Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result

Paul-Éric Chaudru de Raynal (1) , Stéphane Menozzi (2, 3)

1 LAMA - Laboratoire de Mathématiques
2 LaMME - Laboratoire de Mathématiques et Modélisation d'Evry
3 HSE - Vysšaja škola èkonomiki = National Research University Higher School of Economics [Moscow]

We investigate the effects of the propagation of a non-degenerate Brownian noise through a chain of deterministic differential equations whose coefficients are rough and satisfy a weak like Hörmander structure (i.e. a non-degeneracy condition w.r.t. the components which transmit the noise). In particular we characterize, through suitable counterexamples , almost sharp regularity exponents that ensure that weak well posedness holds for the associated SDE. As a by-product of our approach, we also derive some density estimates of Krylov type for the weak solutions of the considered SDEs.

Mots clés: Secondary: 60H30; 34F05; martingale problem; density esti-; Kolmogorov hypoelliptic PDEs; parametrix; MSC: Primary: 60H10; regularization by noise; mates

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

Paul-Éric Chaudru de Raynal, Stéphane Menozzi. Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result. Transactions of the American Mathematical Society, 2021, 375 (01), pp.1-45. ⟨10.1090/tran/7947⟩. ⟨hal-01613679v1⟩