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Smooth properties for semigroups of Lévy processes and their application
Statistics and Probability Letters 89 (2014) 23–30
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Smooth properties for semigroups of Lévy processes and their application Jian Wang ∗ School of Mathematics and Computer Science, Fujian Normal University, 350007, Fuzhou, PR China
article
abstract
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Article history: Received 17 May 2013 Received in revised form 24 November 2013 Accepted 20 February 2014 Available online 26 February 2014
Explicit smooth properties for the semigroup of Lévy processes are derived in terms of its symbol. As an application, we obtain new sufficient conditions for the strong Feller property of stochastic differential equations driven by the additive Lévy process. © 2014 Elsevier B.V. All rights reserved.
MSC: 60J25 60J75 35S05 Keywords: Lévy processes Symbol Strong Feller Stochastic differential equations
1. Main result Let Z = (Zt )t ≥0 be a Lévy process on Rd , which is defined on some stochastic basis (Ω , F , (Ft )t ≥0 , P), continuous in probability, has stationary independent increments, càdlàg trajectories, and satisfies Z0 = 0, P-a.s. It is well known that the characteristic exponent or the symbol Φ of (Zt )t ≥0 , defined by
E ei⟨ξ ,Zt ⟩ = e−t Φ (ξ ) ,
ξ ∈ Rd ,
enjoys the following Lévy–Khintchine representation:
Φ (ξ ) =
1 2
⟨Q ξ , ξ ⟩ + i⟨b, ξ ⟩ +
1 − ei⟨ξ ,z ⟩ + i⟨ξ , z ⟩1{|z |≤1} (z ) ν(dz ),
(1.1)
z ̸=0
where Q ∈ Rd×d is a positive semi-definite matrix, b ∈ Rd is the drift vector and ν is the Lévy measure, that is, a σ -finite d 2 measure on R \ {0} such that z ̸=0 (1 ∧ |z | ) ν(dz ) < ∞. Our standard references for the Lévy process and its symbol are the monographs (Jacob, 2001; Sato, 1999). Let (PtZ )t ≥0 be the semigroup associated with (Zt )t ≥0 . In this short paper, we are concerned with smooth properties for (PtZ )t ≥0 in terms of the symbol Φ (ξ ). For this aim, we need the following notation of function spaces. Denote by Bb (Rd )
∗
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http://dx.doi.org/10.1016/j.spl.2014.02.013 0167-7152/© 2014 Elsevier B.V. All rights reserved.
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J. Wang / Statistics and Probability Letters 89 (2014) 23–30
the set of bounded Borel measurable functions on Rd , and by Lp (Rd ) the standard Lebesgue space with norm ∥ · ∥p for all p ∈ [1, ∞]. For any k ∈ N0 := N ∪ {0, ∞}, let Cbk (Rd ) be the space of continuously differentiable functions with bounded derivatives up to the order k; in particular, Cb (Rd ) = Cb0 (Rd ) is the space of bounded continuous functions on Rd , and Cb∞ (Rd ) is the space of smooth functions with bounded derivatives for all orders. For any k ∈ N0 := N ∪ {0} and f ∈ Cbk (Rd ),
∥f ∥C k (Rd ) := b
|α|≤k
∥∂ α f ∥∞ . For a real non-integer number s > 0, denote by Cbs (Rd ) the subset of Cb[s] (Rd ) consisting of
functions f such that
∥f ∥C s (Rd ) :=
b
∥∂ α f ∥∞ +
sup
|β|=[s] x̸=y
|α|≤[s]
|∂ β f (x) − ∂ β f (y)| < ∞, |x − y|{s}
where s = [s] + {s} with [s] ∈ N0 and 0 < {s} < 1. The main contribution of this paper is as follows. Theorem 1.1. Suppose that lim
|ξ |→∞
Re Φ (ξ ) log(1 + |ξ |)
= ∞,
(1.2)
and there exists a constant c > 0 such that for any t ∈ (0, 1] and n ∈ N0 ,
exp −t Re Φ (ξ ) |ξ |n dξ ≤ ch(t )−n−d ,
(1.3)
