Intrinsic Contractivity of Feynman-Kac Semigroups for Symmetric Jump Processes with Infinite Range Jumps

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by  Xin Chen

INTRINSIC CONTRACTIVITY PROPERTIES OF FEYNMAN-KAC
SEMIGROUPS FOR SYMMETRIC JUMP PROCESSES WITH
INFINITE RANGE JUMPS∗
XIN CHEN JIAN WANG
5 Abstract. Let(Xt)t>0 beasymmetricstrongMarkovprocessgeneratedbynon-
1 local regular Dirichlet form (D,D(D)) as follows
0
2 D(f,g)= f(x)−f(y) g(x)−g(y) J(x,y)dxdy, f,g ∈D(D)

r Rd Rd
Z Z
p whereJ(x,y)isastrict(cid:0)lypositivean(cid:1)(cid:0)dsymmetric(cid:1)measurablefunctiononRd×Rd.
A
Westudytheintrinsichypercontractivity,intrinsicsupercontractivityandintrinsic

4 ultracontractivity for the Feynman-Kac semigroup
2
t
] TtV(f)(x)=Ex exp − 0 V(Xs)ds f(Xt) , x∈Rd,f ∈L2(Rd;dx).
R (cid:18) (cid:16) Z (cid:17) (cid:19)
In particular, we prove that for
P
h. J(x,y)≍|x−y|−d−α1{|x−y|61}+e−|x−y|1{|x−y|>1}
at with α ∈ (0,2) and V(x) = |x|λ with λ > 0, (TtV)t>0 is intrinsically ultracon-
m tractive if and only if λ > 1; and that for symmetric α-stable process (Xt)t>0
[ with α ∈ (0,2) and V(x) = logλ(1+|x|) with some λ > 0, (TtV)t>0 is intrin-
sically ultracontractive (or intrinsically supercontractive) if and only if λ > 1,
v2 and (TtV)t>0 is intrinsically hypercontractive if and only if λ > 1. Besides, we
8 also investigate intrinsic contractivity properties of (TtV)t>0 for the case that
2 liminf|x|→∞V(x)<∞.
1
Keywords: symmetricjumpprocess;Lévyprocess;Dirichletform;Feynman-Kac
6
semigroup; intrinsic contractivity
0
1. MSC 2010: 60G51; 60G52; 60J25;60J75.
0
5
1 1. Introduction and Main Results
:
v
i 1.1. Setting and Assumptions. Let (X ) ,Px be a symmetric strong Markov
X t t>0
process on Rd generated by the following non-local symmetric regular Dirichlet form
r (cid:0) (cid:1)
a
D(f,g) = f(x)−f(y) g(x)−g(y) J(x,y)dxdy,
Rd Rd
Z Z
D(D) = C1(Rd)D(cid:0)1, (cid:1)(cid:0) (cid:1)
c
Here J(x,y) is a strictly positive and symmetric measurable function on Rd × Rd
satisfying that
∗Dedicated to Professor Mu-Fa Chen on the occasion of his 70th birthday.
X. Chen: Department of Mathematics, Shanghai Jiao Tong University, 200240 Shanghai, P.R.
China. [e-mail].
J. Wang: School of Mathematics and Computer Science, Fujian Normal University, 350007
Fuzhou, P.R. China. [e-mail].
1
2 XIN CHEN JIAN WANG
• There exist constants α ,α ∈ (0,2) with α 6 α and positive constants
1 2 1 2
κ,c ,c such that
1 2
(1.1) c |x−y|−d−α1 6 J(x,y) 6 c |x−y|−d−α2, 0 < |x−y| 6 κ,
1 2
(1.2) J(x,y) > 0, |x−y| > κ
and
(1.3) sup J(x,y)dy < ∞;
x∈RdZ{|x−y|>κ}
C1(Rd)denotesthespaceofC1 functionsonRd withcompactsupport,andC1(Rd)D1
c c
denotes the closure of C1(Rd) under the norm kfk := D(f,f)+ f2(x)dx. Ac-
c D1
cording to [1, Theorems 1.1 and 1.2], (Xt)>0 has a symmqetric, boundRed and positive
transition density function p(t,x,y) defined on [0,∞)×Rd ×Rd, whence the asso-
ciated strongly continuous Markov semigroup (T ) is given by
t t>0
T f(x) = Ex f(X ) = p(t,x,y)f(y)dy, x ∈ Rd, t > 0, f ∈ B (Rd),
t t b
Rd
Z
where Ex denotes(cid:0)the ex(cid:1)pectation under the probability measure Px. Throughout
this paper, we further assume that for every t > 0, the function (x,y) 7→ p(t,x,y)
is continuous on Rd ×Rd, see [5, 6, 7, 1, 8] and the references therein for sufficient
conditions ensuring this property. For symmetric Lévy process (X ) , the conti-
t t>0
nuity of density function is equivalent to e−tΨ0(·) ∈ L1(Rd;dx) for any t > 0, where
Ψ is the characteristic exponent or the symbol of Lévy process (X )
0 t t>0
Ex eihξ,Xt−xi = e−tΨ0(ξ), ξ ∈ Rd,t > 0.
