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

Transcription

1 Stability of heat kernel estimates for symmetric non-local Dirichlet forms Zhen-Qing Chen, Takashi Kumagai and Jian Wang 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. AS 200 athematics Subject Classification: Primary 60J35, 35K08, 60J75; Secondary 3C25, 60J25, 60J45. Keywords and phrases: symmetric jump process, metric measure space, heat kernel estimate, stability, Dirichlet form, cut-off Sobolev inequality, capacity, Faber-Krahn inequality, Lévy system, jumping kernel, exit time. Contents Introduction and ain Results 2. Setting Heat kernel Preliminaries 4 3 Implications of heat kernel estimates UHKφ + E, F is conservative = J φ,, and HKφ = J φ UHKφ and E, F is conservative = SCSJφ Research partially supported by NSF grant DS Research partially supported by the Grant-in-Aid for Scientific Research A and 7H0093. Research partially supported by the National Natural Science Foundation of China No , Fok Ying Tung Education Foundation No the JSPS postdoctoral fellowship , National Science Foundation of Fujian Province No. 205J0003, the Program for Nonlinear Analysis and Its Applications No. IRTL206, and Fujian Provincial Key Laboratory of athematical Analysis and its Applications FJKLAA.

2 4 Implications of CSJφ and J φ, 3 4. J φ, = FKφ Caccioppoli and L -mean value inequalities FKφ + J φ, + CSJφ = E φ FKφ + E φ + J φ, = UHKDφ Consequences of condition J φ and mean exit time condition E φ UHKDφ + J φ, + E φ = UHKφ, J φ + E φ = UHKφ J φ + E φ = LHKφ Applications and Example Applications Counterexample Appendix The Lévy system formula eyer s decomposition Some results related to FKφ Some results related to the Dirichlet heat kernel SCSJφ + J φ, = E, F is conservative Introduction and ain Results. Setting Let, d be a locally compact separable metric space, and µ a positive Radon measure on with full support. We will refer to such a triple, d, µ as a metric measure space, and denote by, the inner product in L 2 ; µ. Throughout the paper, we assume that all balls are relatively compact and assume for simplicity that µ =. We would emphasize that in this paper we do not assume to be connected nor, d to be geodesic. We consider a regular Dirichlet form E, F on L 2 ; µ. By the Beurling-Deny formula, such form can be decomposed into three terms the strongly local term, the pure-jump term and the killing term see [FOT, Theorem 4.5.2]. Throughout this paper, we consider the form that consists of the pure-jump term only; namely there exists a symmetric Radon measure J, on \ diag, where diag denotes the diagonal set {x, x : x }, such that Ef, g = fx fygx gy Jdx, dy, f, g F.. \diag Since E, F is regular, each function f F admits a quasi-continuous version f on see [FOT, Theorem 2..3]. Throughout the paper, we will always take a quasicontinuous version of f F without denoting it by f. Let L, DL be the negative definite L 2 -generator of E, F on L 2 ; µ; this is, L is the self-adjoint operator in 2

3 L 2 ; µ whose domain DL consists exactly those of f F that there is some unique u L 2 ; µ so that Ef, g = u, g for all g F, and Lf := u. Let {P t } t 0 be the associated semigroup. Associated with the regular Dirichlet form E, F on L 2 ; µ is a µ-symmetric Hunt process X = {X t, t 0; P x, x \ N }. Here N is a properly exceptional set for E, F in the sense that µn = 0 and \ N is X-invariant; that is, P x X t \ N and X t \ N for all t 0 = 0 for all x \ N with the convention that X 0 := X 0. Here := { } is the onepoint compactification of. This Hunt process is unique up to a properly exceptional set see [FOT, Theorem 4.2.8]. We fix X and N, and write 0 = \ N. While the semigroup {P t } t 0 associated with E is defined on L 2 ; µ, a more precise version with better regularity properties can be obtained, if we set, for any bounded Borel measurable function f on, P t fx = E x fx t, x 0. The heat kernel associated with the semigroup {P t } t 0 if it exists is a measurable function pt, x, y : 0 0 0, for every t > 0, such that E x fx t = P t fx = pt, x, yfy µdy, x 0, f L ; µ,.2 pt, x, y = pt, y, x for all t > 0, x, y 0,.3 ps + t, x, z = ps, x, ypt, y, z µdy for all s > 0, t > 0, x, z 0..4 While.2 only determines pt, x, µ-a.e., using the Chapman-Kolmogorov equation.4 one can regularize pt, x, y so that.2.4 hold for every point in 0. See [BBCK, Theorem 3.] and [GT, Section 2.2] for details. We call pt, x, y the heat kernel on the metric measure Dirichlet space or D space, d, µ, E. By.2, sometime we also call pt, x, y the transition density function with respect to the measure µ for the process X. Note that in some arguments of our paper, we can extend without further mention pt, x, y to all x, y by setting pt, x, y = 0 if x or y is outside 0. The existence of the heat kernel allows to extend the definition of P t f to all measurable functions f by choosing a Borel measurable version of f and noticing that the integral.2 does not change if function f is changed on a set of measure zero. Denote the ball centered at x with radius r by Bx, r and µbx, r by V x, r. When the metric measure space is an Alhfors d-regular set on R n with d 0, n] that is, V x, r r d for all x R n and r 0, ], and the Radon measure Jdx, dy = Jx, y µdx µdy for some non-negative symmetric function Jx, y such that Jx, y, x, y.5 dx, y d+α 3

4 for some 0 < α < 2, it is established in [CK] that the corresponding arkov process X has infinite lifetime, and has a jointly Hölder continuous transition density function pt, x, y with respect to the measure µ, which enjoys the following two-sided estimate pt, x, y t d/α t dx, y d+α.6 for any t, x, y 0, ]. Here for two positive functions f, g, notation f g means f/g is bounded between two positive constants, and a b := min{a, b}. oreover, if is a global d-set; that is, if V x, r r d holds for all x R n and r > 0, then the estimate.6 holds for all t, x, y 0,. We call the above Hunt process X an α-stable-like process on. Note that when = R d and Jx, y = c x y d+α for all x, y R d and some constants α 0, 2 and c > 0, X is a rotationally symmetric α-stable Lévy process on R d. The estimate.6 can be regarded as the jump process counterpart of the celebrated Aronson estimates for diffusions. Since Jx, y is the weak limit of pt, x, y/t as t 0, heat kernel estimate.6 implies.5. Hence the results from [CK] give a stable characterization for α-stable-like heat kernel estimates when α 0, 2 and the metric measure space is a d-set for some constant d > 0. This result has later been extended to mixed stable-like processes on more general metric measure spaces in [CK2] and to diffusions with jumps on Euclidean spaces in [CK3], with some growth condition on the rate function φ such as r 0 s c r2 ds φs φr for all r > 0.7 with some constant c > 0. For α-stable-like processes where φr = r α, condition.7 corresponds exactly to 0 < α < 2. Some of the key methods used in [CK] were inspired by a previous work [BL] on random walks on integer lattice Z d. The notion of d-set arises in the theory of function spaces and in fractal geometry. Geometrically, self-similar sets are typical examples of d-sets. There are many self-similar fractals on which there exist fractal diffusions with walk dimension d w > 2 that is, diffusion processes with scaling relation time space dw. This is the case, for example, for the Sierpinski gasket in R n n 2 which is a d-set with d = logn + / log 2 and has walk dimension d w = logn + 3/ log 2, and for the Sierpinski carpet in R n n 2 which is a d-set with d = log3 n / log 3 and has walk dimension d w > 2; see [B]. A direct calculation shows see [BSS, Sto] that the β-subordination of the fractal diffusions on these fractals are jump processes whose Dirichlet forms E, F are of the form given above with α = βd w and their transition density functions have two-sided estimate.6. Note that as β 0,, α 0, d w so α can be larger than 2. When α > 2, the approach in [CK] ceases to work as it is hopeless to construct good cut-off functions a priori in this case. A long standing open problem in the field is whether estimate.6 holds for generic jump processes with jumping kernel of the form.5 for any α 0, d w. A related open question is to find a characterization for heat kernel estimate.6 that is stable under rough isometries. Do they hold on general metric measure spaces with volume doubling VD and reverse volume doubling RVD properties see Definition. below for these two terminologies? These are the questions we will address in this paper. 4

