The flows generated by vector fields of Sobolev type and the corresponding transformations of probability measures (Q1593897)

Let \(A\) be a vector field and \(\mu\) a probability measure on the space \(X\) which may be \(\mathbb{R}^n\) or a Banach space. The existence and uniqueness of a flow \(U_t\), \(t\in\mathbb{R}\), defined by \(A\) (i.e. the solution \(\dot U= A(U)\)) is studied under weak regularity conditions on \(A\). It is assumed that \(A\) and \(\mu\) are smooth which for the case \(X=\mathbb{R}^n\) means that \(\mu\) has density \(\rho\) and both \(A\) and \(\rho\) belong to a Sobolev space. Then the divergence \(\delta_\mu A\) is defined which for \(\mathbb{R}^n\) equals \(\text{div }A + (A,\nabla\rho/\rho)\). It is shown that the integrability of \(\text{exp}|\delta_\mu A|\) is close to the optimal condition which ensures the existence of absolutely continuous flows \((\mu U^{-1}_t\ll \mu; U_{t+s} = U_t\circ U_s)\) both in the finite- and infinite-dimensional cases. Some estimates on solutions and formulas for the densities of \(\mu U^{-1}_t\) are also given. The results are obtained under weaker assumptions than known from the literature e.g. \textit{R. J. DiPerna} and \textit{P. L. Lions} [Invent. Math. 98, No.~3, 511-547 (1989; Zbl 0696.34049)].