J
i
on
Electr
o
u
a
rn l
o
f
P
c
r
ob
abil
ity
Vol. 11 (2006), Paper no. 6, pages 146–161.
Journal URL
http://www.math.washington.edu/∼ejpecp/
Two-sided estimates on the density of the
Feynman-Kac semigroups of stable-like processes
Renming Song ∗
Department of Mathematics, University of Illinois
Urbana, IL 61801, USA
Email: rsong@math.uiuc.edu
http://www.math.uiuc.edu/∼rsong
Abstract
Suppose that α ∈ (0, 2) and that X is an α-stable-like process on Rd . Let µ be a signed
measure on Rd belonging to the class Kd,α and Aµt be the continuous additive functional of
X associated with µ. In this paper we show that the Feynman-Kac semigroup {Ttµ : t ≥ 0}
defined by
µ
Ttµ f (x) = Ex e−At f (Xt )
has a density q µ and that there exist positive constants c1 , c2 , c3 , c4 such that
d
c1 e−c2 t t− α
1∧
t1/α
|x − y|
d+α
d
≤ q µ (t, x, y) ≤ c3 ec4 t t− α
1∧
t1/α
|x − y|
d+α
for all (t, x, y) ∈ (0, ∞) × Rd × Rd . We also provide similar estimates for the densities of two
other kinds of Feynman-Kac semigroups of X.
Key words: Stable processes, stable-like processes, Kato class, Feynman-Kac semigroups,
continuous additive functionals, continuous additive functionals of zero energy, purely discontinuous additive functionals.
AMS 2000 Subject Classification: Primary: 60J45, 60J35; Secondary: 60J55, 60J57,
60J75, 60G51, 60G52.
Submitted to EJP on November 9, 2004. Final version accepted on February 21, 2006.
∗
The research of this author is supported in part by a joint US-Croatia grant INT 0302167.
146
1
Introduction
Suppose that X = (Xt , Px ) is a Brownian motion on Rd and that V is a function on Rd
belonging to the Kato class of X, i. e., a function satisfying the condition
Z t
|V |(Xs )ds = 0.
lim sup Ex
t↓0 x∈Rd
0
It is well known (see [11] for instance) that the Feynman-Kac semigroup {TtV : t ≥ 0} with
potential V
Z t
TtV f (x) = Ex exp(−
V (Xs )ds)f (Xt ) ,
0
q V (t, x, y)
has a transition density
with respect to the Lebesgue measure and that q V has both
an upper and a lower Gaussian estimates, that is there exist positive constants c1 , c2 , c3 , c4 such
that
d
d
3|x − y|2
|x − y|2
c1 e−c2 t t− 2 exp(−
) ≤ q V (t, x, y) ≤ c3 ec4 t t− 2 exp(−
)
(1.1)
4t
4t
for all (t, x, y) ∈ (0, ∞) × Rd × Rd . This result can be easily generalized (see [2] for instance) to
the case when V is replaced by a signed measure satisfying
Z tZ
d
|x − y|2
lim sup
t− 2 exp(−
)|µ|(dy)ds = 0
t↓0 x∈Rd 0
2t
Rd
and
Rt
0
V (Xs )ds is replaced by the continuous additive functional Aµt of X associated with µ.
Now suppose that α ∈ (0, 2) and that X = (Xt , Px ) is a symmetric α-stable process on Rd . The
question that we are going to address in this paper is the following: can one establish two-sided
estimates for the density of the Feynman-Kac semigroup of the symmetric α-stable process X?
As far as we know, this question has not been addressed in the literature. The proof of (1.1) in
[11] and [2] can not be adapted to the case of discontinuous stable processes. It seems that, to
answer the question above, one has to use some new ideas. In this paper, we are going to tackle
the question above by adapting an idea used in [13] and [14] to establish heat kernel estimates
for diffusions to the present case. Actually, instead of dealing with symmetric stable processes,
we are going to deal with the more general stable-like processes introduced in [4].
The content of this paper is organized as follows. In section 2, we first recall the definition of the
Kato class with respect to symmetric α-stable processes and some basic facts about stable-like
processes, and then we present some preliminary results on Feynman-Kac semigroups. In section
3 we establish two-sided estimates on the density of Feynman-Kac semigroups with potentials
given by measures belonging to the Kato class. In the last section we deal with two other kinds
of Feynman-Kac semigroups of stable-like processes. The first kind consists of Feynman-Kac
semigroups given by purely discontinuous additive functionals, and the second kind consists of
Feynman-Kac semigroups given by continuous additive functionals of zero energy.
147
In this paper we will use the following convention on the labeling of constants. The values
of the constants M1 , M2 , · · · will remain the same throughout this paper, while the values of
the constants C1 , C2 , · · · might change from one appearance to the next. The labeling of the
constants C1 , C2 , · · · starts anew in the statement of each result.