5 For diffusions on manifolds with walk dimension 2, a remarkable fundamental result obtained independently by Grigor yan [Gr2] and Saloff-Coste [Sa] asserts that the following are equivalent: i Aronson-type Gaussian bounds for heat kernel, ii parabolic Harnack equality, and iii VD and Poincaré inequality. This result is then extended to strongly local Dirichlet forms on metric measure spaces in [B, St, St2] and to graphs in [De]. For diffusions on fractals with walk dimension larger than 2, the above equivalence still holds but one needs to replace iii by iii VD, Poincaré inequality and a cut-off Sobolev inequality; see [BB2, BBK, AB]. For heat kernel estimates of symmetric jump processes in general metric measure spaces, as mentioned above, when α 0, 2 and the metric measure space is a d-set, characterizations of α-stable-like heat kernel estimates were obtained in [CK] which are stable under rough isometries; see [CK2, CK3] for further extensions. For the equivalent characterizations of heat kernel estimates for symmetric jump processes analogous to the situation when α 2, there are some efforts such as [BGK, Theorem.2] and [GHL2, Theorem 2.3] but none of these characterizations are stable under rough isometries. In [BGK, Theorem 0.3], assuming that E, F is conservative, V x, r c r d for all x, r > 0 and some constants c, d > 0, and that pt, x, x c 2 t d/α for any x, t > 0 and some constant c 2 > 0, an equivalent characterization for the heat kernel upper bound estimate in.6 is given in terms of certain exit time estimates. Under the assumption that E, F is conservative, the Radon measure Jdx, dy = Jx, y µdx µdy for some non-negative symmetric function Jx, y, and V x, r cr d for all x, r > 0 and some constants c, d > 0, it is shown in [GHL2] that heat kernel upper bound estimate in.6 holds if and only if there are some constants c, c 2 > 0 such that pt, x, x c t d/α and Jx, y c 2 dx, y d+α for all x, y and t > 0, and the following survival estimate holds: there are constants δ, ε 0, so that P x τ Bx,r t ε for all x and r, t > 0 with t /α δr. In both [BGK, GHL2], α can be larger than 2. We note that when α < 2, further equivalent characterizations of heat kernel estimates are given for jump processes on graphs [BBK2, Theorem.5], some of which are stable under rough isometries. Also, when the Dirichlet form of the jump process is parabolic namely the capacity of any non-empty compact subset of is positive [GHL2, Definition 6.3], which is equivalent to that every singleton has positive capacity, an equivalent characterization of heat kernel estimates is given in [GHL2, Theorem 6.7], which is stable under rough isometries..2 Heat kernel In this paper, we are concerned with both upper bound and two-sided estimates on pt, x, y for mixed stable-like processes on general metric measure spaces including α- stable-like processes with α 2. To state our results precisely, we need a number of definitions. Definition.. i We say that, d, µ satisfies the volume doubling property VD if there exists a constant C µ such that for all x and r > 0, V x, 2r C µ V x, r..8 5

6 ii We say that, d, µ satisfies the reverse volume doubling property RVD if there exist constants d > 0, c µ > 0 such that for all x and 0 < r R, V x, R R d. µ V x, r c.9 r VD condition.8 is equivalent to the existence of d 2 > 0 and C µ > 0 so that V x, R V x, r C R d2 µ for all x and 0 < r R,.0 r while RVD condition.9 is equivalent to the existence of l µ > and c µ > so that V x, l µ r c µ V x, r for all x and r > 0.. Since µ has full support on, we have µbx, r > 0 for every x and r > 0. Under VD condition, we have from.0 that for all x and 0 < r R, V x, R V y, r V y, dx, y + R V y, r C dx, y + R d2. µ.2 r On the other hand, under RVD, we have from. that µ Bx 0, l µ r \ Bx 0, r > 0 for each x 0 and r > 0. It is known that VD implies RVD if is connected and unbounded. See, for example [GH, Proposition 5. and Corollary 5.3]. Let R + := [0,, and φ : R + R + be a strictly increasing continuous function with φ0 = 0, φ = and satisfying that there exist constants c, c 2 > 0 and β 2 β > 0 such that R β φr R β2 c c 2 for all 0 < r R..3 r φr r Note that.3 is equivalent to the existence of constants c 3, l 0 > such that c 3 φr φl 0 r c 3 φr for all r > 0. Definition.2. We say J φ holds if there exists a non-negative symmetric function Jx, y so that for µ µ-almost all x, y, and Jdx, dy = Jx, y µdx µdy,.4 c V x, dx, yφdx, y Jx, y c 2 V x, dx, yφdx, y.5 We say that J φ, resp. J φ, if.4 holds and the upper bound resp. lower bound in.5 holds. 6

7 Remark.3. i Since changing the value of Jx, y on a subset of having zero µ µ-measure does not affect the definition of the Dirichlet form E, F on L 2 ; µ, without loss of generality, we may and do assume that in condition J φ J φ, and J φ,, respectively that.5 and the corresponding inequality holds for every x, y. In addition, by the symmetry of J,, we may and do assume that Jx, y = Jy, x for all x, y. ii Note that, under VD, for every λ > 0, there are constants 0 < c < c 2 so that for every r > 0, c V y, r V x, r c 2 V y, r for x, y with dx, y λr..6 Indeed, by.2, we have for every r > 0 and x, y with dx, y λr, C µ + λ d 2 V x, r V y, r C µ + λ d 2. Taking λ = and r = dx, y in.6 shows that, under VD the bounds in condition.5 are consistent with the symmetry of Jx, y. Definition.4. Let U V be open sets of with U U V. We say a non-negative bounded measurable function ϕ is a cut-off function for U V, if ϕ = on U, ϕ = 0 on V c and 0 ϕ on. For f, g F, we define the carré du champ Γf, g for the non-local Dirichlet form E, F by Γf, gdx = fx fygx gy Jdx, dy. y Clearly Ef, g = Γf, g. Let F b = F L, µ. It can be verified see [CKS, Lemma 3.5 and Theorem 3.7] that for any f F b, Γf, f is the unique Borel measure called the energy measure on satisfying g dγf, f = Ef, fg 2 Ef 2, g, f, g F b. Note that the following chain rule holds: for f, g, h F b, dγfg, h = f dγg, h + g dγf, h. Indeed, this can be easily seen by the following equality fxgx fygy = fxgx gy + gyfx fy, x, y. We now introduce a condition that controls the energy of cut-off functions. Definition.5. Let φ be an increasing function on R +. 7