Let V be a non-negati(cid:0)ve measu(cid:1)rable and locally bounded measurable (potential)
function on Rd. Define the Feynman-Kac semigroup (TV)
t t>0
t
TV(f)(x) = Ex exp − V(X )ds f(X ) , x ∈ Rd,f ∈ L2(Rd;dx).
t s t
(cid:18) (cid:16) Z0 (cid:17) (cid:19)
It is easy to check that (TV) is a bounded symmetric semigroup on L2(Rd;dx).
t t>0
By assumptions of (X ) , for each t > 0, TV is also bounded from L1(Rd;dx) to
t t>0 t
L∞(Rd;dx), and there exists a symmetric, bounded and positive transition density
function pV(t,x,y) such that for every t > 0, the function (x,y) 7→ pV(t,x,y) is
continuous, and for every 1 6 p 6 ∞,
TVf(x) = pV(t,x,y)f(y)dy, x ∈ Rd,f ∈ Lp(Rd;dx),
t
Rd
Z
see e.g. [10, Section 3.2]. Suppose that for every r > 0,
(1.4) |{x ∈ Rd : V(x) 6 r}| < ∞,
where |A| denotes the Lebesgue measure of Borel set A. According to [4, Proposi-
tion 1.1] (which is essentially based on [21, Corollary 1.3]), the semigroup (TV)
t t>0
is compact. By general theory of semigroups for compact operators, there exists
an orthonormal basis in L2(Rd;dx) of eigenfunctions {φ }∞ associated with corre-
n n=1
sponding eigenvalues {λ }∞ satisfying 0 < λ < λ 6 λ ··· and lim λ = ∞.
n n=1 1 2 3 n→∞ n
That is, L φ = −λ φ and TVφ = e−λntφ , where (L ,D(L )) is the infinitesi-
V n n n t n n V V
mal generator of the semigroup (TV) . The first eigenfunction φ is called ground
t t>0 1
INTRINSIC CONTRACTIVITY PROPERTIES OF FEYNMAN-KAC SEMIGROUPS 3
state in the literature. Indeed, in our setting there exists a version of φ which is
1
bounded, continuous and strictly positive, e.g. see [4, Proposition 1.2].
In the following, we always assume that (1.1)-(1.4) hold, and that the ground state
φ is bounded, continuous and strictly positive.
1
1.2. Previous Work and Motivation. We are concerned with intrinsic contrac-
tivity properties for the semigroup (TV) . We first recall the definitions of these
t t>0
properties introduced in [11]. The semigroup (TV) is intrinsically ultracontrac-
t t>0
tive if and only if for any t > 0, there exists a constant C > 0 such that for all x,
t
y ∈ Rd,
pV(t,x,y) 6 C φ (x)φ (y).
t 1 1
Define
eλ1t
(1.5) T˜Vf(x) = TV((φ f))(x), t > 0,
t φ (x) t 1
1
which is a Markov semigroup on L2(Rd;φ2(x)dx). Then, (TV) is intrinsically
1 t t>0
ultracontractive if and only if (T˜V) ultracontractive, i.e., for every t > 0, T˜V is
t t>0 t
a bounded operator from L2(Rd;φ2(x)dx) to L∞(Rd;φ2(x)dx). If for every 2 <
1 1
p < ∞, there exists a constant t (p) > 0 such that for all t > t (p), T˜V is a
0 0 t
bounded operator from L2(Rd;φ2(x)dx) to Lp(Rd;φ2(x)dx), then we say (T˜V) is
1 1 t t>0
hypercontractive, and equivalently, (TV) is intrinsically hypercontractive. If one
t t>0
cantaket (p) = 0,thenwesay(T˜V) issupercontractive, andequivalently, (TV)
0 t t>0 t t>0
is intrinsically supercontractive. In particular, the intrinsic ultracontractivity is
stronger than the intrinsic supercontractivity, which is in turn stronger than the
intrinsic hypercontractivity.
