A quasi-sure flow property and the equivalence of capacities for differential equations on the Wiener Space
The author considers the problem \[ d/dt \bigl(U_t (x)\bigr) =A\bigl(U_t(x) \bigr), \quad U_0(x) =x, \tag{1} \] in an abstract Wiener space \((X,H,\mu)\) in which \(A\) is a vector field on \(X\), \(A:X\to H\), \(A\) is smooth in the sense of Malliavin. In an earlier paper the author constructed a solution \(U_t(x)\) of (1) which satisfies the equation quasi-everywhere i.e. except of a ``slim set''. The considered paper deals with refinements of the earlier results. Especially that the mapping \(x\to U_t(x)\) preserves the condition \[ U_t\circ U_s(x) =U_{t+s} (x). \]
differential equation in the Malliavin sense
scientific article; zbMATH DE number 880569