2
Kato Class and Basic Properties of Feynman-Kac Semigroups
In this paper we will always assume that α ∈ (0, 2). We will use X 0 = {Xt0 , P0x } to denote a
symmetric α-stable process in Rd whose transition density p0 (t, x, y) = p0 (t, x − y) with respect
to the Lebesgue measure satisfies
Z
α
eix·ξ p0 (t, x)dx = e−t|ξ| , t > 0.
Rd
It is known (see [3]) that there exist positive constants M1 < M2 such that
M1 t
d
−α
t1/α
1∧
|x − y|
!d+α
≤ p0 (t, x, y) ≤ M2 t
d
−α
t1/α
1∧
|x − y|
!d+α
(2.1)
for all (t, x, y) ∈ (0, ∞) × Rd × Rd . For any λ > 0, we define
Z ∞
0
0
Gλ (x, y) = Gλ (x − y) =
e−λt p0 (t, x − y)dt
0
for all x, y ∈ Rd . When α < d, the process X 0 is transient and its potential density G0 (x, y) =
G0 (x − y) is given by
Z ∞
α −1
d−α
0
0
−α − d2
G (x − y) =
Γ
|x − y|α−d .
p (t, x − y)dt = 2 π Γ
2
2
0
The Dirichlet form (E 0 , F) of X 0 is given by
Z Z
1
(u(x) − u(y))(v(x) − v(y))
0
E (u, v) = A(d, −α)
dxdy
2
|x − y|d+α
Rd Rd
Z Z
(u(x) − u(y))2
2
d
F = u ∈ L (R ) :
dxdy < ∞ ,
|x − y|d+α
Rd Rd
where
A(d, −α) =
|α| Γ( d+α
2 )
.
d/2
1−α
2
π Γ(1 − α2 )
For any function V on Rd and t > 0, we define
Z tZ
NV (t) = sup
p0 (s, x, y)|V (y)|dyds.
x∈Rd
0
Rd
148
By a signed measure we mean in this paper the difference of two nonnegative measures at most
one of which can have infinite total mass. For any signed measure on Rd , we use µ+ and µ− to
denote its positive and negative parts, and |µ| = µ+ + µ− its total variation. For any t > 0, we
define
Z Z
t
p0 (s, x, y)|µ|(dy)ds.
Nµ (t) = sup
x∈Rd
0
Rd
Definition 2.1 We say that a function V on Rd belongs to the Kato class Kd,α if limt↓0 NV (t) =
0. We say that a signed Radon measure µ on Rd belongs to the Kato class Kd,α if limt↓0 Nµ (t) =
0.
Rigorously speaking a function V in Kd,α may not give rise to a signed measure µ in Kd,α since
it may not give rise to a signed measure at all. However, for the sake of simplicity we use the
convention that whenever we write that a signed measure µ belongs to Kd,α we are implicitly
assuming that we are covering the case of all the functions in Kd,α as well.
The following result is well known, see [1] and [12] for instance.
Proposition 2.1 Suppose that µ is a signed measure on Rd . Then µ ∈ Kd,α if and only if
Z
lim sup
G0λ (x, y)|µ|(dy) = 0.
λ→∞ x∈Rd
Rd
When α < d, µ ∈ Kd,α is also equivalent to the condition
Z
|µ|(dy)
lim sup
= 0.
r→0 x∈Rd |x−y|<r |x − y|d−α
We assume from now on that m is a measure on Rd given by m(dx) = M (x)dx, where M is a
function satisfying
M3 ≤ M (x) ≤ M4 , x ∈ Rd
(2.2)
for some positive constants M3 < M4 . We will fix a symmetric function c(x, y) on Rd × Rd
which is bounded between two fixed positive constants. If for any u, v ∈ F we define
Z Z
c(x, y)(u(x) − u(y))(v(x) − v(y))
E(u, v) =
m(dx)m(dy),
|x − y|d+α
Rd Rd
then (E, F) is a regular Dirichlet form on L2 (Rd , m). It is shown in [4] that, associated with
this Dirichlet form, there is an m-symmetric Hunt process X = {Xt , Px } on Rd which can
start from any point x ∈ Rd . We will use {Mt ; t ≥ 0} to denote the natural filtration of X.
The process X is called an α-stable-like process in [4]. It is also shown in [4] that the process
X admits a transition density p(t, x, y) with respect to m and that p is jointly continuous on
(0, ∞) × Rd × Rd and satisfies the condition
!d+α
!d+α
1/α
1/α
d
d
t
t
M5 t− α 1 ∧
≤ p(t, x, y) ≤ M6 t− α 1 ∧
(2.3)
|x − y|
|x − y|
149
for all (t, x, y) ∈ (0, 1] × Rd × Rd . By using the scaling property (see the proof of Proposition
4.1 in [4]), one can show that (2.3) is valid for all (t, x, y) ∈ (0, ∞) × Rd × Rd . We may and do
assume that M5 < 1 < M6 .
