The Annals of Probability

On Differentiability Preserving Properties of Semigroups Associated with One-Dimensional Singular Diffusions

Norio Okada

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Abstract

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.

Article information

Source
Ann. Probab., Volume 13, Number 1 (1985), 206-225.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993076

Digital Object Identifier
doi:10.1214/aop/1176993076

Mathematical Reviews number (MathSciNet)
MR770638

Zentralblatt MATH identifier
0562.60084

JSTOR
links.jstor.org

Subjects
Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

Keywords
Diffusion processes semigroup martingale problem degenerated second order differential operator

Citation

Okada, Norio. On Differentiability Preserving Properties of Semigroups Associated with One-Dimensional Singular Diffusions. Ann. Probab. 13 (1985), no. 1, 206--225. doi:10.1214/aop/1176993076. https://projecteuclid.org/euclid.aop/1176993076


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