The intrinsic ultracontractivity of (TV) associated with pure jump symmetric
t t>0
Lévy process (X ) has been investigated in [15, 13, 14]. The approach of all these
t t>0
cited papers is based on two-sided estimates for ground state φ corresponding to
1
the semigroup (TV) , for which some restrictions on the density function of Lévy
t t>0
measure and the potential function V are needed, see [14, Assumptions 2.1 and 2.5]
or assumptions (H1)–(H3) below. Recently, the authors make use of super Poincaré
inequalities with respect to infinite measure developed in [17, 18] and functional
inequalities for non-local Dirichlet forms recently studied in [20, 24, 3] to investigate
theintrinsic ultracontractivity of Feynman-Kacsemigroups forsymmetric jump pro-
cesses in [4]. The main result [4, Theorem 1.3] applies to symmetric jump process
such that associated jump kernel is given by
J(x,y) ≍ |x−y|−d−α1 +e−|x−y|γ1
{|x−y|61} {|x−y|>1}
with α ∈ (0,2) and γ ∈ (1,∞], for which the approach of [15, 13, 14] does not work.
In particular, when γ = ∞,
J(x,y) ≍ |x−y|−d−α1 ,
{|x−y|61}
which is associated with the truncated symmetric α-stable-like process.
As already mentioned in [4], in the model above finite range jumps play an es-
sential role in the behavior of the associated process. In the present setting, the
argument of [4] may lead to obtain some sufficient conditions for intrinsic ultracon-
tractivity of (TV) . However, as we will see from examples below, the conclusions
t t>0
yielded by the approach of [4] are far from optimality because of the large range
jumps. This explains the motivation of our present paper.
4 XIN CHEN JIAN WANG
Themainpurposeofthispaperistoderiveexplicit andsharpcriterionforintrinsic
contractivity properties of Feynman-Kac semigroups for symmetric jump processes
with infinite range jumps. We will use the intrinsic super Poincaré inequalities
introduced in [17, 18] which have been applied in [16, 22] to investigate the intrinsic
ultracontactivity for diffusion processes on Riemannian manifolds. Our method to
establish the intrinsic super Poincaré inequality is efficient for a large class of jump
processes. Indeed, ourmainresultsnotonlyworkforjumpprocesses ofinfiniterange
jumps without technical restrictions used in [15, 13, 14], but also apply to space-
inhomogeneousjumpprocesses andthecorresponding Feynman-Kacsemigroupwith
potential V(x) not necessarily going to infinity as |x| → ∞.
1.3. Main Results. We assume that (1.1)—(1.4) hold in all results of the paper.
To state our main result, we need some necessary assumptions and notations. For
x ∈ Rd, define
inf J(y,z), |x| > 3,
J∗(x) = y−z∈B(x,3/2)
1 , |x| < 3,
(
and
J∗(x)
V∗(x) = sup V(z), ϕ(x) = .
1+V∗(x)
z∈B(x,1)
For any s, r > 0, set
2 2sup J(x,y)−1
α(r,s) = inf : t 6 r and 0<|x−y|6t 6 s .
|B(0,t)|inf ϕ2(x) |B(0,t)|
( x∈B(0,r+t) )
In particular, by (1.1),
sup J(x,y)−1
0<|x−y|6t
lim = 0,
t↓0 |B(0,t)|
which implies that the set of infimum in the definition of α(r,s) is not empty for all
r,s > 0.
1.3.1. Regular Potential Function: lim V(x) = ∞. Without loss of gener-
|x|→∞
ality, we may and can assume that in the result below inf V(x) = 0, otherwise
x∈Rd
one can take V(x) := V(x)−inf V(z) instead of V.
z∈Rd
Theorem 1.1. Suppose that
e
lim V(x) = ∞.
|x|→∞
For any s,δ ,δ > 0, define
1 2
Φ(s) = inf V(x)
|x|>s
and
4 s∧δ
(1.6) β(s) = β(s;δ ,δ ) = δ α Φ−1 , 2 ,
1 2 1
s∧δ 4
(cid:18) (cid:18) 2(cid:19) (cid:19)
where Φ−1 is the generalized inverse of Φ, i.e. Φ−1(r) = inf{s > 0 : Φ(s) > r}.