From the display above and estimates (2.1) one can easily see that a signed measure µ is in Kd,α
if and only if
Z Z
t
p(s, x, y)|µ|(dy)ds = 0.
lim sup
t↓0 x∈Rd
0
Rd
By repeating the argument in the proof of Theorem 2.1 in [2], we can show the following
Lemma 2.2 Let µ be a signed Radon measure on Rd . Then µ ∈ Kd,α if and only the following
conditions are satisfied:
1. |µ| is a smooth measure in the sense of [9],
2. the continuous additive functional At associated with µ can be defined without exceptional
set,
−
+
−
3. limt↓0 supx∈Rd Ex |A|t = 0, where |A|t = A+
t +At , At and At being the positive continuous
additive functionals of X associated with µ+ and µ− respectively.
Furthermore, if µ ∈ Kd,α and At is the continuous additive functional of X associated with µ,
then
Z tZ
Ex At =
p(s, x, y)µ(dy)ds, ∀(t, x) ∈ (0, ∞) × Rd .
0
Rd
2
Proof. We omit the details.
In the sequel, whenever we have a signed measure µ ∈ Kd,α , we will use Aµt to denote the
continuous additive functional of X associated with µ. Using Khas’minskii’s lemma (see Lemma
2.6 of [2]), we can easily show the following
Lemma 2.3 Suppose that µ ∈ Kd,α and Aµt is the continuous additive functional of X associated
with µ. There exist positive constants C1 and C2 , depending on µ only via the rate at which
Nµ (t) goes to zero, such that
|µ| ≤ C1 eC2 t , t > 0.
sup Ex eAt
x∈Rd
The meaning of the phrase “depending on µ only via the rate at which Nµ (t) goes to zero” will
become clear in the proof of Theorem 3.3. It roughly means that if w(t) is a increasing function
on (0, ∞) with limt→0 w(t) = 0, then there exist positive constants C1 and C2 such that for any
signed measure µ with
Nµ (t) ≤ w(t), t > 0,
150
we have
|µ| ≤ C1 eC2 t ,
sup Ex eAt
t > 0.
x∈Rd
For any µ ∈ Kd,α , we define the Feynman-Kac semigroup {Ttµ : t ≥ 0} with potential µ by
µ
Ttµ f (x) = Ex e−At f (Xt ) .
When µ is given by µ(dx) = U (x)dx for some function U , we will sometimes write Ttµ as TtU .
The following result is well known, see [12] and [5].
Theorem 2.4 Suppose that µ ∈ Kd,α , then
1. For any p ∈ [1, ∞), {Ttµ : t ≥ 0} is a strongly continuous semigroup in Lp (Rd , m);
2. For each t > 0, Ttµ maps L∞ (Rd , m) into bounded continuous functions on Rd ;
3. For any p ∈ [1, ∞) and t > 0, Ttµ maps Lp (Rd , m), p ∈ [0, ∞), into bounded continuous
functions on Rd which converges to zero at infinity, and there exist positive constants C1
and C2 , depending on µ only via the rate at which Nµ (t) goes to zero, such that
kTtµ kp,p ≤ kTtµ k∞,∞ ≤ C1 eC2 t ,
t > 0,
where, for any p, q ∈ [1, ∞], kTtµ kp,q stands for the norm of Ttµ as an operator from
Lp (Rd , m) into Lq (Rd , m).
Using an argument similar to that of the proof of Theorem 3.1 in [2], we can show the following
Theorem 2.5 For any µ ∈ Kd,α , there exists a function q µ (t, x, y) such that
1. q µ is jointly continuous on (0, ∞) × Rd × Rd ;
2. there exist positive constants C1 and C2 depending on µ only via the rate at which Nµ (t)
goes to zero such that
d
0 < q µ (t, x, y) ≤ C1 eC2 t t− α ,
∀(t, x, y) ∈ (0, ∞) × Rd × Rd ;
R
3. Ttµ f (x) = Rd q µ (t, x, y)f (y)m(dy) for all (t, x) ∈ (0, ∞) × Rd and all bounded function f
on Rd ;
R
4. Rd q µ (t, x, z)q µ (s, z, y)m(dz) = q µ (t + s, x, y) for all t, s > 0 and (x, y) ∈ Rd × Rd ;
5. q µ (t, x, y) is symmetric in x and y;
151
6. if f is a bounded function continuous at x ∈ Rd , then
Z
lim
q µ (t, x, y)f (y)m(dy) = f (x).
t↓0
Rd
2
Proof. We omit the details.