8 i Condition CSJφ We say that condition CSJφ holds if there exist constants C 0 0, ] and C, C 2 > 0 such that for every 0 < r R, almost all x 0 and any f F, there exists a cut-off function ϕ F b for Bx 0, R Bx 0, R + r so that the following holds: f 2 dγϕ, ϕ C fx fy Bx 0,R++C 0 r U U 2 Jdx, dy + C.7 2 f 2 dµ, φr Bx 0,R++C 0 r where U = Bx 0, R + r \ Bx 0, R and U = Bx 0, R + + C 0 r \ Bx 0, R C 0 r. ii Condition SCSJφ We say that condition SCSJφ holds if there exist constants C 0 0, ] and C, C 2 > 0 such that for every 0 < r R and almost all x 0, there exists a cut-off function ϕ F b for Bx 0, R Bx 0, R + r so that.7 holds for any f F. Clearly SCSJφ = CSJφ. Remark.6. i SCSJφ is a modification of CSAφ that was introduced in [AB] for strongly local Dirichlet forms as a weaker version of the so called cut-off Sobolev inequality CSφ in [BB2, BBK]. For strongly local Dirichlet forms the inequality corresponding to CSJφ is called generalized capacity condition in [GHL3]. As we will see in Theorem.5 below, SCSJφ and CSJφ are equivalent under FKφ see Definition.8 below and J φ,. ii The main difference between CSJφ here and CSAφ in [AB] is that the integrals in the left hand side and in the second term of the right hand side of the inequality.7 are over Bx, R + + C 0 r containing U instead of over U for [AB]. Note that the integral over U c is zero in the left hand side of.7 for the case of strongly local Dirichlet forms. As we see in the arguments of the stability of heat kernel estimates for jump processes, it is important to fatten the annulus and integrate over U rather than over U. Another difference from CSAφ is that in [AB] the first term of the right hand side is 8 U ϕ2 dγf, f. However, we will prove in Proposition 2.4 that CSJφ implies the stronger inequality CSJφ + under some regular conditions VD,.3 and J φ,. See [AB, Lemma 5.] for the case of strongly local Dirichlet forms. iii As will be proved in Proposition 2.3 4, under VD and.3, if.7 holds for some C 0 0, ], then it holds for all C 0 [C 0, ] with possibly different C 2 > 0. iv By the definition above, it is clear that if φ φ 2, then CSJφ 2 implies CSJφ. v Denote by F loc the space of functions locally in F; that is, f F loc if and only if for any relatively compact open set U there exists g F such that f = g µ-a.e. on U. Since each ball is relatively compact and.7 uses the property of f on Bx 0, R + + C 0 r only, both SCSJφ and CSJφ also hold for any f F loc. 8

9 Remark.7. Under VD,.3 and J φ,, SCSJφ always holds if β 2 < 2, where β 2 is the exponent in.3. In particular, SCSJφ holds for φr = r α with α < 2. Indeed, for any fixed x 0 and r, R > 0, we choose a non-negative cut-off function ϕx = hdx 0, x, where h C [0, such that 0 h, hs = for all s R, hs = 0 for s R + r and h s 2/r for all s 0. Then, by J φ,, for almost every x, dγϕ, ϕ x = dµ ϕx ϕy 2 Jx, y µdy Jx, y µdy + 4 r 2 {dx,y r} {dx,y r} c φr + c r 2 c φr + c 2 φr Jx, y µdy + 4 r 2 i=0 {dx,y r} i=0 V x, 2 i r2 2i r 2 V x, 2 i rφ2 i r i=0 2 i2 β2 c 3 φr, dx, y 2 Jx, y µdy {2 i r<dx,y 2 i r} dx, y 2 Jx, y µdy where in the third inequality we have used Lemma 2. below, and the fourth inequality is due to VD and.3. Thus.7 holds. We next introduce the Faber-Krahn inequality, see [GT, Section 3.3] for more details. For λ > 0, we define E λ f, g = Ef, g + λ fxgx µdx for f, g F. For any open set D, F D is defined to be the E -closure in F of F C c D. Define λ D = inf {Ef, f : f F D with f 2 = },.8 the bottom of the Dirichlet spectrum of L on D. Definition.8. The D space, d, µ, E satisfies the Faber-Krahn inequality FKφ, if there exist positive constants C and ν such that for any ball Bx, r and any open set D Bx, r, λ D C φr V x, r/µdν..9 We remark that since V x, r µd for D Bx, r, if.9 holds for some ν = ν 0 > 0, it holds for every ν 0, ν 0. So without loss of generality, we may and do assume 0 < ν <. Recall that X = {X t } is the Hunt process associated with the regular Dirichlet form E, F on L 2 ; µ with properly exceptional set N, and 0 := \N. For a set A, define the exit time τ A = inf{t > 0 : X t / A}. 9

10 Definition.9. We say that E φ holds if there is a constant c > such that for all r > 0 and all x 0, c φr E x [τ Bx,r ] c φr. We say that E φ, resp. E φ, holds if the upper bound resp. lower bound in the inequality above holds. Under.3, it is easy to see that E φ, and E φ, imply the following statements respectively: E y [τ Bx,r ] c 2 φr for all x, y Bx, r/2 0, r > 0; E y [τ Bx,r ] c 3 φr for all x, y 0, r > 0. Indeed, for y Bx, r/2 0, we have E y [τ Bx,r ] E y [τ By,r/2 ] c φr/2 c 2 φr. Similarly, for y Bx, r 0, we have E y [τ Bx,r ] E y [τ By,2r ] c φ2r c 3 φr and E y [τ Bx,r ] = 0 for y 0 \ Bx, r. Definition.0. We say EP φ, holds if there is a constant c > 0 such that for all r, t > 0 and all x 0, P x τ Bx,r t ct φr. We say EP φ,,ε holds, if there exist constants ε, δ 0, such that for any ball B = Bx 0, r with radius r > 0, P x τ B δφr ε for all x Bx 0, r/4 0. It is clear that EP φ, implies EP φ,,ε. We will prove in Lemma 4.6 below that under.3, E φ implies EP φ,,ε. Definition.. i We say that HKφ holds if there exists a heat kernel pt, x, y of the semigroup {P t } associated with E, F, which has the following estimates for all t > 0 and all x, y 0, c V x, φ t t V x, dx, yφdx, y pt, x, y c 2 V x, φ t t.20, V x, dx, yφdx, y where c, c 2 > 0 are constants independent of x, y 0 and t > 0. Here the inverse function of the strictly increasing function t φt is denoted by φ t. ii We say UHKφ resp. LHKφ holds if the upper bound resp. the lower bound in.20 holds. iii We say UHKDφ holds if there is a constant c > 0 such that for all t > 0 and all x 0, c pt, x, x V x, φ t. 0