1 Let β−1(s) be the generalized inverse of β(s). Then, we have the following three
statements.
1
The set {s > 0 : Φ(s) > r} is not empty for every r > 0 because of lim V(x) = ∞ and
|x|→∞
infx∈RdV(x)=0.
INTRINSIC CONTRACTIVITY PROPERTIES OF FEYNMAN-KAC SEMIGROUPS 5
(1) If for any constants δ and δ > 0,
1 2
∞ β−1(s)
ds < ∞, t > infβ,
s
Zt
then the semigroup (TV) is intrinsically ultracontractive.
t t>0
(2) If for any constants δ and δ > 0,
1 2
limslogβ(s) = 0,
s→0
then the semigroup (TV) is intrinsically supercontractive.
t t>0
(3) If for any constants δ and δ > 0,
1 2
limsupslogβ(s) < ∞
s→0
then the semigroup (TV) is intrinsically hypercontractive.
t t>0
For symmetric Lévy process, due to the space-homogenous property it holds that
J(x,y) = J(0,x− y) = ρ(x − y) for x 6= y, where ρ is the density function of the
associated Lévy measure. Obviously, in this case Theorem 1.1 excludes the following
assumptions used in [14] (see Assumptions 2.1, 2.3 and 2.5 therein):
(B1) There are constants c and c > 1 such that
3 4
c−1 sup ρ(z) 6 ρ(x) 6 c inf ρ(z), |x| > 2 (H1)
3 3
B(x,1) z∈B(x,1)
and
ρ(x−z)ρ(z −y)dz 6 c ρ(x−y), |x−y| > 1. (H2)
4
Z{|z−x|>1,|z−y|>1}
(B2) For all 0 < r < r < r 6 1,
1 2
sup sup G (x,y) < ∞,
B(0,r)
x∈B(0,r1)y∈Bc(0,r2)
where B(0,r) denotes the ball with center 0 and radius r, and G (x,y)
B(0,r)
is the Green function for the killed process of (X ) on domain B(0,r).
t t>0
(B3) There exists a constant c > 1 such that
5
sup V(z) 6 c V(x). (H3)
5
z∈B(x,1)
In particular, assumption (B2) is less explicit as it is given by the Green function
rather than the jump rate.
To illustrate the optimality of Theorem 1.1, we consider the following two exam-
ples.
Example 1.2. Let
J(x,y) ≍ |x−y|−d−α1 +e−|x−y|γ1 ,
{|x−y|61} {|x−y|>1}
where α ∈ (0,2) and γ ∈ (0,1]. Let V(x) = |x|λ for some λ > 0. Then, there is a
constant C > 0 such that for all x ∈ Rd,
1
C
φ (x) > 1 ,
1 (1+|x|)λe|x|γ
6 XIN CHEN JIAN WANG
and the associated semigroup (TV) is intrinsically ultracontractive if and only if
t t>0
λ > γ. Furthermore, if λ > γ and for every x ∈ Rd,
(1.7) |z||J(x,x+z)−J(x,x−z)| dz < ∞,
Z{|z|61}
then there is a constant C > 0 such that for all x ∈ Rd,
2
C
φ (x) 6 2 .
1 (1+|x|)λe|x|γ
Remark 1.3. (1) For symmetric Lévy process, (1.7) is automatically satisfied. (2)
When V(x) = |x|λ with λ > 1, one can also use the argument in [4] to prove
the intrinsic ultracontractivity of (TV) . However, the condition λ > 1 is much
t t>0
stronger than λ > γ ∈ (0,1] required by the first assertion in Example 1.2.
Example 1.4. Let (X ) be a symmetric α-stable process with some α ∈ (0,2),
t t>0
i.e.
J(x,y) = ρ(x−y) := c(d,α)|x−y|−d−α,
where c(d,α) is a constant only depending on d and α. Let V(x) = logλ(1+|x|) for
some λ > 0. Then,
(1) The semigroup (TV) is intrinsically ultracontractive if and only if λ > 1.
t t>0
(2) The semigroup (TV) is intrinsically supercontractive if and only if λ > 1.
t t>0
(3) The semigroup (TV) is intrinsically hypercontractive if and only if λ > 1.
t t>0
1.3.2. Irregular Potential Function: liminf V(x) < ∞. We make the fol-
|x|→∞
lowing assumption as in [4].
