Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms

Heat kernels for reflected diffusions with jumps on inner uniform domains

“…So, when diam(D) = ∞, we can deduce Theorem 1.4 from [CKW3,Theorem 1.13]. Furthermore, as seen from [GHH] and [CKW2,Remark 1.19], all results of the paper [CKW3] continue to hold for bounded state space with obvious localized versions. Thus when diam(D) < ∞, according to [CKW3,Theorem 1.13] again, for any T 0 > 0, one can obtain estimates of q(t, x, y) for t ∈ (0, T 0 ] with constants c i further dependent on T 0 .…”

“…Note that, in the present setting, the fact that the underlying state space D has infinite diameter is equivalent to the fact that D has infinite volume; see [GH,Corollary 5.3]. So, when diam(D) = ∞, we can deduce Theorem 1.4 from [CKW3,Theorem 1.13]. Furthermore, as seen from [GHH] and [CKW2,Remark 1.19], all results of the paper [CKW3] continue to hold for bounded state space with obvious localized versions.…”

“…We now make some comments on the approach of the main result Theorem 1.6. Two-sided heat kernel estimates for symmetric diffusions with stable-like jumps on metric measure spaces were obtained in the recent paper [CKW3] by three of the authors of this paper. In that paper, some powerful tools for the study of stability of heat kernel estimates and parabolic Harnack inequalities for symmetric jump processes developed in [CKW1,CKW2] are adapted; on the other hand, a new self-improvement argument for upper bounds via exit time probability estimates is proposed, see [CKW3,Section 4.3] for details.…”

“…See [CKW3] for the corresponding results for symmetric diffusions with jumps on metric measure spaces where β = 0 but the diffusion part can be sub-diffusive and the upper growth exponent α * for φ in (1.11) can be possibly larger than 2. We further remark that the two-sided heat kernel estimates for diffusions with jumps of the form (1.20)-(1.21) was first given in [CK3].…”

“…In that paper, some powerful tools for the study of stability of heat kernel estimates and parabolic Harnack inequalities for symmetric jump processes developed in [CKW1,CKW2] are adapted; on the other hand, a new self-improvement argument for upper bounds via exit time probability estimates is proposed, see [CKW3,Section 4.3] for details. However, as for symmetric diffusions with (sub-or super-)exponential decay jumps in the present paper, the approach of [CKW3] (in particular, the self-improvement argument for upper bounds as mentioned above) does not work, due to light tails of jumps for the associated process. The novelty of the proof for Theorem 1.6 is twofold.…”