Harnack Inequalities for Functional SDEs Driven by Subordinate Brownian Motions
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Harnack Inequalities for Functional SDEs Driven by Subordinate Brownian Motions Chang-Song Deng1 · Xing Huang2 Received: 7 September 2019 / Accepted: 19 October 2020 / © Springer Nature B.V. 2020
Abstract Using coupling by change of measure and an approximation technique, Wang’s Harnack inequalities are established for a class of functional SDEs driven by subordinate Brownian motions. The results cover the corresponding ones in the case without delay. Keywords Functional SDE · Harnack inequality · Subordinate Brownian motion · Coupling Mathematics Subject Classification (2010) 60H10 · 60H15 · 34K26 · 39B72
1 Introduction The dimension-free Harnack inequality was firstly introduced by Wang [13] to derive the log-Sobolev inequality on Riemannian manifolds. As a weaker version of the powerHarnack inequality, the log-Harnack inequality was considered in [10] for semi-linear SDEs. These two Harnack-type inequalities have been intensively investigated and applied for various finite- and infinite-dimensional SDEs and SPDEs driven by Brownian noise; we refer to the monograph by F.-Y. Wang [14] for a systematic theory on dimension-free Harnack inequalities and applications. For the functional SDEs and SPDEs, the Harnack inequalities are also investigated in [1, 2], see also [12] for SDEs with non-Lipschitz coefficients and [7, 8] for SDEs with Dini drifts. However, the noise in all the above results is assumed to contain a Brownian motion part. The central aim of this work is to establish Harnack
Supported in part by NNSFC (11801406, 11831015) and the Fundamental Research Funds for the Central Universities (2042019kf0236). Chang-Song Deng
[email protected] Xing Huang [email protected] 1
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
2
Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
C.-S. Deng, X. Huang
inequalities for functional SDEs driven by subordinate Brownian motions, which form a very large class of L´evy processes. It turns out that our results cover the corresponding ones in the case without delay derived by J. Wang and F.-Y. Wang [15] (cf. [4] for an improved estimate). Fix a constant r0 ≥ 0. Denote by C the family of all right continuous functions f : [−r0 , 0] → Rd with left limits. To characterize the state space, equip C with the norm · 2 given by 0 |ξ(s)|2 ds + |ξ(0)|2 , ξ ∈ C . ξ 22 := −r0
For f : [−r0 , ∞) → by
Rd ,
we will denote ft ∈ C , t ≥ 0, the corresponding segment process, ft (s) := f (t + s),
s ∈ [−r0 , 0].
Let S = (S(t))t≥0 be a subordinator (without killing), i.e. a nondecreasing L´evy process on [0, ∞) starting at S(0) = 0. Due to the independent and stationary increments property, it is uniquely determined by the Laplace transform E e−uS(t) = e−tφ(u) ,
u > 0, t ≥ 0,
where the characteristic (Laplace) exponent φ : (0, ∞) → (0, ∞) is a Bernstein function with φ(0+) := limr↓0 φ(r) = 0, i.e. a C ∞ -function such that (−1)n−1 φ (n) ≥ 0 for all n ∈ N. Every such φ has a unique L´evy–Khintchine representation (cf. [11,
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