Gradient Estimate for Ornstein-Uhlenbeck Jump Processes

Abstract

By using absolutely continuous lower bounds of the Lévy measure, explicit gradient estimates are derived for the semigroup of the corresponding Lévy process with a linear drift. A derivative formula is presented for the conditional distribution of the process at time $t$ under the condition that the process jumps before $t$. Finally, by using bounded perturbations of the Lévy measure, the resulting gradient estimates are extended to linear SDEs driven by Lévy-type processes.

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References in corpus (6)

• Formulae for the derivatives of heat semigroups
• Densities for Ornstein-Uhlenbeck processes with jumps
• Coupling for Ornstein--Uhlenbeck processes with jumps
• A note on the existence of transition probability densities for Lévy processes
• Coupling and Strong Feller for Jump Processes on Banach Spaces
• Transition density estimates for a class of Lévy and Lévy-type processes

Cited by in corpus (14)

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• Coupling property and gradient estimates of Lévy processes via the symbol
• Exponential Ergodicity of stochastic Burgers equations driven by $\alpha$-stable processes
• Derivative Formula and Harnack Inequality for Linear SDEs Driven by Lévy Processes
• Coupling and Strong Feller for Jump Processes on Banach Spaces
• Harnack Inequalities for Stochastic Equations Driven by Lévy Noise
• Gradient estimates for SDEs Driven by Multiplicative Lévy Noise
• Exponential ergodicity and regularity for equations with Lévy noise
• Exponential mixing for SDEs forced by degenerate Levy noises
• Constructions of Coupling Processes for Lévy Processes
• Exponential Convergence for Semilinear SDEs Driven by Lévy Processes on Hilbert Spaces
• On the Coupling Property and the Liouville Theorem for Ornstein-Uhlenbeck Processes
• On some smoothening effects of the transition semigroup of a Lévy process
• Linear Evolution Equations with Cylindrical Lévy Noise: Gradient Estimates and Exponential Ergodicity