We study a class of stochastic differential equations with non-Lipschitz coefficients. A unique strong solution is obtained and the non confluence of the solutions of stochastic differential equations is proved. The dependence with respect to the initial values is investigated. To obtain a continuous version of solutions, the modulus of continuity of coefficients is assumed to be less than |x − y| log 1 |x−y| . Finally a large deviation principle of Freidlin-Wentzell type is also established in the paper.

We consider the Itô stochastic differential equation dX t = m j =1 A j (X t ) dw j t + A 0 (X t ) dt on R d . The diffusion coefficients A 1 , . . . , A m are supposed to be in the Sobolev space W 1,p loc (R d ) with p > d, and to have linear growth. For the drift coefficient A 0 , we distinguish two cases: (i) A 0 is a continuous vector field whose distributional divergence δ(A 0 ) with respect to the Gaussian measure γ d exists, (ii) A 0 has Sobolev regularity W 1,p loc for some p > 1. Assume R d exp[λ 0 (|δ(A 0 )| + m j =1 (|δ(A j )| 2 + |∇A j | 2 ))] dγ d < +∞ for some λ 0 > 0. In case (i), if the pathwise uniqueness of solutions holds, then the push-forward (X t ) # γ d admits a density with respect to γ d . In particular, if the coefficients are bounded Lipschitz continuous, then X t leaves the Lebesgue measure Leb d quasi-invariant. In case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish existence and uniqueness of stochastic flow of maps.

No abstract

For infinitesimal data given on the group of diffeomorphism of the circle with respect to the metric H 3=2 , the associated Brownian motion has been constructed by Malliavin (C.R. Acad. Sci. Paris t.329 (1999), 325-329). In this work, we shall give another approach and prove the invariance of heat measures under the adjoint action of S 1 . # 2002 Elsevier Science (USA)

We consider stochastic differential equations driven by Wiener processes. The vector fields are supposed to satisfy only local Lipschitz conditions. The Lipschitz constants of the drift vector field, valid on balls of radius R, are supposed to grow not faster than log R, while those of the diffusion vector fields are supposed to grow not faster than √ log R. We regularize the stochastic differential equations by associating with them approximating ordinary differential equations obtained by discretization of the increments of the Wiener process on small intervals. By showing that the flow associated with a regularized equation converges uniformly to the solution of the stochastic differential equation, we simultaneously establish the existence of a global flow for the stochastic equation under local Lipschitz conditions.Introduction. Let A 0 , A 1 , . . . , A N be N + 1 vector fields on the Euclidean space R d and (w t ) t≥0 be an R N -valued standard Brownian motion. Consider the following Stratonovich stochastic differential equation: