TY - JOUR

T1 - Heat kernel estimates for Δ+Δα/2 in C 1, 1 open sets

AU - Chen, Zhen Qing

AU - Kim, Panki

AU - Song, Renming

N1 - Funding Information:
The research of Zhen-Qing Chen was partially supported by NSF Grant DMS-0906743. The research of Panki Kim was supported by the Mid-career Researcher Program through NRF grant funded by the MEST (No. 2010-0027491).

PY - 2011/8

Y1 - 2011/8

N2 - We consider a family of pseudo differential operators Δ+aΔ /2; a∈(0, 1Ş on d for every d ≥ 1 that evolves continuously from Δ to Δ+Δ/2, where ∈(0, 2). It gives rise to a family of Lévy processes Xa, a∈(0, 1Ş in d, where Xa is the sum of a Brownian motion and an independent symmetric -stable process with weight a. We establish sharp two-sided estimates for the heat kernel of Δ + aΔ/2 with zero exterior condition in a family of open subsets, including bounded C1, 1 (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric -stable process with weight a in such open sets. Our result is the first sharp two-sided estimates for the transition density of a Markov process with both diffusion and jump components in open sets. Moreover, our result is uniform in a in the sense that the constants in the estimates are independent of a∈(0, 1Ş so that it recovers the Dirichlet heat kernel estimates for Brownian motion by taking a → 0. Integrating the heat kernel estimates in time t, we recover the two-sided sharp uniform Green function estimates of Xa in bounded C1, 1 open sets in d, which were recently established in (Z.-Q. Chen, P. Kim, R. Song and Z. Vondracek, 'Sharp Green function estimates for Δ+Δ /2 in C1, 1 open sets and their applications', Illinois J. Math., to appear) using a completely different approach. 2011 London Mathematical Society2011

AB - We consider a family of pseudo differential operators Δ+aΔ /2; a∈(0, 1Ş on d for every d ≥ 1 that evolves continuously from Δ to Δ+Δ/2, where ∈(0, 2). It gives rise to a family of Lévy processes Xa, a∈(0, 1Ş in d, where Xa is the sum of a Brownian motion and an independent symmetric -stable process with weight a. We establish sharp two-sided estimates for the heat kernel of Δ + aΔ/2 with zero exterior condition in a family of open subsets, including bounded C1, 1 (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric -stable process with weight a in such open sets. Our result is the first sharp two-sided estimates for the transition density of a Markov process with both diffusion and jump components in open sets. Moreover, our result is uniform in a in the sense that the constants in the estimates are independent of a∈(0, 1Ş so that it recovers the Dirichlet heat kernel estimates for Brownian motion by taking a → 0. Integrating the heat kernel estimates in time t, we recover the two-sided sharp uniform Green function estimates of Xa in bounded C1, 1 open sets in d, which were recently established in (Z.-Q. Chen, P. Kim, R. Song and Z. Vondracek, 'Sharp Green function estimates for Δ+Δ /2 in C1, 1 open sets and their applications', Illinois J. Math., to appear) using a completely different approach. 2011 London Mathematical Society2011

UR - http://www.scopus.com/inward/record.url?scp=79960794000&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960794000&partnerID=8YFLogxK

U2 - 10.1112/jlms/jdq102

DO - 10.1112/jlms/jdq102

M3 - Article

AN - SCOPUS:79960794000

VL - 84

SP - 58

EP - 80

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 1

ER -