11 Remark.2. We have three remarks about this definition. i First, note that under VD V y, φ t t V y, dx, yφdx, y V x, φ t t V x, dx, yφdx, y..2 Therefore we can replace V x, dx, y by V y, dx, y in.20 by modifying the values of c and c 2. This is because V x, φ t if and only if dx, y φ t, and by.2, C µ + This together with.6 yields.2. t V x, dx, yφdx, y dx, y d2 V x, φ t φ t V y, φ t C µ + dx, y d2. φ t ii By the Cauchy-Schwarz inequality, one can easily see that UHKDφ is equivalent to the existence of c > 0 so that pt, x, y c V x, φ tv y, φ t for x, y 0 and t > 0. Consequently, by Remark.3ii, under VD, UHKDφ implies that for every c > 0 there is a constant c 2 > 0 so that pt, x, y c 2 V x, φ t for x, y 0 with dx, y c φ t. iii It will be implied by Theorem.3 and Lemma 5.6 below that if VD,.3 and HKφ hold, then the heat kernel pt, x, y is Hölder continuous on x, y for every t > 0, and so.20 holds for all x, y. In the following, we say E, F is conservative if its associated Hunt process X has infinite lifetime. This is equivalent to P t = a.e. on 0 for every t > 0. It follows from Proposition 3.2 that LHKφ implies that E, F is conservative. We can now state the stability of the heat kernel estimates HKφ. The following is the main result of this paper. Theorem.3. Assume that the metric measure space, d, µ satisfies VD and RVD, and φ satisfies.3. Then the following are equivalent: HKφ. 2 J φ and E φ. 3 J φ and SCSJφ. 4 J φ and CSJφ.

12 Remark.4. i When φ satisfies.3 with β 2 < 2, by Remark.7, SCSJφ holds and so in this case we have by Theorem.3 that HKφ J φ. Thus Theorem.3 not only recovers but also extends the main results in [CK, CK2] except for the cases where Jx, y decays exponentially when dx, y is large, in the sense that the underlying spaces here are general metric measure spaces satisfying VD and RVD. ii A new point of Theorem.3 is that it gives us the stability of heat kernel estimates for general symmetric jump processes of mixed-type, including α-stable-like processes with α 2, on general metric measure spaces when the underlying spaces have walk dimension larger than 2. In particular, if, d, µ is a metric measure space on which there is an anomalous diffusion with walk dimension d w > 2 such as Sierpinski gaskets or carpets, one can deduce from the subordinate anomalous diffusion the two-sided heat kernel estimates of any symmetric jump processes with jumping kernel Jx, y of α-stable type or mixed stable type; see Section 6 for details. This in particular answers a long standing problem in the field. In the process of establishing Theorem.3, we also obtain the following characterizations for UHKφ. Theorem.5. Assume that the metric measure space, d, µ satisfies VD and RVD, and φ satisfies.3. Then the following are equivalent: UHKφ and E, F is conservative. 2 UHKDφ, J φ, and E φ. 3 FKφ, J φ, and SCSJφ. 4 FKφ, J φ, and CSJφ. We point out that UHKφ alone does not imply the conservativeness of the associated Dirichlet form E, F. For example, censored also called resurrected α-stable processes in upper half spaces with α, 2 enjoy UHKφ with φr = r α but have finite lifetime; see [CT, Theorem.2]. We also note that RVD are only used in the proofs of UHKDφ = FKφ and J φ, = FKφ. We emphasize again that in our main results above, the underlying metric measure space, d, µ is only assumed to satisfy the general VD and RVD. We do not assume the uniform comparability of volume of balls; that is, we do not assume the existence of a non-decreasing function V on [0, with V 0 = 0 so that µbx, r V r for all x and r > 0. Neither do we assume to be connected nor, d to be geodesic. As mentioned earlier, parabolic Harnack inequality is equivalent to the two-sided Aronson type heat kernel estimates for diffusion processes. In subsequent papers [CKW, CKW2], we study stability of parabolic Harnack inequality and elliptic Harnack inequality respectively for symmetric jump processes on metric measure spaces. Let Z := {V s, X s } s 0 be the space-time process where V s = V 0 s corresponding to X. The filtration generated by Z satisfying the usual conditions will be denoted by { F s ; s 0}. The law of the space-time process s Z s starting from t, x will be denoted by P t,x. For every open subset D of [0,, define τ D = inf{s > 0 : Z s / D}. 2

13 Definition.6. i We say that a Borel measurable function ut, x on [0, is parabolic or caloric on D = a, b Bx 0, r for the process X if there is a properly exceptional set N u associated with the process X so that for every relatively compact open subset U of D, ut, x = E t,x uz τu for every t, x U [0, \N u. ii We say that the parabolic Harnack inequality PHIφ holds for the process X, if there exist constants 0 < C < C 2 < C 3 < C 4, C 5 > and C 6 > 0 such that for every x 0, t 0 0, R > 0 and for every non-negative function u = ut, x on [0, that is parabolic on cylinder Qt 0, x 0, φc 4 R, C 5 R := t 0, t 0 +φc 4 R Bx 0, C 5 R, ess sup Q u C 6 ess inf Q+ u,.22 where Q := t 0 + φc R, t 0 + φc 2 R Bx 0, R and Q + := t 0 + φc 3 R, t 0 + φc 4 R Bx 0, R. We note that the above PHIφ is called a weak parabolic Harnack inequality in [BGK2], in the sense that.22 holds for some C,, C 5. It is called a parabolic Harnack inequality in [BGK2] if.22 holds for any choice of positive constants C,, C 5 with C 6 = C 6 C,..., C 5 <. Since our underlying metric measure space may not be geodesic, one can not expect to deduce parabolic Harnack inequality from weak parabolic Harnack inequality. As a consequence of Theorem.3 and various equivalent characterizations of parabolic Harnack inequality established in [CKW], we have the following. Theorem.7. Suppose that the metric measure space, d, µ satisfies VD and RVD, and φ satisfies.3. Then HKφ PHIφ + J φ,. Thus for symmetric jump processes, parabolic Harnack inequality PHIφ is strictly weaker than HKφ. This fact was proved for symmetric jump processes on graphs with V x, r r d and φr = r α for all x, r > 0 and some d, α 0, 2 in [BBK2, Theorem.5]. Some of the main results of this paper were presented at the 38th Conference on Stochastic Processes and their Applications held at the University of Oxford, UK from July 3-7, 205 and at the International Conference on Stochastic Analysis and Related Topics held at Wuhan University, China from August 3-8, 205. While we were at the final stage of finalizing this paper, we received a copy of [S, S2] from. urugan. Stability of discrete-time long range random walks of stable-like jumps on infinite connected locally finite graphs is studied in [S2]. Their results are quite similar to ours when specialized to the case of φr = r α but the techniques and the settings are somewhat different. They work on discrete-time random walks on infinite connected locally finite graphs equipped with graph distance, while we work on continuous-time symmetric jump processes on general metric measure space and with much more general jumping mechanisms. oreover, it is assumed in [S2] that there is a constant c so that c µ{x} c for every x and the d-set condition that there are constants C 3