where h(t ) = ϕ −11(1/t ) and ϕ(ρ) = sup|ξ |≤ρ Re Φ (ξ ). Then, for any p ∈ [1, ∞] and s, t > 0, the semigroup PtZ maps Lp (Rd ) into Cbs (Rd ). More explicitly, for all p ∈ [1, ∞] and s > 0, there is a constant C > 0 such that for any t > 0,
∥PtZ f ∥C s (Rd ) ≤ C ∥f ∥p h(t ∧ 1)−s−d/p . b
The assumption (1.2) is called Hartman–Wintner’s condition in the literature; see Hartman and Wintner (1942) for classical results or Knopova and Schilling (2013) for a recent study. We will present some comments on Theorem 1.1. First, as a direct consequence of Theorem 1.1, we can conclude that under (1.2) and (1.3), the semigroup PtZ maps Bb (Rd ) into Cbs (Rd ) for any s and t > 0. In particular, the semigroup PtZ is strong Feller, i.e., for any t > 0 and f ∈ Bb (Rd ), PtZ f ∈ Cb (Rd ). Second, Theorem 1.1 extends Kusuoka and Marinelli (in press, Proposition 3.8), where the Lévy process Z is assumed to be decomposed into two independent parts, and one of them is a subordinate Brownian motion. Third, according to Theorem 1.1 and the proof of Wang (2013a, Proposition 1.3), we have the following corollary, which reduces (1.2) and (1.3) into the asymptotic behavior of Re Φ (ξ ) near infinity. Corollary 1.2. Assume that Re Φ (ξ ) ≍ g (ξ ) as |ξ | → ∞, where g is a strictly increasing function, which is differentiable on (s0 , ∞) for some constant s0 > 0, and which satisfies that lims→∞ g (s)/ log s = ∞ and lims→∞ g −1 (2s)/g −1 (s) < ∞. Then, for all p ∈ [1, ∞] and s > 0, there is a constant C > 0 such that for any t > 0, s+d/p 1 Z −1 . ∥Pt f ∥C s (Rd ) ≤ C ∥f ∥p g b t ∧1 To close this section, we will present the following two examples to illustrate Theorem 1.1 and Corollary 1.2. Example 1.3. Let Z be a subordinate Brownian motion with symbol f (|ξ |2 ), where f (λ) = λα/2 (log(1 + λ))β/2 , α ∈ (0, 2) and β ∈ (−α, 2 − α). Then, for all p ∈ [1, ∞] and s > 0, there is a constant C > 0 such that for any t > 0,
−β/2 (s+d/p)/α ∥PtZ f ∥C s (Rd ) ≤ C ∥f ∥p (t ∧ 1)−1 log(1 + (t ∧ 1)−1 ) . b
Example 1.4. Let µ be a finite nonnegative measure on the unit sphere S and assume that µ is nondegenerate in the sense that its support is not contained in any proper linear subspace of Rd . Let α ∈ (0, 2), β ∈ (0, ∞] and assume that the Lévy measure ν satisfies that for some constant r0 > 0 and any A ∈ B (Rd ),
ν(A) ≥
r0
0
1A (sθ )s
−1−α
ds µ(dθ ) +
S
∞ r0
1A (sθ )s−1−β ds µ(dθ ). S
Then, for all p ∈ [1, ∞] and s > 0, there is a constant C > 0 such that for any t > 0,
∥PtZ f ∥C s (Rd ) ≤ C ∥f ∥p (t ∧ 1)−(s+d/p)/α . b
J. Wang / Statistics and Probability Letters 89 (2014) 23–30
25
2. Proofs Proof of Theorem 1.1. Step 1: For any t > 0, let µt be the law of Zt . Then, for any f ∈ Bb (Rd ), PtZ f (x) = Ex f (Zt ) =
f (x + z ) µt (dz ).
For every r > 0, let {µrt , t ≥ 0} be the family of infinitely divisible probability measures on Rd whose Fourier transforms
rt (ξ ) = exp(−t Φr (ξ )), where are of the form µ Φr (ξ ) =
1 − ei⟨ξ ,z ⟩ + i⟨ξ , z ⟩ ν(dz ),
ξ ∈ Rd .
|z |≤r
Then, according to (1.2), (1.3) and Schilling et al. (2012, Proposition 2.3), we know that for all r , t > 0, the measure µrt has a density prt ∈ Cb∞ (Rd ), and there is a constant t1 > 0 such that for all β ∈ Nd0 , y ∈ Rd , n ∈ N0 and t ∈ (0, t1 ].
β h(t ) ∂ pt (y) ≤ C (n, d, |β|, Φ ) h(t )−(d+|β|) 1 + h(t )−1 |y| −n . In particular, (2.4) implies that for any β ∈
Nd0 ,
(2.4)
l ∈ [1, ∞] and t ∈ (0, t1 ],
∥∂ β pht (t ) ∥l ≤ C (l, d, |β|, Φ )h(t )−(d+|β|)+ l . d
(2.5)
For r > 0 and ξ ∈ R , define d
Ψr (ξ ) := Φ (ξ ) − Φr (ξ ) =
1 − ei⟨ξ ,z ⟩ ν(dz ) − i ξ ,
|z |>r
1<|z |≤r
z ν(dz ) − b.
