Strong regularization by Brownian noise propagating through a weak Hörmander structure

Paul-Éric Chaudru de Raynal (1) , Igor Honoré (2) , Stéphane Menozzi (2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 LaMME - Laboratoire de Mathématiques et Modélisation d'Evry

We establish strong uniqueness for a class of degenerate SDEs of weak Hörmander type under suitable Hölder regularity conditions for the associated drift term. Our approach relies on the Zvonkin transform which requires to exhibit good smoothing properties of the underlying parabolic PDE with rough, here Hölder, drift coefficients and source term. Such regularizing effects are established through a perturbation technique (forward parametrix approach) which also heavily relies on appropriate duality properties on Besov spaces. For the method employed, we exhibit some sharp thresholds on the Hölder exponents for the strong uniqueness to hold.

Domains: Probability [math.PR]; Analysis of PDEs [math.AP]

Paul-Éric Chaudru de Raynal, Igor Honoré, Stéphane Menozzi. Strong regularization by Brownian noise propagating through a weak Hörmander structure. Probability Theory and Related Fields, 2022, 184 (1-2), pp.1-83. ⟨10.1007/s00440-022-01150-z⟩. ⟨hal-01906576v2⟩