14 and d f > 0 so that C r d f V x, r Cr d f for every x and r, while we only assume general VD and RVD. Technically, their approach is to generalize the so-called Davies method to obtain the off-diagonal heat kernel upper bound from the on-diagonal upper bound to be applicable when α > 2 under the assumption of cut-off Sobolev inequalities. Quite recently, we also learned from A. Grigor yan [GHH] that they are also working on the same topic of this paper on metric measure spaces with the d-set condition and the conservativeness assumption on E, F. Their results are also quite similar to ours, again specialized to the case of φr = r α, but the techniques are also somewhat different. Their approach [GHH] is to deduce a kind of weak Harnack inequalities first from J φ and CSJφ, which they call generalized capacity condition. They then obtain uniform Hölder continuity of harmonic functions, which plays the key role for them to obtain the near-diagonal lower heat kernel bound that corresponds to 3.2. As we see below, our approach is different from theirs. We emphasize here that in this paper we do not assume a priori that E, F is conservative. The rest of the paper is organized as follows. In the next section, we present some preliminary results about J φ, and CSJφ. In particular, in Proposition 2.4 we show that the leading constant in CSJφ is self-improving. Sections 3, 4 and 5 are devoted to the proofs of = 3, 4 = 2 and 2 = in Theorems.3 and.5, respectively. Among them, Section 4 is the most difficult part, where in Subsection 4.2 we establish the Caccioppoli inequality and the L p -mean value inequality for subharmonic functions associated with symmetric jump processes, and in Subsection 4.4 eyer s decomposition is realized for jump processes in the VD setting. Both subsections are of interest in their own. In Section 6, some examples are given to illustrate the applications of our results, and a counterexample is also given to indicate that CSJφ is necessary for HKφ in general setting. For reader s convenience, some known facts used in this paper are streamlined and collected in Subsections of the Appendix. In connection with the implication of 3 = in Theorem.5, we show in Subsection 7.5 that SCSJφ+J φ, = E, F is conservative; in other words FKφ is not needed for establishing the conservativeness of E, F. We remark that, in order to increase the readability of the paper, we have tried to make the paper as self-contained as possible. Figure illustrates implications of various conditions and flow of our proofs. Throughout this paper, we will use c, with or without subscripts, to denote strictly positive finite constants whose values are insignificant and may change from line to line. For p [, ], we will use f p to denote the L p -norm in L p ; µ. For B = Bx 0, r and a > 0, we use ab to denote the ball Bx 0, ar, and B := {x : dx, x 0 r}. For any subset D of, D c denotes the complement of D in. 2 Preliminaries For basic properties and definitions related to Dirichlet forms, such as the relation between regular Dirichlet forms and Hunt processes, associated semigroups, resolvents, capacity and quasi-continuity, we refer the reader to [CF, FOT]. We begin with the following estimate, which is essentially given in [CK2, Lemma 2.]. 4

15 4. Lem4.5 J φ, FKφ E φ, Prop7.6 J φ, 4.4 CSJφ E φ ζ= UHKDφ 4.3 Lem LHKφ 5.2 J φ 5. UHKφ SCSJφ Prop ζ= Figure : diagram Lemma 2.. Assume that VD and.3 hold. Then there exists a constant c 0 > 0 such that Bx,r V x, dx, y φdx, y µdy c 0 for every x and r > φr c Thus if, in addition, J φ, holds, then there exists a constant c > 0 such that Jx, y µdy c for every x and r > 0. Bx,r φr c Proof. For completeness, we present a proof here. By J φ, and VD, we have for every x and r > 0, Bx,r V x, dx, y φdx, y µdy c = V x, dx, y φdx, y µdy i=0 c 2 i=0 Bx,2 i+ r\bx,2 i r V x, 2 i r φ2 i r V x, 2i+ r i=0 φ2 i r c 3 φr i=0 2 iβ c 4 φr, where the lower bound in.3 is used in the second to the last inequality. 5

16 Fix ρ > 0 and define a bilinear form E ρ, F by E ρ u, v = ux uyvx vy {dx,y ρ} Jdx, dy. 2.2 Clearly, the form E ρ u, v is well defined for u, v F, and E ρ u, u Eu, u for all u F. Assume that VD,.3 and J φ, hold. Then we have by Lemma 2. that for all u F, Eu, u E ρ u, u = ux uy 2 {dx,y>ρ} Jdx, dy 4 u 2 x µdx Jx, y µdy c 0 u Bx,ρ φρ. c Thus E u, u is equivalent to E ρ u, u := E ρ u, u + u 2 2 for every u F. Hence E ρ, F is a regular Dirichlet form on L 2 ; µ. Throughout this paper, we call E ρ, F ρ-truncated Dirichlet form. The Hunt process associated with E ρ, F can be identified in distribution with the Hunt process of the original Dirichlet form E, F by removing those jumps of size larger than ρ. Assume that J φ, holds, and in particular.4 holds. Define Jx, dy = Jx, y µdy. Let J ρ dx, dy = {dx,y ρ} Jdx, dy, J ρ x, dy = {dx,y ρ} Jx, dy, and Γ ρ f, g be the carré du champ of the ρ-truncated Dirichlet form E ρ, F; namely, E ρ f, g = µdx fx fygx gy J ρ x, dy =: dγ ρ f, g. We now define variants of CSJφ. Definition 2.2. Let φ be an increasing function on R + with φ0 = 0, and C 0 0, ]. For any x 0 and 0 < r R, set U = Bx 0, R + r \ Bx 0, R, U = Bx 0, R + + C 0 r \ Bx 0, R C 0 r and U = Bx 0, R + 2r \ Bx 0, R r. i We say that condition CSJ ρ φ holds if the following holds: there exist constants C 0 0, ] and C, C 2 > 0 such that for every 0 < r R, almost all x 0 and any f F, there exists a cut-off function ϕ F b for Bx 0, R Bx 0, R + r so that the following holds for all ρ > 0: f 2 dγ ρ ϕ, ϕ C fx fy Bx 0,R++C 0 r U U 2 J ρ dx, dy + C f 2 dµ. φr ρ Bx 0,R++C 0 r ii We say that condition CSAJφ holds if there exist constants C 0 0, ] and C, C 2 > 0 such that for every 0 < r R, almost all x 0 and any f F, there exists a cut-off function ϕ F b for Bx 0, R Bx 0, R + r so that the following holds for all ρ > 0: U f 2 dγϕ, ϕ C U U fx fy 2 Jdx, dy + C 2 φr 6 U f 2 dµ. 2.5