Since Ψr is given by a Lévy–Khintchine formula, it is the characteristic exponent of some d-dimensional infinitely divisible random variable. Let {πtr , t ≥ 0} be the family of infinitely divisible measures, whose Fourier transforms are of the form
πtr (ξ ) = exp(−t Ψr (ξ )) for ξ ∈ Rd . Clearly, µt = µrt ∗ πtr for all t , r > 0. Therefore, for all f ∈ Bb (Rd ), we have Z Pt f (x) = f x + z µt (dz ) = f x + z µrt ∗ πtr (dz ) = f x + z1 + z2 πtr (dz1 ) µrt (dz2 ). Taking r = h(t ), we get, using the conclusions above, that for all t ≤ t1 , PtZ f (x) =
h(t )
pt
(z2 ) dz2
h(t )
f x + z1 + z2 πt
(dz1 ) =
h(t )
pt
z2 − x dz2
h(t )
f (z1 + z2 ) πt
(dz1 ).
According to (2.4) and the dominated convergence theorem, for any β ∈ Nd0 and x ∈ Rd ,
∂ β PtZ f (x) =
∂xβ pht (t ) z2 − x dz2
h(t )
f (z1 + z2 ) πt
(dz1 ).
(2.6)
Step 2: We begin with the proof of the required assertion for all s ∈ N0 . For any p ∈ [1, ∞] and f ∈ Lp (Rd ), set
f (x) :=
h(t )
(dz1 ). Then, for any p ∈ [1, ∞), by the Hölder inequality, p 1/p 1/p h(t ) h(t ) p ∥f ∥p = ≤ |f (z1 + x)| πt (dz1 ) dx f (z1 + x) πt (dz1 ) dx
f (z1 + x) πt
=
h(t ) t
|f (z1 + x)| dx π p
(dz1 )
1/p
= ∥f ∥p . It is easy to check that the inequality above also holds for p = ∞. Applying the Young inequality, we get from (2.5) and (2.6) that for any p ∈ [1, ∞], t ≤ t1 and α ∈ Nd0 with |α| ≤ s,
α Z ∂ P f (x) = ∂ α pht (t ) z2 − x f˜ (z2 ) dz2 x t ≤ ∥∂ α pht (t ) ∥q ∥f ∥p
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J. Wang / Statistics and Probability Letters 89 (2014) 23–30
≤
C (q, d, |α|, Φ ) h(t )
d+|α|− dq
C
≤
s+ dp
h(t )
∥f ∥p
∥f ∥p ,
where in the first inequality q ∈ [1, ∞] such that p−1 + q−1 = 1, and the last inequality we have used the fact that h(t ) → 0 as t → 0. Next, we treat any non-integer s > 0. It suffices to consider the term
|∂ β PtZ f (x) − ∂ β PtZ f (y)| , |x − y|{s}
It ,s,β (x, y) :=
where β ∈ Nd0 with |β| = [s]. Let the constants C below vary from line to line. According to (2.5), (2.6), the Hölder inequality and the Cr -inequality, we get that for any t ∈ (0, t1 ] and p ∈ (1, ∞],
∂ β pht (t ) z2 − x − ∂ β pht (t ) z2 − y ˜f (z2 ) dz2 |x − y|{s} β h(t ) ∂ pt z2 − x − ∂ β pht (t ) z2 − y q{s} ≤ ∥f ∥p |x − y| 1/q q(1−{s}) × ∂ β pht (t ) z2 − x − ∂ β pht (t ) z2 − y dz2 β h(t ) {s}/q ∂ pt z2 − x − ∂ β pht (t ) z2 − y q dz2 ≤ C ∥f ∥p |x − y| (1−{s})/q β h(t ) q ∂ pt z2 − x − ∂ β pht (t ) z2 − y dz2 ×
It ,s,β (x, y) ≤
{s}/q β h(t ) q ∥∂ β pht (t ) ∥1q−{s} ≤ C ∥f ∥p ∇∂ pt z2 − x − θ (y − x) dz2 ≤ C ∥f ∥p ∥∇∂ β pht (t ) ∥{qs} ∥∂ β pht (t ) ∥1q−{s} ≤ C ∥f ∥p h(t )−(1−{s})(d+|β|−d/q)−{s}(d+|β|+1−d/q) ≤ C ∥f ∥p h(t )−(1−{s})([s]+d/p)−{s}([s]+1+d/p) = C ∥f ∥p h(t )−s−d/p , where in the second inequality the constant q ∈ [1, ∞) is such that p−1 + q−1 = 1, and in the fourth inequality we have used the mean value theorem and θ ∈ [0, 1]. Note that, according to (2.5) and the Hölder inequality, one can easily verify that the estimate above also holds for p = 1. Therefore, the required assertion for t ∈ (0, t1 ] follows from all the conclusions above and the definition of ∥f ∥C s (Rd ) . b
For any t > t1 , one can write PtZ f = PtZ1 PtZ−t1 f . Then, we have
∥PtZ f ∥C s (Rd ) ≤ C ∥PtZ−t1 f ∥p h(t1 )−s−d/p b
≤ C ∥f ∥p h(t1 )−s−d/p . The proof is completed.
