Stability of heat kernel estimates for symmetric non-local Dirichlet forms

Abstract

In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cut-off Sobolev inequalities, and the Faber-Krahn inequalities. In particular, we establish stability of heat kernel estimates for α-stable-like processes even with α ≥ 2 when the underlying spaces have walk dimensions larger than 2, which has been one of the major open problems in this area.

Heat kernels for reflected diffusions with jumps on inner uniform domains

“…Lower bound estimates for q(t, x, y) are derived in Theorem 5.4 of Section 5 under conditions (J φ1,0+,≤ ) and (J φ2,β * ,≥ ) with β * ∈ (0, ∞] and possibly different strictly increasing functions φ 1 and φ 2 . Its proof uses the approaches from [CKW1,CKW2,CKW3]. The main result of this paper, Theorem 1.6, follows directly by combining Theorem 4.4 with Theorem 5.4 as well as Theorem 1.5.…”

“…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. 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.…”

“…• While we rely heavily on techniques from [CKW1,CKW2,CKW3] to study lower bounds for the heat kernel q(t, x, y), a new observation here is that parabolic Harnack inequalities for full ranges do not hold. In fact under conditions (J φ1,β * ,≤ ) and (J φ2,β * ,≥ ) with β * < β * in {0 + } ∪ (0, ∞], the jumping kernel J(x, y) may not satisfy the UJS condition, see Subsection 6.2.…”

“…We further mention that, in Theorems 1.4 and 1.5, we do not require the underlying state space D to have infinite volume (equivalently, D to have infinite diameter). To handle the case that diam(D) is bounded, we use localized versions of results in [CKW1,CKW2,CKW3] to obtain heat kernel estimates in bounded time intervals; we then combine a modification of Doeblin's celebrated result with the small time heat kernel bounds to get estimates for large times.…”

“…In the proof of this part, we use the Poincaré inequality and cut-off Sobolev inequality for general Dirichlet forms. The readers are referred to [AB,CKW1,CKW2,CKW3,GHL] for these definitions (see also Subsection 3.1 for some details). First, we write…”