17 iii We say that condition CSAJ ρ φ holds if the following holds: there exist constants C 0 0, ] and C, C 2 > 0 such that for every 0 < r R, almost all x 0 and any f F, there exists a cut-off function ϕ F b for Bx 0, R Bx 0, R + r so that the following holds for all ρ > 0: U f 2 dγ ρ ϕ, ϕ c U U fx fy 2 J ρ dx, dy + C 2 φr ρ U f 2 dµ. iv We say that condition CSJ ρ φ + holds if the following holds: for any ε > 0, there exists a constant c ε > 0 such that for every 0 < r R, almost all x 0 and any f F, there exists a cut-off function ϕ F b for Bx 0, R Bx 0, R + r so that the following holds for all ρ > 0: f 2 dγ ρ ϕ, ϕ ε ϕ 2 xfx fy 2 J ρ dx, dy Bx 0,R+2r U U + c 2.6 ε f 2 dµ. φr ρ Bx 0,R+2r v We say that condition CSAJ ρ φ + holds if the following holds: for any ε > 0, there exists a constant c ε > 0 such that for every 0 < r R, almost all x 0 and any f F, there exists a cut-off function ϕ F b for Bx 0, R Bx 0, R + r so that the following holds for all ρ > 0: f 2 dγ ρ ϕ, ϕ ε ϕ 2 x fx fy 2 J ρ dx, dy + c ε f 2 dµ. U U U φr ρ U For open subsets A and B of with A B, and for any ρ > 0, define Cap ρ A, B = inf{e ρ ϕ, ϕ : ϕ F, ϕ A =, ϕ B c = 0}. Proposition 2.3. Let φ be an increasing function on R +. Assume that VD,.3 and J φ, hold. The following hold. CSJφ is equivalent to CSJ ρ φ. 2 CSJφ is implied by CSAJφ. 3 CSAJφ is equivalent to CSAJ ρ φ. 4 If CSJ ρ φ resp. CSAJ ρ φ holds for some C 0 0, ], then for any C 0 [C 0, ], there exist constants C, C 2 > 0 where C 2 depends on C 0 such that CSJ ρ φ resp. CSAJ ρ φ holds for C 0. 5 If CSJφ holds, then there is a constant c 0 > 0 such that for every 0 < r R, ρ > 0 and almost all x, Cap ρ V x, R + r Bx, R, Bx, R + r c 0. φr ρ 7

18 In particular, we have V x, R + r CapBx, R, Bx, R + r c φr Proof. Letting ρ, we see that 2.4 implies.7. Now, let ρ > 0 and assume that.7 holds. Then there exist constants C 0 0, ] and C, C 2 > 0 such that for every 0 < r R, almost all x 0 and any f F, there exists a cut-off function ϕ F b for Bx 0, R Bx 0, R + r such that f 2 dγ ρ ϕ, ϕ Bx 0,R++C 0 r f 2 dγϕ, ϕ Bx 0,R++C 0 r C fx fy 2 Jdx, dy + C 2 f 2 dµ U U φr Bx 0,R++C 0 r C fx fy U U 2 J ρ dx, dy + 2C f 2 x + f 2 y {dx,y>ρ} Jdx, dy U U + C 2 f 2 dµ φr Bx 0,R++C 0 r C fx fy 2 J ρ dx, dy + C 3 f 2 dµ, U U φr ρ Bx 0,R++C 0 r where Lemma 2. is used in the last inequality. 2 Fix x 0, 0 < r R and C 0 0, ]. Let ϕ F b be a cut-off function for Bx 0, R Bx 0, R + r. Since ϕx = on x Bx 0, R, we have for f F, f 2 dγϕ, ϕ = f 2 x µdx ϕy 2 Jx, y µdy Bx 0,R C 0 r Bx 0,R C 0 r f 2 x µdx Jx, y µdy Bx 0,R C 0 r Bx 0,R c f 2 x µdx Jx, y µdy Bx 0,R C 0 r Bx,C 0 r c c f 2 dµ φc 0 r c 2 φr Bx 0,R C 0 r Bx 0,R C 0 r f 2 dµ, where we used Lemma 2. and.3 in the last two inequalities. This together with 2.5 gives us the desired conclusion. 3 This can be proved in the same way as. 4 This is easy. Indeed, for x 0, 0 < r R, C 0 0, ] and C 0 [C 0, ], set D = Bx 0, R++C 0r\Bx 0, R++C 0 r and D 2 = Bx 0, R C 0 r\bx 0, R C 0r. 8

19 Let ϕ F b be a cut-off function for Bx 0, R Bx 0, R + r. Then for any f F and ρ > 0, f 2 dγ ρ ϕ, ϕ = f 2 x µdx ϕ 2 yj ρ x, y µdy D D Bx 0,R+r f 2 x µdx Jx, y µdy D Bx,C 0 r c c f 2 dµ, φr D where Lemma 2. and.3 are used in the last inequality. Similarly, for any f F and ρ > 0, f 2 dγ ρ ϕ, ϕ c 2 f 2 dµ. D 2 φr D 2 From both inequalities above we can get the desired assertion for C 0 C 0. 5 In view of and 4, CSJ ρ φ holds for every ρ > 0 and we can and do take C 0 = in.7. Fix x 0 and write B s := Bx 0, s for s 0. Let f F with 0 f such that f BR+2r = and f B c R+3r = 0. For any ρ > 0 and 0 < r R, let ϕ F b be the cut-off function for B R B R+r associated with f in CSJ ρ φ. Then Cap ρ B R, B R+r dγ ρ ϕ, ϕ + dγ ρ ϕ, ϕ B R+2r BR+2r c = f 2 dγ ρ ϕ, ϕ + dγ ρ ϕ, ϕ B R+2r BR+2r c c fx fy 2 J ρ dx, dy B R+r \B R B R+2r \B R r c 2 + f 2 dµ + µdx ϕ 2 yjx, y µdy φr ρ B R+2r B R+r c 2µB R+2r φr ρ c 4µB R+r φr ρ, + c 3µB R+r φr B c R+2r where we used CSJ ρ φ in the second inequality and Lemma 2. with VD in the third inequality. Now let f ρ be the potential whose E ρ -norm gives the capacity. Then the Cesàro mean of a subsequence of f ρ converges in E -norm, say to f, and Ef, f is no less than the capacity corresponding to ρ =. So 2.7 is proved. CSAJ ρ φ is self- We next show that the leading constant in CSJ ρ φ resp. improving in the following sense. Proposition 2.4. Suppose that VD,.3 and J φ, hold. Then the following hold. 9

