Harnack Inequality and Applications for SDEs Driven by G -Brownian Motion
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Acta Mathemacae Applicatae Sinica, English Series The Editorial Office of AMAS & Springer-Verlag GmbH Germany 2020
Harnack Inequality and Applications for SDEs Driven by G-Brownian Motion Fen-fen YANG Center for Applied Mathematics, Tianjin University, Tianjin 300072, China (E-mail: [email protected])
Abstract In this paper, Wang’s Harnack and shift Harnack inequality for a class of stochastic differential equations driven by G-Brownian motion are established. The results generalize the ones in the linear expectation setting. Moreover, some applications are also given. Keywords Harnack inequality; shift Harnack inequality; stochastic differential equations; G-Brownian motion; G-expectation. 2000 MR Subject Classification
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60H10; 60H15
Introduction
Since Wang[20] introduced dimensional-free Harnack inequality for diffusions on Riemannian manifold, this Harnack inequality has been extensively investigated. This type Harnack inequality acts as a powerful tool in the study of functional inequalities (see [1, 15, 16, 21, 22]), heat kernel estimates (see [6]), high order eigenvalues (see [9, 18]), transportation cost inequalities (see [4]), and short-time behavior of transition probabilities (see [2, 3, 9]). To establish Wang’s Harnack inequality, Wang and co-authors introduced the coupling by change of measures (see Wang [19] and references within for details). On the other hand, for the potential applications in uncertainty problems, risk measures and the superhedging in finance, the theory of nonlinear expectation has been developed. Especially, Peng [12, 13] established the fundamental theory of G-expectation theory, G-Brownian motion and stochastic differential equations driven by G-Brownian motion (G-SDEs, in short). To establish Wang’s Harnack inequality using coupling by change of measures in the linear probability setting, the Girsanov transform plays a crucial role. In [7, 11, 23], the Girsanov’s theorem has been extended to the G-framework, and Girsanov’s formula has been derived for G-Brownian motion. Song [17] firstly derived the gradient estimates for nonlinear diffusion semigroups by using the method of Wang’s coupling by change of measure. Recently, Hu et al. [8] studied the invariant and ergodic nonlinear expectations for G-diffusion processes. In this paper, we investigate Wang’s Harnack and shift Harnack inequality and applications for the following G-SDE dXt = b(Xt )dt + dBt , (1.1) where Bt is a G-Brownian motion. The paper is organized as follows. In Section 2, we recall some preliminaries on G-Brownian motion, related stochastic calculus and transformation for G-expection. In Section 3, Wang’s Harnack inequality and shift Harnack inequality are established for the nonlinear Markov operator associated with (1.1). In addition, we give some applications of Harnack inequality. Manuscript received August 31, 2018. Accepted on November 14, 2019. This paper is supported by the National Natural Science Foundation of China (Nos. 11801406).
F.F. YANG
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Preliminaries
2.1
Sublinear Expectation S
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