Corollary 2.6 For any µ ∈ Kd,α , the function q µ in the theorem above satisfies the equation
Z tZ
µ
q (t, x, y) = p(t, x, y) −
p(s, x, z)q µ (t − s, z, y)µ(dz)ds,
(2.4)
0
Rd
for all (t, x, y) ∈ (0, ∞) × Rd × Rd .
Proof. Since for any t > 0
−Aµ
t
e
t
Z
=1−
µ
µ
e−(At −As ) dAµs ,
0
we have
−Aµ
t
Ex e
Z t
−(Aµ
−Aµ
µ
s)
t
e
f (Xt ) = Ex f (Xt ) − Ex f (Xt )
dAs
0
(0, ∞)×Rd
Rd .
and all bounded functions f on
Now the conclusion of the corollary
for all (t, x) ∈
follows easily from the Markov property, Fubini’s theorem and the two theorems above.
2
When the measure µ is given by µ(dx) = U (x)dx form some function U , we will sometimes write
q µ as q U .
3
Two-sided Estimates for Densities of Local Feynman-Kac
transforms
In this section we shall establish two-sided estimates for the densities of Feynman-Kac semigroups
with potentials belonging to Kd,α . The following elementary lemma will play an important role.
Lemma 3.1 Suppose that a, b, c, d are positive numbers. If a < d and c < b, then we have
a
d
a
d
(1 ∧ )(1 ∧ ) ≤ (1 ∧ )(1 ∧ ).
b
c
c
b
(3.1)
Proof. We prove this lemma by looking at all the different cases.
In the first case we assume that a ≥ b. In this case we have c < b ≤ a < d, so the left and right
hand sides of (3.1) are both equal to 1. Thus (3.1) is valid in this case.
In the second case we assume that c ≤ a ≤ b. We further divide this case into two subcases. In
the first subcase we assume that c ≤ a ≤ b ≤ d. In this subcase the left hand side of (3.1) is equal
152
to ab and the right hand side is equal to 1. In the second subcase we assume that c ≤ a < d ≤ b.
In this subcase the left hand side of (3.1) is equal to ab and the right hand side is equal to db .
Thus (3.1) is valid in this case.
In the third case we assume that a ≤ c. We further divide this case into three subcases. In the
first subcase we assume that a ≤ c < b ≤ d. In this subcase the left hand side of (3.1) is equal to
a
a
b and the right hand side is equal to c . In the second subcase we assume that a ≤ c ≤ d ≤ b. In
a
this subcase the left hand side of (3.1) is equal to ab and the right hand side is equal to ad
bc ≥ b .
In the third subcase we assume that a < d ≤ c < b. In this subcase the left and right hand sides
of (3.1) are both equal to ad
2
bc . Thus (3.1) is also valid in this case.
The following lemma is similar to Lemma 3.1 of [14] and is crucial in establishing the main
estimates of this paper.
Lemma 3.2 There exists a positive constant C depending only on d and α such that for any
measure ν on Rd and (t, x, y) ∈ (0, ∞) × Rd × Rd ,
Z tZ
s1/α
1∧
|x − z|
d
−α
s
Rd
0
≤
!d+α
(t − s)
!d+α
t1/α
1∧
|x − y|
d
CM1−1 t− α
(t − s)1/α
1∧
|z − y|
d
−α
!d+α
ν(dz)ds
t
Nν ( ).
2
Proof. Put
Z tZ
s1/α
1∧
|x − z|
d
−α
J(t, x, y) =
s
0
Rd
!d+α
d
−α
(t − s)
(t − s)1/α
1∧
|z − y|
!d+α
ν(dz)ds.
We can rewrite J as
Z
J(t, x, y) =
0
t
2
Z t! Z
· · · ν(dz)ds := J1 (t, x, y) + J2 (t, x, y).
+
t
2
Rd
We estimate J1 (t, x, y) by estimating the integrand separately on the region
1
t
R1 := {(s, z) : s ∈ (0, ), |z − y| ≥ |x − y|}
2
2
and the region
t
1
R2 := {(s, z) : s ∈ (0, ), |z − y| < |x − y|}.
2
2
153
On R1 we have
!d+α
(t − s)1/α
s
(t − s)
1∧
|z − y|
!d+α
!d+α
1/α
1/α
d
d
d
s
2t
≤ 2 α s− α 1 ∧
t− α 1 ∧
|x − z|
|x − y|
!d+α
!d+α
1/α
1/α
d
d
d
s
t
t− α 1 ∧
.