(A) There exists a constant K > 0 such that
lim Φ (R) = ∞,
K
R→∞
where
Φ (R) = inf V(x), R > 0.
K
|x|>R,V(x)>K
Let
Θ (R) = {x ∈ Rd : |x| > R,V(x) 6 K} , R > 0,
K
where K is the constant given in (A). Then, by (1.4), lim Θ (R) = 0. Similar
(cid:12) (cid:12) R→∞ K
to Theorem 1.1, in Theore(cid:12)m 1.5 below we can assume tha(cid:12)t inf V(x) = 0,
x∈Rd,V(x)>K
otherwise V is replaced by V(x) := V(x)− inf V(z) 1 (x).
z∈Rd,V(z)>K {z∈Rd,V(z)>K}
In particular, under such assumption and (A), for any r > 0 the set {s > 0 :
(cid:0) (cid:1)
Φ (s) > r} is not empty. e
K
Theorem 1.5. Suppose that assumption (A) holds, and that d > α , where α ∈
1 1
(0,2) is given in (1.1). For any s,δ > 0 with 1 6 i 6 4, define
i
8 s∧δ
(1.8) βˆ(s) = βˆ(s;δ ,δ ,δ ,δ ) = δ α Ψ−1 ∧δ , 2 ,
1 2 3 4 1 K s∧δ 3 8
(cid:18) (cid:18) 2(cid:19) (cid:19)
where
−1
1
Ψ (R) = +δ Θ (R)α1/d , Φ−1(r) = inf{s > 0 : Φ (s) > r}
K Φ (R) 4 K K K
(cid:20) K (cid:21)
and Φ−1 denotes the generalized inverse of Φ . Then all assertions in Theorem 1.1
K K
ˆ
hold with β(s) replaced by β(s).
INTRINSIC CONTRACTIVITY PROPERTIES OF FEYNMAN-KAC SEMIGROUPS 7
Note that, when lim V(x) = ∞, for any constant K > 0 there exists R > 0
|x|→∞ 0
such that Θ (R) = 0 and Ψ (R) = Φ (R) for R > R . Therefore, by (1.8) and
K K K 0
(1.6), in this case Theorem 1.5 reduces to Theorem 1.1. To show that Theorem
1.5 is sharp, we reconsider symmetric α-stable process both with irregular potential
function.
Example 1.6. Let (X ) be a symmetric α-stable process on Rd with d > α, and
t t>0
let V be a nonnegative measurable function defined by
logλ(1+|x|), x ∈/ A,
(1.9) V(x) =
1, x ∈ A,
(
where λ > 1 and A is a unbounded set on Rd such that inf V(x) = 0.
x∈/A
(1) Suppose that
c
|A∩B(0,R)c| 6 1 , R > 1
logθR
holds with some constants c ,θ > 0. Then, the associated semigroup (TV)
1 t t>0
is intrinsicallyultracontractive (and also intrinsicallysupercontractive) if θ >
d; (TV) is intrinsically hypercontractive if θ > d.
α t t>0 α
(2) For any ε > 0, let
∞
A = B(x ,r ),
m m
m=1
[
where x ∈ Rd with |x | = emk0, and r = m−k0+1 for some k > 2. Then,
m m m α d 0 ε
c
(1.10) |A∩B(0,R)c| 6 2 , R > 1
logαd−εR
holds for some constant c > 0; however, the semigroup (TV) is not in-
2 t t>0
trinsically ultracontractive.
The reminder of this paper is arranged as follows. In the next section, we will
present some preliminary results, including lower bound estimate for the ground
state and intrinsic local super Poincaré inequalities for non-local Dirichlet forms
with infinite range jumps. Section 3 is devoted to the proofs of all the theorems and
examples.
2. Some Technical Estimates
2.1. Lower bound for the ground state. In this subsection, we consider lower
bound estimate for the ground state φ . Recall that for x ∈ Rd
1
inf J(y,z), |x| > 3,
J∗(x) = y−z∈B(x,3/2)
1 , |x| < 3,
(
and
J∗(x)
V∗(x) = sup V(z), ϕ(x) = .
