and Their Probabilistic Interpretation via the FBSDEs

Abstract

The article is devoted to representation of weak solutions (in Sobolev sense) of degenerate parabolic partial differential equations through forward-backward stochastic differential equations. Before, we prove a weak version of a norm equivalence result.

Keywords:

• Forward backward stochastic differential equation
• Semilinear partial differential equation
• Weighted Sobolev spaces

Mathematics Subject Classification:

• 60H10
• 60H30
• 35K55

ACKNOWLEDGMENT

The authors thank the referee for a careful reading that improved the first version of the article.