Proof of Example 1.3. The symbol of the subordinate Brownian motion Z here satisfies Re Φ (ξ ) = |ξ |α log(1 + |ξ |2 )
β/2
.
1/α For r > 0, set g (r ) = r α (log(1 + r 2 ))β/2 and h(r ) = r (log(1 + r ))−β/2 . Then, for r → ∞, we have g h(r ) = r log(1 + r 2 )
≍ r (log r )−β/2
−β/2
log 1 + r (log(1 + r ))−β/2
2 log r − β log log r
α
β/2
2/α β/2
J. Wang / Statistics and Probability Letters 89 (2014) 23–30
=r
2 log r − β log log r
α log r
≍ r.
This shows that g −1 (r ) ≍ h(r ) for r → ∞, and now Corollary 1.2 applies. Proof of Example 1.4. Let Z1
ν (A) :=
r0
27
β/2
0
Zt1
and
1A (sθ )s
Zt2
−1−α
be Lévy processes whose Lévy measures are given by ds µ(dθ ) +
∞
r0
S
1A (sθ )s−1−β ds µ(dθ ) S
and 2
1
ν Z (dz ) := ν(dz ) − ν Z (dz ) ≥ 0, respectively. After some elementary calculations, we see that the symbol Φ Z of Zt1 satisfies Re Φ Z (ξ ) ≍ |ξ |α as |ξ | → ∞. 1
1
Z1
Let Pt denote the semigroup of Zt1 . According to Theorem 1.1 or Corollary 1.2, we can prove the claim first for the Lévy process Zt1 . To come back to the original semigroup PtZ we can now use the following facts that for any p ∈ [1, ∞] and s > 0, 1
2
2
∥PtZ f ∥C s (Rd ) = ∥PtZ PtZ f ∥C s (Rd ) and ∥PtZ f ∥p ≤ ∥f ∥p . b
b
Combining with all the conclusions above, we prove the desired assertion.
3. Extension and application 3.1. Extension: smooth properties for semigroups of Ornstein–Uhlenbeck processes Let (Xt (x))t ≥0 be a d-dimensional Ornstein–Uhlenbeck process, which is defined as the unique strong solution of the following stochastic differential equation dXt = AXt dt + dZt ,
X0 = x ∈ R d .
(3.7)
Here A is a real d × d matrix, and Z is a Lévy process in R with symbol Φ given by (1.1). Let Pt be the semigroup of the ddimensional Ornstein–Uhlenbeck process (Xt (x))t ≥0 given by (3.7). Then, we have the following statement, which extends Theorem 1.1 to Ornstein–Uhlenbeck processes driven by Lévy processes. d
Proposition 3.1. Assume that for every t > 0,
t 0
lim inf
⊤
Re Φ esA ξ ds
log(1 + |ξ |)
|ξ |→∞
= ∞,
and there exists a constant c > 0 such that for any t ∈ (0, 1] and all n ∈ N0 ,
t
⊤
n+d
Re Φ esA ξ ds |ξ |n dξ ≤ c ϕt−1 (1)
exp −
,
0
where A⊤ is the transpose of A, and
ϕt (ρ) := sup
t
|ξ |≤ρ
⊤
Re Φ esA ξ ds.
0
Then, for all p ∈ [1, ∞] and s > 0, there is a constant C > 0 such that for any t ∈ [0, 1],
s+d/p ∥Pt f ∥C s (Rd ) ≤ C ∥f ∥p ϕt−∧11 (1) . b
Sketch of the proof of Proposition 3.1. The Ornstein–Uhlenbeck process (Xt (x))t ≥0 determined by (3.7) is of the following explicit form Xtx
t
tA
=e x+
e(t −s)A dZs ,
t ≥ 0.
0
For t > 0, denote by µt the law of Xt0 := characteristic exponent of µt is given by
Φt (ξ ) :=
0
t
⊤ Φ esA ξ ds,
ξ ∈ Rd .
t 0
e(t −s)A dZs . Then, µt is an infinitely divisible probability distribution, and the
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J. Wang / Statistics and Probability Letters 89 (2014) 23–30
Thus, for all f ∈ Bb (Rd ), Pt f (x) =
f etA x + z µt (dz ).
Now, the required assertion essentially follows from the arguments of Theorem 1.1 and Wang (2013a, Theorem 1.1).
