Heat Kernel Estimates for Non-symmetric Finite Range Jump Processes
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Heat Kernel Estimates for Non-symmetric Finite Range Jump Processes Jie Ming WANG Department of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P. R. China E-mail : [email protected] Abstract In this paper, we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation S b := Δ
α/2
+ b · ∇,
α/2
is the truncated fractional Laplacian, α ∈ (1, 2) and b ∈ Kdα−1 . In the second part, for where Δ a more general finite range jump process, we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance |x − y| in short time. Keywords Heat kernel, transition density function, gradient estimate, finite range jump process, truncated fractional Laplacian, martingale problem MR(2010) Subject Classification
1
60J35, 47G20, 60J75, 47D07
Introduction
Heat kernel is an important study object in partial differential equations and probability theory. It is known that the heat kernel for an operator L is the fundamental solution of the parabolic heat equation of L. On the other hand, when the operator is associated with a Markov process X, the heat kernel is just the transition density function of X. The classical heat kernel estimates for second order elliptic differential operators have a long history and there are many deep results on this topic. In recent years, the two sided heat kernel estimates for non-local operators and discontinuous processes have been studied intensively due to their importance in applications and theory. See the references [3, 5, 8, 10, 11, 15] and [22] etc. for the study of symmetric non-local operators and see [6, 14, 17, 19–21] and [26] etc. for the non-symmetric case. Finite range jump processes are an important class of discontinuous processes, which could model the stochastic system in applications with the jump size allowed to be up to a certain size. In [1], Barlow, Bass, Chen and Kassmann studied the upper and lower bound estimates of heat kernel and parabolic Harnack principle for a class of symmetric finite range operator with the jump kernel between |x − y|−d−α 1{|x−y|≤1} and |x − y|−d−β 1{|x−y|≤1} with 0 < α < β < 2. Received October 16, 2019, accepted June 5, 2020 Partially supported by NSFC (Grant Nos. 11731009 and 11401025)
Wang J. M.
2
Chen, Kim and Kumagai in [7] established sharp two-sided heat kernel estimates and parabolic Harnack principle for symmetric finite range stable-like operator c(x, y) Sf (x) = lim (f (y) − f (x)) 1{|x−y|≤a} dy, ε↓0 {ε 1 in short time (see [7] and (1.3) below). In the second type function ( |x−y| part, for a more general non-symmetric finite range jump process, under some mild conditions, we improve the estimates of the heat kernel in the form of t/φ(|x − y|) to the Poisson type function (t/|x − y|)|x−y| for large distance |x − y| in short ti
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