≤ 2 α +d+α s− α 1 ∧
|x − z|
|x − y|
d
−α
s1/α
1∧
|x − z|
!d+α
d
−α
On R2 we have |x − z| ≥ |x − y| − |y − z| ≥ 21 |x − y|. So by applying Lemma 3.1 with a = s1/α ,
b = |x − z|, c = |z − y| and d = (t − s)1/α we get that
!d+α
!d+α
1/α
1/α
d
d
s
(t
−
s)
s− α 1 ∧
(t − s)− α 1 ∧
|x − z|
|z − y|
!d+α
!d+α
d
d
s1/α
(t − s)1/α
−α
−α
(t − s)
1∧
1∧
≤s
|z − y|
|x − z|
!d+α
!d+α
d
d
d
s1/α
2t1/α
−
−
≤ 2α s α 1 ∧
t α 1∧
|y − z|
|x − y|
!d+α
!d+α
1/α
1/α
d
d
d
s
t
≤ 2 α +d+α s− α 1 ∧
t− α 1 ∧
.
|y − z|
|x − y|
Thus we have
J1 (t, x, y)
≤2
d
+d+α
α
≤ M1−1 2
t
d
−α
t1/α
1∧
|x − y|
d
+d+α+1
α
!d+α Z
d
t
Nν ( )t− α
2
0
t
2
Z
!d+α
s1/α
s1/α
(1 ∧
) + (1 ∧
)
ν(dz)ds
|x − z|
|y − z|
d
−α
s
Rd
t1/α
1∧
|x − y|
!d+α
.
By a similar argument we get
d
t
Nν ( )t− α
2
t1/α
1∧
|x − y|
!d+α
d
d
t
M1−1 2 α +d+α+2 Nν ( )t− α
t1/α
1∧
|x − y|
!d+α
J2 (t, x, y) ≤ M1−1 2
d
+d+α+1
α
.
Consequently we have
J(t, x, y) ≤
2
(3.2)
2
for all (t, x, y) ∈ (0, ∞) × Rd × Rd .
154
Theorem 3.3 For any µ ∈ Kd,α , there exists a positive constant T , depending on µ only via
the rate at which Nµ (t) goes to zero, such that
!d+α
!d+α
d
d
t1/α
t1/α
µ
−α
−α
≤ q (t, x, y) ≤ C2 t
C1 t
1∧
1∧
|x − y|
|x − y|
for some constants C1 and C2 depend only on M5 and for all (t, x, y) ∈ (0, T ] × Rd × Rd .
Proof. For (t, x, y) ∈ (0, ∞) × Rd × Rd , we define In (t, x, y) recursively for n ≥ 0 by
I0 (t, x, y) = p(t, x, y),
Z tZ
In+1 =
p(s, x, z)In (t − s, z, y)µ(dz)ds.
0
Rd
We claim that there exists a positive constant T , depending on µ only via the rate at which
Nµ (t) goes to zero, such that for all n ≥ 1 and (t, x, y) ∈ (0, T ] × Rd × Rd
!d+α
M5 n − d
t1/α
.
(3.3)
) t α 1∧
In (t, x, y) ≤ (
2
|x − y|
We will prove this claim by induction. In fact, for n = 1, we have
Z tZ
|I1 (t, x, y)| = |
p(s, x, z)p(t − s, z, y)µ(dz)ds|
Rd
0
≤ M62
Z tZ
s1/α
1∧
|x − z|
d
−α
s
0
Rd
!d+α
(t − s)1/α
1∧
|z − y|
d
−α
(t − s)
!d+α
|µ|(dz)ds.
Applying Lemma 3.2 we get that there exists a constant c1 > 0 depending only on d and α such
that
!d+α
1/α
d
t
t
.
|I1 (t, x, y)| ≤ c1 M1−1 M62 Nµ ( )t− α 1 ∧
2
|x − y|
Take T > 0 small enough so that
M5
t
c1 M1−1 M62 Nµ ( ) ≤
, t ≤ T.
2
2
Obviously, this T depends on µ only via the rate at which Nµ (t) goes to zero and
!d+α
M5 − d
t1/α
|I1 (t, x, y)| ≤
t α 1∧
2
|x − y|
for all (t, x, y) ∈ (0, T ] × Rd × Rd . Thus the claim above is valid for n = 1. Now suppose that
the claim is valid for n. Then we have
Z tZ
|In+1 (t, x, y)| = |
p(s, x, z)In (t − s, z, y)µ(dz)ds|
Rd
0
M5 n
≤ M6 (
)
2
Z tZ
d
−α
s
0
Rd
s1/α
1∧
|x − z|
!d+α
d
−α
(t − s)
155
(t − s)1/α
1∧
|z − y|
!d+α
|µ|(dz)ds.
Applying Lemma 3.2 again we get that
d
t
M5 n
t1/α
) Nµ ( )t− α 1 ∧
|In+1 (t, x, y)| ≤ c1 M1−1 M6 (
2
2
|x − y|
!d+α
M5 n+1 − d
t1/α
)
t α 1∧
≤ (
2
|x − y|
!d+α
for all (t, x, y) ∈ (0, T ] × Rd × Rd . Therefore the claim above is valid.