3.2. Application: strong Feller property of stochastic differential equations driven by additive Lévy processes We consider the following stochastic differential equation (SDE) dXt = b(Xt ) dt + dZt ,
X 0 = x,
(3.8)
where (Zt )t ≥0 is a Lévy process with symbol Φ satisfying (1.2) and (1.3), and b : R → R is a continuous function such that for any x, y ∈ Rd , d
⟨b(x) − b(y), x − y⟩ ≤ K |x − y|2
d
(3.9)
holds for some constant K > 0. It is known that (e.g. see the proofs of Applebaum (2004, Theorems 6.2.3, 6.2.9 and 6.2.11)) there exists a unique strong solution to the SDE (3.8), which will be denoted by (Xt (x))t ≥0 . Let (PtX )t ≥0 be the semigroup corresponding to (Xt (x))t ≥0 . As an application of Theorem 1.1, we have the following statement for the strong Feller property of (PtX )t ≥0 , which means that for any t > 0 and f ∈ Bb (Rd ), PtX f ∈ Cb (Rd ). Proposition 3.2. Let ν be the Lévy measure corresponding to the symbol Φ given by (1.1), and h(t ) be the positive measurable function defined in Theorem 1.1. If 1
1 h(t )
0
dt < ∞
and there exists a constant α ∈ (0, 2) such that {|z |≥1} |z |α ν(dz ) < ∞, then the semigroup (PtX )t ≥0 associated with the process (Xt (x))t ≥0 above is strong Feller.
Two equivalent characterizations for the strong Feller property of Markov semigroups have been studied in Schilling and Wang (2012). In particular, motivated by Schilling and Wang (2012, Theorem 2.8), we have obtained in Wang (2013b, Proposition 3.2) some conditions for the strong Feller property of stochastic differential equations driven by additive Lévy processes. The proof of Wang (2013b, Proposition 3.2) is based on the positive maximum principle and lower-boundedness of the associated generator, while the approach of Proposition 3.2 depends on Duhamel’s formula. Note that Proposition 3.2 strengthens Wang (2013b, Proposition 3.2) and Kusuoka and Marinelli (in press, Theorem 4.5), where the drift term is required to be differentiable with bounded derivatives or to be bounded, respectively. The point is important, because stochastic differential equations with unbounded drift are more interesting in applications; see e.g. Wang and Wang (2014) and Wang (2013c). Proof of Proposition 3.2. (1) We first assume that b is a bounded continuous function. According to Wang (2010, Theorem 2.1), the semigroup (PtX )t ≥0 is Cb -Feller, i.e., for any f ∈ Cb (Rd ) and t ≥ 0, PtX f ∈ Cb (Rd ). Let f ∈ Bb (Rd ) and t > 0. It is obvious that PtX f is a bounded measurable function, and, by Duhamel’s formula (see e.g., Chen et al. (2012, (1.1))), PtX f = PtZ f +
t
PtX−s ⟨b, ∂ PsZ f ⟩ ds.
(3.10)
0
Since the symbol of Lévy process (Zt )t ≥0 satisfies (1.2) and (1.3), Theorem 1.1 yields that for any t > 0, PtZ f ∈ Cb1 (Rd ) and ∥∂ PtZ f ∥∞ ≤ Ch(t ∧ 1)−1 ∥f ∥∞ holds with some constant C > 0. Note that, b ∈ Cb (Rd ) implies that ⟨b, ∂ PtZ f ⟩ ∈ Cb (Rd ); since
1
h(t )−1 dt < ∞ and ∥PtX f ∥∞ ≤ ∥f ∥∞ , the sup-norm of the integral is finite. This, along with the dominated convergence theorem, gives us that PtX f is a bounded continuous function. (2) Next, we consider the general drift term b. For any l ≥ 1, let Bl = {x ∈ Rd : |x| ≤ l}, and hl : Rd → [0, 1] be a smooth function with compact support such that hl = 1 on Bl . Define bl = bhl . Consider the following stochastic differential equation (SDE) 0
(l)
dXt
= bl (Xt(l) ) dt + dZt ,
(l)
X0 = x.