20 CSJ ρ φ is equivalent to CSJ ρ φ +. 2 CSAJ ρ φ is equivalent to CSAJ ρ φ +. Proof. We only prove, since 2 can be verified similarly. It is clear that CSJ ρ φ + implies CSJ ρ φ. Below, we assume that CSJ ρ φ holds. Fix x 0, 0 < r R and f F. For s > 0, set B s = Bx 0, s. The goal is to construct a cut-off function ϕ F b for B R B R+r so that 2.6 holds. Without loss of generality, in the following we may and do assume that B R+2r f 2 dµ > 0; otherwise, 2.6 holds trivially. For λ > 0 whose exact value to be determined later, let s n = c 0 re nλ/2β 2, where c 0 := c 0 λ is chosen so that n= s n = r and β 2 is given in.3. Set r 0 = 0 and r n = n s k, n. k= Clearly, R < R + r < R + r 2 < < R + r. For any n 0, define U n := B R+rn+ \ B R+rn, and Un := B R+rn+ +s n+ \ B R+rn sn+. Let θ > 0, whose value also to be determined later, and define f θ := f + θ. By CSJ ρ φ with C 0 = ; see Proposition 2.3 4, there exists a cut-off function ϕ n for B R+rn B R+rn+ such that fθ 2 dγ ρ ϕ n, ϕ n C f θ x f θ y 2 J ρ dx, dy B R+rn+ +s n+ + U n U n C 2 φs n+ ρ B R+rn+ +s n+ f 2 θ dµ, 2.8 where C, C 2 are positive constants independent of f θ and ϕ n. Here, we mention that since E, F is a regular Dirichlet form on L 2, µ, f θ F loc, and so, by Remark.6v, CSJ ρ φ can apply to f θ. Let b n = e nλ and define ϕ = b n b n ϕ n. 2.9 n= Then ϕ is a cut-off function for B R B R+r, because ϕ = on B R and ϕ = 0 on B c R+r. On U n we have ϕ = b n b n ϕ n + b n, so that b n ϕ b n on U n. In particular, on U n b n b n ϕb n b n b n = e λ ϕ. 2.0 Below, we verify that the function ϕ defined by 2.9 satisfies 2.6 and ϕ F b. For this, we will make a non-trivial and substantial modification of the proof of [AB, Lemma 5.]. Set F n,m x, y = f 2 θ xϕ n x ϕ n yϕ m x ϕ m y 20

21 for any n, m. Then fθ 2 dγ ρ ϕ, ϕ = B R+2r fθ 2 x B R+2r B R+2r [ 2 2 b n b n ϕ n x ϕ n y J ρ dx, dy n= n 2 b n b n b m b m F n,m x, y n=3 m= = : I + I 2 + I 3. b n b n b n 2 b n F n,n x, y n=2 ] b n b n 2 F n,n x, y J ρ dx, dy n= For n m + 2, since F n,m x, y = 0 for x, y B R+rn or x, y / B R+rm+, we can deduce that F n,m x, y 0 only if x B R+rm+, y / B R+rn or x / B R+rn, y B R+rm+. Since F n,m x, y fθ 2 x, using Lemma 2., we have F n,m x, y J ρ dx, dy B R+2r = + B R+2r B R+rm+ BR+rn c B R+2r BR+rn c B R+rm+ 2. c φ n k=m+2 s fθ 2 x µdx k B R+2r c fθ 2 x µdx. φs m+2 B R+2r Note that, according to.3, we have Therefore, φr φs k+2 r β2 c c 0 λre k+2λ/2β 2 = c eλ e kλ/2 c 0 λ = c e λ e λ /2 β 2 c 0 λ β 2 bk b k. /2 This together with 2. implies b k b k /2 φs k+2 c λφr. 2.2 n 2 c I 2 b n b n b m b m fθ 2 x µdx φs n=3 m= m+2 B R+2r n 2 b n b n b m b m /2 c 2λ fθ 2 x µdx φr n=3 m= B R+2r c 3λ fθ 2 x µdx, φr B R+2r 2

22 because m= b m b m /2 = c 4 λ and n= b n b n =. For I 2, by the Cauchy- Schwarz inequality, we have I 2 2 n=2 2 I 3, B R+2r B R+2r /2 b n b n 2 F n,n x, y 2 J ρ dx, dy /2 b n 2 b n 2 F n,n x, y 2 J ρ dx, dy where we used 2ab /2 a + b for a, b 0 in the last inequality. For I 3, F n,n x, y J ρ dx, dy B R+2r = + F n,n x, y J ρ dx, dy B R+rn+ +s n+ B R+2r \B R+rn+ +s n+ F n,n x, y J ρ c dx, dy + fθ 2 x µdx B R+rn+ +s n+ φs n+ B R+2r C f θ x f θ y 2 J ρ dx, dy + c + C 2 fθ 2 x µdx, φs n+ ρ B R+2r U n U n where we used Lemma 2. in the second line and 2.8 in the last line. Using 2.0 and 2.2, and noting that s k+ s k+2 and m= b m b m 3/2 + m= b m b m 2 = c 5 λ, we have I 3 C 3 e λ 2 f θ x f θ y 2 J ρ dx, dy + c 6λ fθ 2 dµ, U U φr ρ B R+2r where we used the facts that {U n ; n } are disjoint, n= U n = U, and Un U for all n. For any ε > 0, we now choose λ so that 3C 3 e λ 2 = ε, and obtain 2.6 for f θ, i.e., fθ 2 dγ ρ ϕ, ϕ ε f θ x f θ y 2 J ρ dx, dy B R+2r U U + C 4ε φr ρ B R+2r f 2 θ x µdx, 2.3 where the positive constant C 4 ε is independent of θ. It is clear that the left hand side of 2.3 is bigger than B R+2r f 2 dγ ρ ϕ, ϕ. On the other hand, since for any x, y and θ > 0, f θ x f θ y f x f y fx fy, it holds that f θ x f θ y 2 J ρ dx, dy fx fy 2 J ρ dx, dy. U U U U Note that B R+2r f 2 θ dµ 2 B R+2r f 2 dµ + θ 2 µb R+2r 22.

23 Then, by choosing we have θ = B R+2r f 2 dµ µb R+2r /2 > 0, fθ 2 dµ 4 f 2 dµ. B R+2r B R+2r Hence, for this choice of θ, we know that the left hand side of 2.3 is smaller than ε fx fy 2 J ρ dx, dy + 4C 4ε f 2 x µdx. U U φr ρ B R+2r Combining both estimates above, we prove that 2.6 holds for f. Next, we prove that ϕ F b. Let ϕ i = i n= b n b n ϕ n for i. It is clear that ϕ i F b and ϕ i ϕ as i. So in order to prove ϕ F b, it suffices to verify that lim i,j Eϕi ϕ j, ϕ i ϕ j = Indeed, for any i > j, we can follow the arguments above and obtain that B R+2r dγϕ i ϕ j, ϕ i ϕ j θ 2 B R+2r f 2 θ dγϕ i ϕ j, ϕ i ϕ j θ 2 e c jλ 7 λ f θ x f θ y 2 Jdx, dy + c 8λ fθ 2 x µdx. U U φr B R+2r On the other hand, by Lemma 2. and the fact that supp ϕ i ϕ j B R+r, B c R+2r dγϕ i ϕ j, ϕ i ϕ j i n=j+ b n b n e jλ c 9λ φr µb R+r. 2 B c R+2r B R+r Jx, y µdy µdx Combining with both inequalities above, we obtain 2.4. As a direct consequence of Proposition 2.3 and Proposition 2.4, we have the following corollary. Corollary 2.5. Suppose that VD,.3, J φ, and CSJφ hold. Then there exists a constant c > 0 such that for every 0 < r R, almost all x 0 and any f F, there 23