It follows from the claim above that, for (t, x, y) ∈ (0, T ] × Rd × Rd , the series
is uniformly absolutely convergent and
∞
X
n=0
∞
X
M5 n − d
|In (t, x, y)| ≤
(
) t α
2
n=0
t1/α
1∧
|x − y|
!d+α
:= c2 t
d
−α
P∞
n=0 |In (t, x, y)|
t1/α
1∧
|x − y|
!d+α
.
Using Corollary 2.6 and Lemma 3.2 we see that
µ
q (t, x, y) =
∞
X
n
(−1) In (t, x, y) ≤ c2 t
n=0
d
−α
t1/α
1∧
|x − y|
!d+α
for all (t, x, y) ∈ (0, T ] × Rd × Rd .
Using the claim above again we get that, for (t, x, y) ∈ (0, T ] × Rd × Rd ,
∞
X
n=1
M5 − d
|In (t, x, y)| ≤
t α
2 − M5
t1/α
1∧
|x − y|
Therefore we have
1 − M5 − d
t α
q (t, x, y) ≥
2 − M5
µ
t1/α
1∧
|x − y|
!d+α
.
!d+α
2
for all (t, x, y) ∈ (0, T ] × Rd × Rd .
As a consequence of the theorem above and the semigroup property (Theorem 2.5.4), we immediately get the following
Theorem 3.4 For any µ ∈ Kd,α , there exist positive constant C1 , C2 , C3 , C4 , depending on µ
only via the rate at which Nµ (t) goes to zero, such that
C1 e−C2 t t
d
−α
t1/α
1∧
|x − y|
!d+α
≤ q µ (t, x, y) ≤ C3 eC4 t t
for all (t, x, y) ∈ (0, ∞) × Rd × Rd .
156
d
−α
t1/α
1∧
|x − y|
!d+α
4
Feynman-Kac Semigroups Given by Discontinuous Additive
Functionals and Continuous Additive Functionals of Zero Energy
We first deal with a class of Feynman-Kac semigroups given by a purely discontinuous additive
functional. To do this, we need to recall a definition and introduce some notations.
Definition 4.1 Suppose that F is a function on Rd × Rd . We say that F belongs to the class
Jd,α if F is bounded, vanishing on the diagonal, and the function
Z
|F (x, y)|
dy
x 7→
d+α
Rd |x − y|
belongs to Kd,α .
It is easy to see from the definition above that if F ∈ Jd,α , then e−F is also in Jd,α .
The process X has a Lévy system (N, H) given by Ht = t and
N (x, dy) = 2c(x, y)|x − y|−(d+α) m(dy),
that is, for any nonnegative function f on Rd × Rd vanishing on the diagonal


Z tZ
X
2c(Xs , y)f (Xs , y)


f (Xs− , Xs ) = Ex
Ex
m(dy)ds
|Xs − y|d+α
d
0
R
s≤t
for every x ∈ Rd and t > 0.
For any F belonging to Jd,α , we put
BtF =
X
F (Xs− , Xs ),
t ≥ 0.
s≤t
We can define the following so-called non-local Feynman-Kac semigroup
F
StF f (x) = Ex e−Bt f (Xt ) .
This semigroup has been studied in [12] and [6].
Theorem 4.1 Suppose that F ∈ Jd,α is a symmetric function. The semigroup {StF , t ≥ 0}
admits a density k F (t, x, y) with respect to m and that k F is jointly continuous on (0, ∞) × Rd ×
Rd . Furthermore, there exist positive constants C1 , C2 , C3 and C4 such that
!d+α
!d+α
d
d
t1/α
t1/α
F
C4 t − α
−C2 t − α
C1 e
t
1∧
≤ k (t, x, y) ≤ C3 e t
1∧
|x − y|
|x − y|
for all (t, x, y) ∈ (0, ∞) × Rd × Rd .
157
Proof. Put G = e−F − 1 and
Z
V (x) =
Rd
2c(x, y)G(x, y)
m(dy).
|x − y|d+α
Using the definition of Lévy systems we can see that BtG − AVt is a Px -martingale for every
x ∈ Rd . It follows from the Doleans-Dade formula that
F −AV
t
Mt = e−Bt
(4.1)
is a local martingale under Px for every x ∈ Rd . Mt is a clearly a multiplicative functional, so
Mt is supermartingale multiplicative functional of X. Therefore by Theorem 62.19 of [10], Mt
defines a family of probability measures {P̃x , x ∈ Rd } by dP̃x = Mt dPx on Mt . We will use
X̃ = (Xt , P̃x ) denote this new process. It follows from [6] that X̃ is a symmetric Hunt process
on Rd whose Dirichlet form (Ẽ, F̃) is given by F̃ = F and
Z Z
e−F (x,y) c(x, y)(u(x) − u(y))2
Ẽ(u, u) =
m(dx)m(dy), u ∈ F.
|x − y|d+α
Rd Rd
Thus X̃ is an α-stable-like process in the sense of [4]. It follows from (4.1) that for any nonnegative function f on Rd , any t > 0 and any x ∈ Rd we have
V
F
Ex e−Bt f (Xt ) = Ẽx eAt f (Xt ) .