1+V∗(x)
z∈B(x,1)
Proposition 2.1. Let ϕ be the function defined above. Then there exists a constant
C > 0 such that for all x ∈ Rd,
0
(2.11) C φ (x) > ϕ(x).
0 1
8 XIN CHEN JIAN WANG
The proof of Proposition 2.1 is mainly based on the argument of [15, Theorem 1.6]
(in particular, see [15, pp. 5054-5055]). For the sake of completeness, we present the
details here. First, for any Borel set D ⊆ Rd, let τ := inf{t > 0 : X ∈/ D} be the
D t
first exit time from D of the process (X ) . The following result is a consequence
t t>0
of [2, Theorem 2.1], and the reader can refer to [4, Lemma 3.1] for the proof of it.
Lemma 2.2. There exist constants c := c (κ) > 0 and r := r (κ) ∈ (0,1] such
0 0 0 0
that for every r ∈ (0,r ] and x ∈ Rd,
0
Px τB(x,r) > c0rα2+(α2−α1α1)d > 1.
2
(cid:18) (cid:19)
In the following, we will fix r ,c in Lemma 2.2 and set t = c rα2+(α2−α1α1)d.
0 0 0 0 0
Lemma 2.3. Let 0 6 t < t 6 t , x ∈ Rd with |x| > 3, D = B(0,r ) and
1 2 0 0
B = B(x,r ). We have
0
(2.12) Px(X ∈ D/2,t 6 τ < t ) > c (t −t )J∗(x)
τB 1 B 2 1 2 1
for some constant c > 0.
1
Proof. Denote by p (t,x,y) the density of the process (X ) killed on exiting the
B t t>0
set B, i.e.
p (t,x,y) = p(t,x,y)−Ex(τ 6 t;p(t−τ ,X(τ ),y)).
B B B B
According to the Ikeda-Watanabe formula for (X ) (see e.g. [15, Proposition 2.5]),
t t>0
we have
Px(X(τ ) ∈ D/2,t 6 τ < t )
B 1 B 2
t2
= p (s,x,y)ds J(y,z)dzdy
B
ZBZt1 ZD/2
t2
>|D/2| inf J(y,z) p (s,x,y)dyds
B
y−z∈B(x,3r0/2) Zt1 ZB
t2
>c inf J(y,z) Px(τ > s)ds
2 B
y−z∈B(x,3r0/2) Zt1
>c inf Pz(τ > t ) (t −t ) inf J(y,z)
2 |z|>3 B(z,r0) 0 2 1 y−z∈B(x,3r0/2)
c h i
> 2(t −t ) inf J(y,z)
2 1
2 y−z∈B(x,3/2)
c
> 2(t −t )J∗(x),
2 1
2
which in the forth inequality we have used Lemma 2.2 and the fact that r 6 1.
0
This completes the proof. (cid:3)
Now, we are in a position to present the
Proof of Proposition 2.1. We only need to consider x ∈ Rd with |x| > 3. Still let
B = B(x,r ) and D = B(0,r ). First, we have
0 0
φ (x) = eλ1t0TV(φ )(x) > eλ1t0TV(1 φ )(x)
1 t0 1 t0 D 1
> eλ1t0(inf φ (x))TV(1 )(x) > c TV(1 )(x),
x∈D 1 t0 D 2 t0 D
INTRINSIC CONTRACTIVITY PROPERTIES OF FEYNMAN-KAC SEMIGROUPS 9
where in the last inequality we have used the fact that φ is strictly positive and
1
continuous.
Second, by the strong Markov property, it holds that
TV(1 )(x)
t0 D
= Ex(Xt0 ∈ D;e−R0t0V(Xs)ds)
> Ex(XτB ∈ D/2,τB < t0,Xs ∈ D for all s ∈ [τB,t0];e−R0τBV(Xs)ds−RτtB0 V(Xs)ds)
> e−t0supz∈DV(z)Ex(XτB ∈ D/2,τB < t0,Xs ∈ D for all s ∈ [τB,t0];e−R0τBV(Xs)ds)
> e−t0supz∈DV(z)Ex(XτB ∈ D/2,τB < t0;e−R0τBV(Xs)ds ·PXτB(τD > t0))
> e−t0supz∈DV(z)(cid:18)|z|i6nrf0/2Pz(τB(z,r0/2) > t0)(cid:19)Ex(XτB ∈ D/2,τB < t0;e−R0τBV(Xs)ds)
> c3Ex(XτB ∈ D/2,τB < t0;e−R0τBV(Xs)ds),
where in the last inequality we have used Lemma 2.2.
