On the Onsager-Machlup functional of diffusion processes around non \(C^ 2\) curves
For a solution of the stochastic differential equation \(dX_ t=f(X_ t)dt+dW_ t\), where W is the standard n-dimensional Brownian motion, \(f_ i\in C_ b^ 2({\mathbb{R}}^ n)\), \(i=1,2,...,n\), the asymptotic behaviour of \[ P\{\| \phi -x\| <\epsilon \}/P\{\| W\| \leq \epsilon \} \] is computed as \(\epsilon\) \(\to 0\) for \(\phi \in C^{1+\alpha}\), \(\alpha >0\), deterministic on [0,T]. Here \(\| \psi \| =| \max_{u\in [0,T]}| \psi (u)|,\) and \(| \cdot |\) is the Euclidean norm in \(R^ N\).
scientific article; zbMATH DE number 4140964