Therefore StF can be regarded as a Feynman-Kac semigroup of X̃ with a potential −V . Now
our assertion follows immediately from Theorem 3.4.
2
Now we deal with Feynman-Kac semigroups given by continuous additive functionals of zero
energy. To do this, we need to recall some facts from the theory of Dirichlet forms.
We denote by Fe the family of functions u on Rd that is finite almost everywhere and there is
an E-Cauchy sequence {un } ⊂ F such that limn→∞ un = u almost everywhere on Rd . (E, Fe )
is called the extended Dirichlet space of (E, F). It is well known that any u ∈ Fe has a quasicontinuous version ũ. In this paper, whenever we talk about a function u ∈ Fe , we implicitly
assume that we are dealing with its quasi-continuous version. It is known (see [9]) that, for any
u ∈ Fe , u(Xt ) has the following Fukushima’s decomposition
u(Xt ) = u(X0 ) + Mtu + Ntu ,
t ≥ 0.
Here Mtu is a martingale additive functional of X and Ntu is a continuous additive functional of
X with zero quadratic variation. Note that in general, Ntu is not a process of finite variation.
The martingale part Mtu is given by

X
Mtu = lim
(u(Xs ) − u(Xs− ))1{|u(Xs )−u(Xs− )|>1/n}
n→∞ 
0<s≤t
! )
Z t Z
2c(Xs , y)(u(y) − u(Xs ))
−
m(dy) ds .
|Xs − y|d+α
0
{y∈Rd :|u(y)−u(Xs )|>1/n}
158
Let µ<u> be the Revuz measure associated with the sharp bracket positive continuous additive
functional < M u >. Then
Z
2c(x, y)(u(x) − u(y))2
m(dy)m(dx).
µ<u> (dx) =
|x − y|d+α
Rd
It follows from [8] that when u ∈ Fe satisfies the condition µ<u> ∈ Kd,α , the additive functionals
Mtu and Ntu can be taken as additive functionals in the strict sense.
For any quasi-continuous function u ∈ Fe with µ<u> ∈ Kd,α , we will consider the following
Feynman-Kac semigroup {Rtu : t ≥ 0}:
u
Rtu f (x) = Ex eNt f (Xt ) , t ≥ 0.
This semigroup has been studied in [7].
Theorem 4.2 Suppose that u is bounded quasi-continuous function belonging to Fe and that
µ<u> ∈ Kd,α . The semigroup {Rtu , t ≥ 0} admits a density ru (t, x, y) with respect to m and ru
is jointly continuous on (0, ∞) × Rd × Rd . Furthermore, there exist positive constants C1 , C2 , C3
and C4 such that
!d+α
!d+α
d
d
t1/α
t1/α
−C2 t − α
u
C4 t − α
C1 e
t
≤ r (t, x, y) ≤ C3 e t
1∧
1∧
|x − y|
|x − y|
for all (t, x, y) ∈ (0, ∞) × Rd × Rd .
Proof. Put ρ(x) = e−u(x) and ρ(∂) = 1. It is easy to check that ρ − 1 ∈ Fe . Thus if we define
M ρ := M ρ−1 and N ρ := N ρ−1 , then we have the Fukushima’s decomposition for ρ(Xt ):
ρ(Xt ) = ρ(X0 ) + Mtρ + Ntρ .
Define a square integrable martingale M by
Z
Mt =
t
0
1
dMsρ .
ρ(Xs− )
Let Lρt be the solution of the following SDE:
Lρt
Z
t
=1+
Lρs− dMs .
0
It follows from the Doleans-Dade formula that
Y
Lρt = exp(Mt )
(1 + Ms − Ms− )e−(Ms −Ms− )
0<s≤t
ρ(Xs )
ρ(Xs )
exp 1 −
= exp(Mt )
ρ(Xs− )
ρ(Xs− )
0<s≤t
!
