Stochastic flows acting on Schwartz distributions
Consider a Schwartz distribution \(T\) and the stochastic flow \(\varphi_{s,t}\) generated by a stochastic differential equation; the composition of \(T\) with \(\varphi\) is defined as a random distribution; the spatial regularity of this variable is studied in terms of Sobolev spaces. For the time regularity, after defining a stochastic integral, a generalized Itô's formula is proved when \(T = T(t)\) is a semimartingale. These results are applied to the regularity of semigroups, and to the existence and spatial regularity of a local time for a one-dimensional flow. Moreover, the relation with the pull-back defined by Watanabe is discussed.
scientific article; zbMATH DE number 566134
scientific article