Heat semi-group and generalized flows on complete Riemannian manifolds

Generalized stochastic flow associated to the Itô SDE with partially Sobolev coefficients and its application

“…The generalization of this theory to the infinite dimensional Wiener space has been done in [3,14], see also [21] in which we studied the Fokker-Planck type equations on the Wiener space. In [10], the authors gave a rather sketchy argument of how to extend the Di Perna-Lions theory to compact Riemannian manifolds; by proving a commutator estimate involving the heat semi-group and Sobolev vector fields on manifolds, this theory was recently generalized in [12] to complete Riemannian manifolds under suitable conditions on the lower bound of the Ricci curvature. Using the pointwise characterization of Sobolev functions, Crippa and De Lellis gave in [8] direct proofs to many of the results in the Di Perna-Lions theory.…”

The Itô SDEs and Fokker–Planck equations with Osgood and Sobolev coefficients

“…Applying Lemmas 4.1 and 4.4 to the flows X n t gives us 16) in which C depends on d, T, R, Λ 1,T and b L 1 (B r+1 ) . Therefore,…”

“…We also prove the uniqueness of solutions to the corresponding Fokker-Planck equation by using the probabilistic method. space, [13,43,16] for studies on Riemannian manifolds. The readers can find a survey of some of these results in [3].…”

Shift Harnack inequality and integration by parts formula for semilinear stochastic partial differential equations

“…These formulas can be applied to study the quasi-invariance of the transition probability measure, integrability of the heat kernel, to characterize the logarithmic derivative of the heat kernel, and so on (see [6,7,[19][20][21] and references therein). Combining the integration by parts formula and the Bismut formula, one can estimate the commutator of the gradient operator and the diffusion semigroup which is important in the study of flow properties (see [11]). The key point of the new type coupling is that the two marginal processes have the same initial point and at the given time, their difference reaches a given quantity.…”