Z tZ
u(Xs )−u(y) )
2c(X
,
y)(u(X
)
−
u(y)
+
1
−
e
s
s
= exp Mt−u +
)m(dy)ds , (4.2)
|Xs − y|d+α
0
Rd
Y
159
where the last equality is shown on page 487 of [7]. Lρt is a nonnegative local martingale and
therefore a supermartingale multiplicative functional of X. Therefore by Theorem 62.19 of [10]
Lρt defines a family of probability measures {P̃x , x ∈ E} by dP̃x = Lρt dPx on Mt . We will use
X̃ = (Xt , P̃x ) denote this new process. Put ν(dx) = ρ2 (x)m(dx). It follows from [7] that X̃ is
a ν-symmetric Hunt process on Rd whose Dirichlet form (Ẽ, F̃) on L2 (Rd , ν) is given by F̃ = F
and
Z Z
ρ(x)ρ(y)c(x, y)(u(x) − u(y))2
Ẽ(u, u) =
m(dx)m(dy), u ∈ F.
|x − y|d+α
Rd Rd
Thus X̃ is an α-stable-like process in the sense of [4]. Using the boundedness of u and the
assumption µ<u> ∈ Kd,α we can easily check that the function
Z
V (x) =
Rd
2c(x, y)(u(x) − u(y) + 1 − eu(x)−u(y) )
m(dy)
|x − y|d+α
belongs to Kd,α . It follows from (4.2) that for any nonnegative function f on Rd , any t > 0 and
any x ∈ Rd we have
Z t
ρ
Ntu
−u(x)
u
Ex e f (Xt ) = e
Ex Lt exp(−
V (Xs )ds)(f e )(Xt )
0
Z t
−u(x)
u
= e
Ẽx exp(−
V (Xs )ds)(f e )(Xt ) .
0
Now applying Theorem 3.4 to the process X̃ and the potential V , we see that there is a function
r(t, x, y) defined on (0, ∞) × Rd × Rd such that
Z
Ntu
−u(x)
Ex e f (Xt ) = e
r(t, x, y)(f eu )(y)ν(dy)
d
ZR
−u(x)
= e
r(t, x, y)e−u(y) f (y)m(dy)
(4.3)
Rd
for all (t, x) ∈ (0, ∞) × Rd and all nonnegative function f on Rd and that there exist positive
constants c1 , c2 , c3 , c4 such that
c1 e−c2 t t
d
−α
t1/α
1∧
|x − y|
!d+α
≤ r(t, x, y) ≤ c3 ec4 t t
d
−α
t1/α
1∧
|x − y|
!d+α
for all (t, x, y) ∈ (0, ∞) × Rd × Rd . It follows from (4.3) that Rtu admits a density with respect
to m given by
ru (t, x, y) = e−u(x) r(t, x, y)e−u(y)
for all (t, x, y) ∈ (0, ∞) × Rd × Rd . The last assertion of the theorem follows easily from the
boundedness of u.
2
160
Remark 4.3 Of course, one can combine Theorems 3.4, 4.1 and 4.2 into one theorem about
the density of Feynman-Kac semigroup given by additive functionals involving all three components: a continuous part with finite variation, a continuous part of zero energy and a purely
discontinuous part. We leave this to the reader.
Acknowledgment: I thank Panki Kim, Murali Rao and Zoran Vondracek for their helpful
comments on the first version of this paper.
References
[1] S. Albeverio and Z. M. Ma, Perturbation of Dirichlet forms–lower semiboundedness, closability and
form cores, J. Funct. Anal., 99(1991), 332–356.
[2] Ph. Blanchard and Z. M. Ma, Semigroup of Schrödinger operators with potentials given by Radon
measures, In Stochastic processes, physics and geometry, 160–195, World Sci. Publishing, Teaneck,
NJ, 1990.
[3] R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc.,
95(1960), 263–273.
[4] Z.-Q. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on d-sets, Stochastic
Process. Appl., 108(2003), 27–62.
[5] Z.-Q. Chen and R. Song, Intrinsic ultracontractivity and conditional gauge for symmetric stable
processes, J. Funct. Anal., 150 (1997), 204–239.
[6] Z.-Q. Chen and R. Song, Conditional gauge theorem for non-local Feynman-Kac transforms, Probab.
Th. rel. Fields,125(2003), 45–72.
[7] Z.-Q. Chen and T. Zhang, Girsanov and Feynman-Kac type transformations for symmetric Markov
processes, Ann. Inst. H. Poincaré Probab. Statist., 38(2002), 475–505.
[8] M. Fukushima, On a decomposition of additive functionals in the strict sense for a symmetric Markov
process. In Dirichlet forms and stochastic processes (Beijing, 1993), 155–169, de Gruyter, Berlin,
1995.
[9] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet forms and symmetric Markov processes, Walter
De Gruyter, Berlin, 1994.
[10] M. Sharpe, General Theory of Markov Processes, Academic Press, Boston, 1988.
[11] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.), 7(1982), 447–526.
[12] R. Song, Feynman-Kac semigroup with discontinuous additive functionals, J. Theoret. Probab., 8
(1995), 727–762.
[13] Qi S. Zhang, A Harnack inequality for the equation ∇(a∇u) + b∇u = 0, when |b| ∈ Kn+1 ,
Manuscripta Math., 89 (1996), 61–77.
[14] Qi S. Zhang, Gaussian bounds for the fundamental solutions of ∇(A∇u)+B∇u−ut = 0, Manuscripta
Math., 93 (1997), 381–390.
161