JournalszaaVol. 19, No. 3pp. 801–830

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

  • N. Jacob

    Universität Erlangen-Nürnberg, Germany
  • R.L. Schilling

    Nottingham Trent University, UK
Fractional Derivatives Non-Symmetric and Time-Dependent Dirichlet Forms and the Drift Form cover
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Abstract

Using fractional derivatives we show that the drift form u(x)dv(x)dxdx\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 Rn\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 τ\tau-direction) and a mixture of τ\tau-parametrized Dirichlet forms over Rn\mathbb R^n, our results show that time-dependent Dirichlet forms arise as limits of ordinary non-symmetric Dirichlet forms in R×Rn\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.

Cite this article

N. Jacob, R.L. Schilling, Fractional Derivatives Non-Symmetric and Time-Dependent Dirichlet Forms and the Drift Form. Z. Anal. Anwend. 19 (2000), no. 3, pp. 801–830

DOI 10.4171/ZAA/981