In this paper we investigate the differentiability preserving properties of the semigroup $\{T_t: t \geq 0\}$ whose infinitesimal generator is a closed extension of the one-dimensional diffusion operator $L = a(x)d^2/dx^2 + b(x)d/dx$ acting on $C^2(I)$, where $I$ is a closed and bounded interval. Especially we treat the case in which the smoothness of the diffusion coefficient fails at the boundary. We get that $\{T_t: t \geq 0\}$ preserves the one and two-times differentiabilities but does not the three-times one of sufficiently many initial data.
Publié le : 1985-02-14
Classification:
Diffusion processes,
semigroup,
martingale problem,
degenerated second order differential operator,
60J60,
60H10,
60J35
@article{1176993076,
author = {Okada, Norio},
title = {On Differentiability Preserving Properties of Semigroups Associated with One-Dimensional Singular Diffusions},
journal = {Ann. Probab.},
volume = {13},
number = {4},
year = {1985},
pages = { 206-225},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993076}
}
Okada, Norio. On Differentiability Preserving Properties of Semigroups Associated with One-Dimensional Singular Diffusions. Ann. Probab., Tome 13 (1985) no. 4, pp. 206-225. http://gdmltest.u-ga.fr/item/1176993076/