On Brownian motion with irregular drift (Q1079297)

The stochastic equation \(X_ t=W_ t+\int_{R}L^ a_ t(X)d\beta (a)\), \(t\geq 0\), is considered, where W is a given standard Brownian motion, \(L^ a_ t(X)\) is the local time of X at state \(a\in R\), and \(\beta\) denotes a function of locally bounded variation on R. Under minimal assumptions on the function \(\beta\), there exists a strong unique solution to the above equation, which is shown to be one of Feller's one- dimensional diffusions with generalized second order differential generator \(D_ mD^+_ p\). In this manner, it is possible to describe a class of Feller's diffusion processes with nonsmooth natural scale p as solutions to stochastic differential equations. For a somewhat broader class of diffusions characterized as weak solutions of stochastic equations see the author's recent papers, Math. Nachr. 122, 157-165 (1985; Zbl 0575.60074) and ''Stochastic analysis for Feller's one-dimensional diffusions.'' Preprint No.641, Sonderforschungsbereich 72 'Approximation und Optimierung', Univ. Bonn (1984).