(3.11)
It is easy to check that under (3.9) for any x ∈ Rd ,
⟨bl (x), x⟩ = hl (x)⟨b(x), x⟩ ≤ ⟨b(x), x⟩ ∨ 0 ≤ K |x|2 + |b(0)||x| ≤ K1 (1 + |x|2 ), and any x, y ∈ Rd with |x − y| ≤ 1,
⟨bl (x) − bl (y), x − y⟩ ≤ hl (x)⟨b(x) − b(y), x − y⟩ + |b(y)||hl (x) − hl (y)||x − y| ≤ K + ∥∇ h∥∞ sup |b(y)| |x − y|2 , |y|≤l+1
J. Wang / Statistics and Probability Letters 89 (2014) 23–30
29
where in the last inequality we have used the fact that for any x, y ∈ Rd with |y| ≤ l + 1 and |x − y| ≤ 1, |hl (x) − hl (y)| = 0. Therefore, according to the proofs of Applebaum (2004, Theorems 6.2.3, 6.2.9 and 6.2.11) and Wang (2010, Theorem 2.1) (l) again, there exists a unique strong solution to the SDE (3.11), which is denoted by X (l) (x) = (Xt (x))t ≥0 , and the semigroup (l) associated with X (x) is Cb -Feller. On the other hand, since the function bl is bounded and continuous, by (1) we know that the semigroup associated with X (l) (x) is also strong Feller. For any l ≥ 1, define the stopping time
τl (x) := inf{s ≥ 0 : Xs (x) ̸∈ Bl }. (l)
It is clear that for any x ∈ Bl and t ≤ τl (x), Xt (x) = Xt (x), and for any x ∈ Rd , liml→∞ τl (x) = ∞. Now, for any f ∈ Bb (Rd ) d and x0 ∈ Rd , we choose a sequence {xk }∞ k=1 ⊆ R such that xk → x0 as k → ∞. Thus, for n large enough such that x0 ∈ Bn−1 and for k large enough,
|Ef (Xt (xk )) − Ef (Xt (x0 ))| ≤ |Ef (Xt(n) (xk )) − Ef (Xt(n) (x0 ))| + |Ef (Xt(n) (xk )) − Ef (Xt (xk ))| + |Ef (Xt(n) (x0 )) − Ef (Xt (x0 ))| ≤ |Ef (Xt(n) (xk )) − Ef (Xt(n) (x0 ))| + ∥f ∥∞ P(τn (xk ) ≤ t ) + P(τn (x0 ) ≤β t ) supk≥0 E sups∈[0,t ] |Xs (xk )| ≤ |Ef (Xt(n) (xk )) − Ef (Xt(n) (x0 ))| + 2∥f ∥∞ β
(3.12)
n
where β ∈ (0, α) and α is the constant in the assumptions of Proposition 3.2. On the other hand, let φ ∈ C 2 (Rd ) such that φ(x) = |x|β for |x| ≥ 1. Let L be the generator of the process (Xt (x))t ≥0 . For any f ∈ Cc∞ (Rd ), Lf (x) = ⟨b(x), ∇ f (x)⟩ +
f (x + z ) − f (x) − ∇ f (x) · z 1{|z |≤1} ν(dz ).
It is easy to see that under (3.9) and the assumptions of Proposition 3.2, there is a constant c > 0 such that Lφ ≤ c φ, which together with the Ito’s formula gives us sup E sup |Xs (xk )|β < ∞.
k≥0
s∈[0,t ]
Therefore, first letting k → ∞ and then n → ∞ in (3.12), we can show that Pt f is continuous. The proof is complete.
Remark 3.3. There are a few papers that deal with stochastic differential equations from the point-of-view of the symbol, see e.g. Schilling and Schnurr (2010) and Schnurr (2013). Unfortunately it seems that one cannot adapt the argument of Proposition 3.2 to study the strong Feller property of the following stochastic differential equation with multiplicative noise dXt = b(Xt ) dt + σ (Xt ) dZt , where σ : Rd → Rd ⊗ Rd and b : Rd → Rd are continuous. The reason is essentially due to that in this case the relation such like (3.10) between the semigroup of Xt and that of Zt is not available. Acknowledgments Financial support through the National Natural Science Foundation of China (No. 11201073), and the Program for New Century Excellent Talents in Universities of Fujian (No. JA12053) and the Program for Nonlinear Analysis and Its Applications (No. IRTL1206) is gratefully acknowledged. References Applebaum, D., 2004. Lévy Processes and Stochastic Calculus. Cambridge Univ. Press, Cambridge. Chen, Z.-Q., Kim, P., Song, R., 2012. Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation. Ann. Probab. 40, 2483–2538. Hartman, P., Wintner, A., 1942. On the infinitesimal generators of integral convolutions. Am. J. Math. 64, 273–298. Jacob, N., 2001. Pseudo differential operators and Markov processes. In: Fourier Analysis and Semigroups, vol. 1. Imperial College Press, London. Knopova, V., Schilling, R.L., 2013. A note on the existence of transition probability densities for Lévy processes. Forum Math. 25, 125–149. Kusuoka, S., Marinelli, C., 2013. On smoothing properties of transition semigroups associated to a class of SDEs with jumps. Ann. Inst. H. Poincaré Probab. Statist. see arXiv:1208.2860 (in press). Sato, K., 1999. Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge. Schilling, R.L., Schnurr, A., 2010. The symbol associated with the solution of a stochastic differential equation. Electr. J. Probab. 15, 1369–1393. Schilling, R.L., Sztonyk, P., Wang, J., 2012. Coupling property and gradient estimates of Lévy processes via the symbol. Bernoulli 18, 1128–1149. Schilling, R.L., Wang, J., 2012. Strong Feller continuity of Feller processes and semigroups. Infin. Dimens. Anal. Quandum Probab. 15, 1250010-1–125001028. Schnurr, A., 2013. Generalization of the Blumenthal–Getoor index to the class of homogeneous diffusions with jumps and some applications. Bernoulli 19, 2010–2032. Wang, J., 2010. Regularity of semigroups generated by Lévy type operators via coupling. Stoc. Proc. Appl. 120, 1680–1700. Wang, J., 2013a. Regularity for semigroups of Ornstein–Uhlenbeck processes. Positivity 17, 205–221.
