Fractional Derivatives Non-Symmetric and Time-Dependent Dirichlet Forms and the Drift Form

Abstract

Using fractional derivatives we show that the drift form ${\int }_{–\infty }^{\infty }u(x)\frac{dv(x)}{dx}dx$ can be approximated by non-symmetric Dirichlet forms. A similar result holds for the drift form in ${\mathbb{R}}^n$ with variable coefficients if the coefficient functions satisfy certain regularity and commutator conditions. Since time-dependent Dirichlet forms (in the sense of Y. Oshima) can be interpreted as sums of a drift form (in $\tau$-direction) and a mixture of $\tau$-parametrized Dirichlet forms over ${\mathbb{R}}^n$, our results show that time-dependent Dirichlet forms arise as limits of ordinary non-symmetric Dirichlet forms in $\mathbb{R}\times {\mathbb{R}}^n$-space. An abstract result on fractional powers of Markov generators allows to extend this observation to generalized Dirichlet forms. Another consequence is that the bilinear form induced by an arbitrary Lévy process is the limit of non-symmetric Dirichlet forms.