Third, according to (2.12),
Ex(XτB ∈ D/2,τB < t0;e−R0τBV(Xs)ds)
∞
> Ex XτB ∈ D/2, j +t0 1 6 τB < tj0;e−R0τBV(Xs)ds
Xj=1 (cid:16) (cid:17)
∞
> e−tj0 supz∈B(x,r0)V(z)Ex XτB ∈ D/2, j +t0 1 6 τB < tj0
Xj=1 (cid:16) (cid:17)
∞
> c1J∗(x) t0 e−tj0 supz∈B(x,r0)V(z)
j(j +1)
j=1
X
c J∗(x)
> 4
1+sup V(z)
z∈B(x,r0)
c J∗(x)
> 4 ,
1+sup V(z)
z∈B(x,1)
where the forth inequality follows from [15, Lemma 5.2], i.e.
∞ e−r/j e−1
> , r > 0.
j(j +1) r +1
j=1
X
Combining all the conclusions above, we prove the desired assertion. (cid:3)
2.2. Intrinsic local super Poincaré inequality. In this subsection, we are con-
cerned with the local intrinsic super Poincaré inequality for DV(f,f).
Proposition 2.4. Let ϕ be a strictly positive measurable function on Rd. Then for
any s, r > 0 and any f ∈ C2(Rd),
c
2
(2.13) f2(x)dx 6 sDV(f,f)+α r,s |f|(x)ϕ(x)dx ,
ZB(0,r) (cid:16)Z (cid:17)
(cid:0) (cid:1)
10 XIN CHEN JIAN WANG
where
2 2sup J(x,y)−1
α(r,s) = inf : t 6 rand 0<|x−y|6t 6 s .
|B(0,t)|inf ϕ2(x) |B(0,t)|
( x∈B(0,r+t) )
Proof. Since V > 0,
D(f,f) = (f(x)−f(y))2J(x,y)dxdy 6 DV(f,f), f ∈ C2(Rd),
c
Rd Rd
Z Z
it suffices to prove (2.13) with DV(f,f) replaced by D(f,f).
We can follow step (1) of the proof of [6, Theorem 3.1] or [24, Lemma 2.1] to
verify that for any 0 < s 6 r and f ∈ C2(Rd),
c
f2(x)dx
ZB(0,r)
2sup J(x,y)−1
(2.14) 6 0<|x−y|6s (f(x)−f(y))2J(x,y)dxdy
|B(0,s)|
(cid:18) (cid:19)ZZ{|x−y|6s}
2
2
+ |f(x)|dx .
|B(0,s)|
(cid:18)ZB(0,r+s) (cid:19)
Note that, if (2.14) holds, then for any 0 < s 6 r and f ∈ C2(Rd)
c
2sup J(x,y)−1
f2(x)dx 6 0<|x−y|6s D(f,f)
|B(0,s)|
ZB(0,r) (cid:18) (cid:19)
2
2
+ |f(x)|ϕ(x)dx .
|B(0,s)|inf ϕ2(x)
x∈B(0,r+s) (cid:18)ZB(0,r+s) (cid:19)
This immediately yields (2.13) by the definition of α(s,r).
Next, we turn to the proof of (2.14). For any 0 < s 6 r and f ∈ C2(Rd), define
c
1
f (x) = f(z)dz, x ∈ B(0,r).
s
|B(0,s)|
ZB(x,s)
We have
1
sup |f (x)| 6 |f(z)|dz,
s
|B(0,s)|
x∈B(0,r) ZB(0,r+s)
and
1
|f (x)|dx 6 |f(z)|dzdx
s
|B(0,s)|
ZB(0,r) ZB(0,r) ZB(x,s)
1
6 dx |f(z)|dz 6 |f(z)|dz.
|B(0,s)|
ZB(0,r+s)(cid:18) ZB(z,s) (cid:19) ZB(0,r+s)
Thus,
f2(x)dx 6 sup |f (x)| |f (x)|dx
s s s
ZB(0,r) (cid:16)x∈B(0,r) (cid:17)ZB(0,r)
2
1
6 |f(z)|dz .
|B(0,s)|
(cid:18)ZB(0,r+s) (cid:19)