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Wang, J., 2013b. Sub-markovian C0 -semigroups generated by fractional Laplacian with gradient perturbation. Integral Equations Operator Theory 76, 151–161. Wang, J., 2013c. Exponential ergodicity and strong ergodicity for SDEs driven by symmetric α -stable processes. Appl. Math. Lett. 26, 654–658. Wang, F.-Y., Wang, J., 2014. Harnack inequalties for stochastic equations driven by Lévy noise. J. Math. Anal. Appl. 410, 513–523.
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Smooth properties for semigroups of Lévy processes and their application Jian Wang ∗ School of Mathematics and Computer Science, Fujian Normal University, 350007, Fuzhou, PR China
article
abstract
info
Article history: Received 17 May 2013 Received in revised form 24 November 2013 Accepted 20 February 2014 Available online 26 February 2014
Explicit smooth properties for the semigroup of Lévy processes are derived in terms of its symbol. As an application, we obtain new sufficient conditions for the strong Feller property of stochastic differential equations driven by the additive Lévy process. © 2014 Elsevier B.V. All rights reserved.
MSC: 60J25 60J75 35S05 Keywords: Lévy processes Symbol Strong Feller Stochastic differential equations
1. Main result Let Z = (Zt )t ≥0 be a Lévy process on Rd , which is defined on some stochastic basis (Ω , F , (Ft )t ≥0 , P), continuous in probability, has stationary independent increments, càdlàg trajectories, and satisfies Z0 = 0, P-a.s. It is well known that the characteristic exponent or the symbol Φ of (Zt )t ≥0 , defined by
E ei⟨ξ ,Zt ⟩ = e−t Φ (ξ ) ,
ξ ∈ Rd ,
enjoys the following Lévy–Khintchine representation:
Φ (ξ ) =
1 2
⟨Q ξ , ξ ⟩ + i⟨b, ξ ⟩ +
1 − ei⟨ξ ,z ⟩ + i⟨ξ , z ⟩1{|z |≤1} (z ) ν(dz ),
(1.1)
z ̸=0
where Q ∈ Rd×d is a positive semi-definite matrix, b ∈ Rd is the drift vector and ν is the Lévy measure, that is, a σ -finite d 2 measure on R \ {0} such that z ̸=0 (1 ∧ |z | ) ν(dz ) < ∞. Our standard references for the Lévy process and its symbol are the monographs (Jacob, 2001; Sato, 1999). Let (PtZ )t ≥0 be the semigroup associated with (Zt )t ≥0 . In this short paper, we are concerned with smooth properties for (PtZ )t ≥0 in terms of the symbol Φ (ξ ). For this aim, we need the following notation of function spaces. Denote by Bb (Rd )
∗
Tel.: +86 13774590233. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.spl.2014.02.013 0167-7152/© 2014 Elsevier B.V. All rights reserved.
24
J. Wang / Statistics and Probability Letters 89 (2014) 23–30
the set of bounded Borel measurable functions on Rd , and by Lp (Rd ) the standard Lebesgue space with norm ∥ · ∥p for all p ∈ [1, ∞]. For any k ∈ N0 := N ∪ {0, ∞}, let Cbk (Rd ) be the space of continuously differentiable functions with bounded derivatives up to the order k; in particular, Cb (Rd ) = Cb0 (Rd ) is the space of bounded continuous functions on Rd , and Cb∞ (Rd ) is the space of smooth functions with bounded derivatives for all orders. For any k ∈ N0 := N ∪ {0} and f ∈ Cbk (Rd ),
∥f ∥C k (Rd ) := b
|α|≤k
∥∂ α f ∥∞ . For a real non-integer number s > 0, denote by Cbs (Rd ) the subset of Cb[s] (Rd ) consisting of
functions f such that
∥f ∥C s (Rd ) :=
b
∥∂ α f ∥∞ +
sup
|β|=[s] x̸=y
|α|≤[s]
|∂ β f (x) − ∂ β f (y)| < ∞, |x − y|{s}
where s = [s] + {s} with [s] ∈ N0 and 0 < {s} < 1. The main contribution of this paper is as follows. Theorem 1.1. Suppose that lim
|ξ |→∞
Re Φ (ξ ) log(1 + |ξ |)
= ∞,
(1.2)
