Heat kernels for non-symmetric diffusion operators with jumps

For d⩾2, we prove the existence and uniqueness of heat kernels to the following time-dependent second order diffusion operator with jumps:Lt:=12∑i,j=1daij(t,x)∂ij2+∑i=1dbi(t,x)∂i+Ltκ, where a=(aij) is a uniformly bounded, elliptic, and Hölder continuous matrix-valued function, b belongs to some suit...

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Bibliographic Details
Published in:Journal of Differential Equations 2017-11, Vol.263 (10), p.6576-6634
Main Authors: Chen, Zhen-Qing, Hu, Eryan, Xie, Longjie, Zhang, Xicheng
Format: Article
Language:English
Subjects:
Gradient estimate
Heat kernel
Kato class
Lévy system
Non-local operator
Transition density
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Summary:For d⩾2, we prove the existence and uniqueness of heat kernels to the following time-dependent second order diffusion operator with jumps:Lt:=12∑i,j=1daij(t,x)∂ij2+∑i=1dbi(t,x)∂i+Ltκ, where a=(aij) is a uniformly bounded, elliptic, and Hölder continuous matrix-valued function, b belongs to some suitable Kato's class, and Ltκ is a non-local α-stable-type operator with bounded kernel κ. Moreover, we establish sharp two-sided estimates, gradient estimate and fractional derivative estimate for the heat kernel under some mild conditions.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2017.07.023