24 exists a cut-off function ϕ F b for Bx 0, R Bx 0, R + r so that the following holds for all ρ 0, ]: f 2 dγ ρ ϕ, ϕ ϕ 2 xfx fy 2 J ρ dx, dy Bx 0,R+2r 8 U U 2.5 c + f 2 dµ, φr ρ Bx 0,R+2r where U = Bx 0, R + r \ Bx 0, R and U = Bx 0, R + 2r \ Bx 0, R r. Remark 2.6. According to all the arguments above, we can easily obtain that Propositions 2.3, 2.4 and Corollary 2.5 with small modifications i.e. the cut-off function ϕ F b can be chosen to be independent of f F hold for SCSJφ. We close this subsection by the following statement. Lemma 2.7. Assume that VD,.3 and UHKφ hold and that E, F is conservative. Then EP φ, holds. Proof. We first verify that there is a constant c > 0 such that for each t, r > 0 and for almost all x, pt, x, y µdy c t Bx,r φr. c Indeed, we only need to consider the case that φr > t; otherwise, the inequality above holds trivially with c =. According to UHKφ, VD and.3, for any t, r > 0 with φr > t and almost all x, Bx,r c pt, x, y µdy = i=0 i=0 Bx,2 i+ r\bx,2 i r c 2 tv x, 2 i+ r V x, 2 i rφ2 i r c 3t φr pt, x, y µdy i=0 2 iβ c 4t φr. Now, since E, F is conservative, by the strong arkov property, for any each t, r > 0 and for almost all x, P x τ Bx,r t = P x τ Bx,r t, X 2t Bx, r/2 c + P x τ Bx,r t, X 2t Bx, r/2 P x X 2t Bx, r/2 c + c 5t φr, sup P z X 2t s Bz, r/2 c z / Bx,r c,s t which yields EP φ,. Note that the conservativeness of E, F is used in the equality above. Indeed, without the conservativeness, there must be an extra term P x τ Bx,r t, ζ 2t in the right hand side of the above equality, where ζ is the lifetime of X. 24

25 3 Implications of heat kernel estimates In this section, we will prove = 3 in Theorems.3 and.5. We point out that, under VD, RVD and.3, UHKφ = FKφ is given in Proposition 7.6 in the Appendix. 3. UHKφ + E, F is conservative = J φ,, and HKφ = J φ We first show the following, where, for future reference, it is formulated for a general Hunt process Y that admits no killings inside. Proposition 3.. Suppose that Y = {Y t, t 0, P x, x E} is an arbitrary Hunt process on a locally compact separable metric space E that admits no killings inside E. Denote its lifetime by ζ. If there is a constant c 0 > 0 so that then P x ζ = = for every x E. P x ζ = c 0 for every x E, 3. 2 Suppose that VD holds, the heat kernel pt, x, y of the process Y exists, and there exist constants ε 0, and c > 0 such that for any x E and t > 0, pt, x, y c V x, φ t for y Bx, εφ t, 3.2 where φ : R + R + is a strictly increasing continuous function with φ0 = 0. Then P x ζ = = for every x E. In particular, LHKφ implies ζ = a.s. Proof. Let {Ft Y ; t 0} be the minimal augmented filtration generated by the Hunt process Y, and set ux := P x ζ =. Then we have ux c 0 > 0 for x E. Note that uy t = {ζ>t} uy t = E [ ] x {ζ= } Ft Y is a bounded martingale with lim t uy t = {ζ= }. Let {K j ; j } be an increasing sequence of compact sets so that j=k j = E and define τ j = inf{t 0 : Y t / K j }. Since the Hunt process Y admits no killings inside E, we have τ j < ζ a.s. for every j. Clearly lim j τ j = ζ. By the optional stopping theorem, we have for x E, ux = lim j E x uy τj = E x [ lim j uy τ j = E x [ lim j uy τ j {ζ< } + lim t uy t {ζ= } c 0 P x ζ < + P x ζ = = c 0 P x ζ < + ux. ] ] 25

26 It follows that P x ζ < = 0 for every x E. 2 By 3.2 and the equivalent characterization.0 of VD, we have for every x E and t > 0, P x ζ > t Bx,εφ t pt, x, y µdy Bx,εφ t c V x, φ t µdy c 2 > 0. Passing t, we get P x ζ = c 2 for every x E. The conclusion now follows immediately from. Remark 3.2. i The condition that Y admits no killings inside E is needed for Proposition 3. to hold. That is, condition 3. alone does not guarantee Y is conservative. Here is a counterexample. Let Y be the process obtained from a Brownian motion W = {W t } in R 3 killed according to the potential qx := B0, x. That is, for f 0 on R 3, t ] E x [fy t ] = E [fw x t exp B0, W s ds. 3.3 Denote by ζ the lifetime of Y. We claim that 3. holds for Y. Indeed, for threedimensional Brownian motion W, we have inf P x σ x R 3 B0, W = = : x 2 0 sup P x σb0, W < = x R 3 : x 2 where σ W B0, = inf{t 0 : W t B0, }. Clearly for x B0, 2 c, sup x R 3 : x 2 x = 2, P x ζ = P x σ W B0, = On the other hand, if we use pt, x, y and p 0 t, x, y to denote the transition density function of Y and W with respect to the Lebesgue measure on R 3 respectively, then we have by 3.3 that e t p 0 t, x, y pt, x, y p 0 t, x, y for t > 0 and x, y R 3. Hence there is a constant c 0, so that P x Y R 3 \ B0, 2 c for every x B0,. Using the arkov property of Y at time, we have from 3.4 that P x ζ = c /2 for every x B0,. This establishes 3. with c 0 = c /2. However P x ζ < > 0 for every x R 3. ii In the setting of this paper, X is the symmetric Hunt process associated with the regular Dirichlet form E, F given by. that has no killing term. So X always admits no killings inside. 26

27 The next proposition in particular shows that UHKφ implies.4. Proposition 3.3. Under VD and.3, UHKφ and E, F is conservative = J φ,, and HKφ = J φ. Proof. The proof is easy and standard, and we only consider HKφ = J φ for simplicity. Consider the form E t f, g := f P t f, g /t. Since E, F is conservative by Proposition 3.2, we can write E t f, g = fx fygx gypt, x, y µdx µdy. 2t It is well known that lim t 0 E t f, g = Ef, g for all f, g F. Let A, B be disjoint compact sets, and take f, g F such that supp f A and supp g B. Then E t f, g = fxgypt, x, y µdy µdx t 0 fxgy Jdx, dy. t Using HKφ, we obtain fxgy Jdx, dy A B A B A B fxgy µdy µdx, V x, dx, yφdx, y for all f, g F such that supp f A and supp g B. Since A, B are arbitrary disjoint compact sets, it follows that Jdx, dy is absolutely continuous w.r.t. µdx µdy, and J φ holds. A B 3.2 UHKφ and E, F is conservative = SCSJφ In this subsection, we give the proof that UHKφ and the conservativeness of E, F imply SCSJφ. For D and λ > 0, define τd G D λ fx = E x e λt fx t dt, x 0. 0 Lemma 3.4. Suppose that VD,.3 and UHKφ hold, and that E, F is conservative. Let x 0, 0 < r R, and define D 0 = Bx 0, R + 9r/0 \ Bx 0, R + r/0, D = Bx 0, R + 4r/5 \ Bx 0, R + r/5, D 2 = Bx 0, R + 3r/5 \ Bx 0, R + 2r/5. Let λ = φr, and set h = G D 0 λ D. Then h F D0 and hx φr for all x 0. oreover, there exists a constant c > 0, independent of x 0, r and R, so that hx c φr for all x D