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

Zhen Qing Chen, Panki Kim, Renming Song

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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

Original languageEnglish (US)
Pages (from-to)58-80
Number of pages23
JournalJournal of the London Mathematical Society
Issue number1
StatePublished - Aug 2011

ASJC Scopus subject areas

  • Mathematics(all)


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