Some SDEs with distributional drift. I: General calculus
Let \[ X_t=x_0+\int^t_0\sigma(X_s)\,dW_s+ \int^t_0b'(X_s)\,ds, \] where \(b,\sigma\) \((\sigma>0)\) are continuous functions, be a stochastic differential equation and \[ Lf=\frac{\sigma^2} {2}f''+b'f'' \] be an operator associated with \(X\). The authors cosider a martingale problem assiociated with \(L\). They show that the solution of the martingale problem will solve some stochastic differential equation. They also prove the theorem on the Dirichlet-Fukushima structure of \(f(X)\) for \(f\in W_{\text{loc}}^{1,2}\) \((f\) absolutely continuous with \(f'\) square integrable on compact subsets).
scientific article; zbMATH DE number 2002270