ROUGH PATHS AND 1D SDE WITH A TIME DEPENDENT DISTRIBUTIONAL DRIFT. APPLICATION TO POLYMERS.

François Delarue (1) , Roland Diel (2)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 LJAD - Laboratoire Jean Alexandre Dieudonné

Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory. Existence and uniqueness are established in the weak sense when the drift reads as the derivative of a Hölder continuous function. Regularity of the drift part is investigated carefully and a related stochastic calculus is also proposed, which makes the structure of the solutions more explicit than within the earlier framework of Dirichlet processes.

Mots clés: Rough paths; Singular SDEs; Polymer measure; KPZ equation

Domaines: Probabilités [math.PR]

François Delarue, Roland Diel. ROUGH PATHS AND 1D SDE WITH A TIME DEPENDENT DISTRIBUTIONAL DRIFT. APPLICATION TO POLYMERS.. Probability Theory and Related Fields, 2015, pp.1-63. ⟨10.1007/s00440-015-0626-8⟩. ⟨hal-00947201v3⟩