Electronic Communications in Probability

Estimates for the Derivative of Diffusion Semigroups

L. Rincon

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Let $\{P_t\}_{t\ge 0}$ be the transition semigroup of a diffusion process. It is known that $P_t$ sends continuous functions into differentiable functions so we can write $DP_tf$. But what happens with this derivative when $t\to 0$ and $P_0f=f$ is only continuous?. We give estimates for the supremum norm of the Frechet derivative of the semigroups associated with the operators ${\cal A}+V$ and ${\cal A}+Z\cdot\nabla$ where ${\cal A}$ is the generator of a diffusion process, $V$ is a potential and $Z$ is a vector field.

Article information

Electron. Commun. Probab., Volume 3 (1998), paper no. 8, 65-74.

Accepted: 18 August 1998
First available in Project Euclid: 2 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}
Secondary: 60J55: Local time and additive functionals 60H10: Stochastic ordinary differential equations [See also 34F05]

Diffusion Semigroups Diffusion Processes Stochastic Differential Equations

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Rincon, L. Estimates for the Derivative of Diffusion Semigroups. Electron. Commun. Probab. 3 (1998), paper no. 8, 65--74. doi:10.1214/ECP.v3-994. https://projecteuclid.org/euclid.ecp/1456935914

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  • Da Prato, G. and Zabczyk, J. (1992) Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications 44. Cambridge University Press.
  • Da Prato, G. and Zabczyk, J. (1997) Differentiability of the Feynman-Kac Semigroup and a Control Application. Rend. Mat. Acc. Lincei s. 9, v. 8, 183-188.
  • Elworthy, K. D. (1982) Stochastic Differential Equations on Manifolds. Cambridge University Press.
  • Elworthy, K. D. and Li, X. M. (1994) Formulae for the Derivative of Heat Semigroups. Journal of Functional Analysis 125, 252-286.
  • Elworthy, K. D. and Li, X. M. (1993) Differentiation of Heat Semigroups and applications. Warwick Preprint 77.
  • Li, X.-M. (1992) Stochastic Flows on Noncompact Manifolds. Warwick University Ph.D. Thesis.
  • Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag. New York.
  • Rincon, L. A. (1994) Some Formulae and Estimates for the Derivative of Diffusion Semigroups. Warwick MSc. Dissertation.