Heat kernel estimates for Δ+Δ ^{α /2} in C ^{1, 1} open sets

Abstract

We consider a family of pseudo differential operators {Δ + a α Δ α/2 ; a ∈ (0, 1]} on R d for every d 1 that evolves continuously from Δ to Δ + Δ α/2 , where α ∈ (0, 2). It gives rise to a family of Lévy processes {X a , a ∈ (0, 1]} in R d , where X a is the sum of a Brownian motion and an independent symmetric α-stable process with weight a. We establish sharp two-sided estimates for the heat kernel of Δ + a α Δ α/2 with zero exterior condition in a family of open subsets, including bounded C 1,1 (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric α-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, 1] 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 X a in bounded C

Symmetric jump processes and their heat kernel estimates

“…Sharp two-sided Dirichlet heat kernel estimates for fractional Laplacian ∆ α/2 := −(−∆) α/2 in C 1,1 open subsets of R d with α ∈ (0, 2) have first been obtained in [11]. Since then, there are many works in extending it to certain classes of symmetric Markov processes and their lower order perturbations as well as to more general open sets; see, for example, [1,7,8,9,12,13,14,15,19,21,26,29] and the references therein. In all these works, the jump measures of the Markov processes are absolutely continuous with respect to the Lebesgue measure.…”

“…with its origin at z such that For an open set D ⊂ R d and x ∈ D, let δ D (x) be the Euclidean distance between x and D c . We say D satisfies the uniform interior ball condition with radius [12,20]. Similarly, we can define the uniform exterior ball condition.…”

A Keller–Segel system involving mixed local and nonlocal operators with logistic term on ℝN

“…We denote heat kernel for mixed local and nonlocal operators as K N,s t , which defined in (2.3). Thanks to the two-sided estimates (2.4) of mixed heat kernel demonstrated in Chen et al [42] (see Lemma 2.1), we overcome the difficulty in the estimate of mixed heat kernel which cannot use scaling transformation (see Zhang et al [36]) to handle with and prove that the semigroup {T(x)} t>0 generated by Δ − (−Δ) s − I is an analytical semigroup (see Theorem 2.5). Moreover, we also obtain that the analytic semigroup {T(x)} t>0 generated by Δ − (−Δ) s − I on L p ( R N ) enjoys the following L p -L q priori estimates (2.20)…”

“…In 2011, Chen et al [42] considered a family of pseudo-differential operators {Δ + a 2s Δ s , a ∈ (0, 1]} on R N for every N > 1 that evolves continuously from Δ to Δ + Δ s , where s ∈ (0, 1). They found sharp two-sided estimates for the heat kernel K N,s,a t (x) of Δ+a 2s Δ s with zero exterior condition in a family of open subsets, which includes bounded C 1,1 open sets, in which C 1,1 is possibly disconnected.…”