and there exists a constant c > 0 such that for any t ∈ (0, 1] and n ∈ N0 ,
exp −t Re Φ (ξ ) |ξ |n dξ ≤ ch(t )−n−d ,
(1.3)
where h(t ) = ϕ −11(1/t ) and ϕ(ρ) = sup|ξ |≤ρ Re Φ (ξ ). Then, for any p ∈ [1, ∞] and s, t > 0, the semigroup PtZ maps Lp (Rd ) into Cbs (Rd ). More explicitly, for all p ∈ [1, ∞] and s > 0, there is a constant C > 0 such that for any t > 0,
∥PtZ f ∥C s (Rd ) ≤ C ∥f ∥p h(t ∧ 1)−s−d/p . b
The assumption (1.2) is called Hartman–Wintner’s condition in the literature; see Hartman and Wintner (1942) for classical results or Knopova and Schilling (2013) for a recent study. We will present some comments on Theorem 1.1. First, as a direct consequence of Theorem 1.1, we can conclude that under (1.2) and (1.3), the semigroup PtZ maps Bb (Rd ) into Cbs (Rd ) for any s and t > 0. In particular, the semigroup PtZ is strong Feller, i.e., for any t > 0 and f ∈ Bb (Rd ), PtZ f ∈ Cb (Rd ). Second, Theorem 1.1 extends Kusuoka and Marinelli (in press, Proposition 3.8), where the Lévy process Z is assumed to be decomposed into two independent parts, and one of them is a subordinate Brownian motion. Third, according to Theorem 1.1 and the proof of Wang (2013a, Proposition 1.3), we have the following corollary, which reduces (1.2) and (1.3) into the asymptotic behavior of Re Φ (ξ ) near infinity. Corollary 1.2. Assume that Re Φ (ξ ) ≍ g (ξ ) as |ξ | → ∞, where g is a strictly increasing function, which is differentiable on (s0 , ∞) for some constant s0 > 0, and which satisfies that lims→∞ g (s)/ log s = ∞ and lims→∞ g −1 (2s)/g −1 (s) < ∞. Then, for all p ∈ [1, ∞] and s > 0, there is a constant C > 0 such that for any t > 0, s+d/p 1 Z −1 . ∥Pt f ∥C s (Rd ) ≤ C ∥f ∥p g b t ∧1 To close this section, we will present the following two examples to illustrate Theorem 1.1 and Corollary 1.2. Example 1.3. Let Z be a subordinate Brownian motion with symbol f (|ξ |2 ), where f (λ) = λα/2 (log(1 + λ))β/2 , α ∈ (0, 2) and β ∈ (−α, 2 − α). Then, for all p ∈ [1, ∞] and s > 0, there is a constant C > 0 such that for any t > 0,
−β/2 (s+d/p)/α ∥PtZ f ∥C s (Rd ) ≤ C ∥f ∥p (t ∧ 1)−1 log(1 + (t ∧ 1)−1 ) . b
Example 1.4. Let µ be a finite nonnegative measure on the unit sphere S and assume that µ is nondegenerate in the sense that its support is not contained in any proper linear subspace of Rd . Let α ∈ (0, 2), β ∈ (0, ∞] and assume that the Lévy measure ν satisfies that for some constant r0 > 0 and any A ∈ B (Rd ),
ν(A) ≥
r0
0
1A (sθ )s
−1−α
ds µ(dθ ) +
S
∞ r0
1A (sθ )s−1−β ds µ(dθ ). S
Then, for all p ∈ [1, ∞] and s > 0, there is a constant C > 0 such that for any t > 0,
∥PtZ f ∥C s (Rd ) ≤ C ∥f ∥p (t ∧ 1)−(s+d/p)/α . b
J. Wang / Statistics and Probability Letters 89 (2014) 23–30
25
2. Proofs Proof of Theorem 1.1. Step 1: For any t > 0, let µt be the law of Zt . Then, for any f ∈ Bb (Rd ), PtZ f (x) = Ex f (Zt ) =
f (x + z ) µt (dz ).
For every r > 0, let {µrt , t ≥ 0} be the family of infinitely divisible probability measures on Rd whose Fourier transforms
rt (ξ ) = exp(−t Φr (ξ )), where are of the form µ Φr (ξ ) =
1 − ei⟨ξ ,z ⟩ + i⟨ξ , z ⟩ ν(dz ),
ξ ∈ Rd .
|z |≤r
Then, according to (1.2), (1.3) and Schilling et al. (2012, Proposition 2.3), we know that for all r , t > 0, the measure µrt has a density prt ∈ Cb∞ (Rd ), and there is a constant t1 > 0 such that for all β ∈ Nd0 , y ∈ Rd , n ∈ N0 and t ∈ (0, t1 ].
β h(t ) ∂ pt (y) ≤ C (n, d, |β|, Φ ) h(t )−(d+|β|) 1 + h(t )−1 |y| −n . In particular, (2.4) implies that for any β ∈
Nd0 ,
(2.4)
l ∈ [1, ∞] and t ∈ (0, t1 ],
∥∂ β pht (t ) ∥l ≤ C (l, d, |β|, Φ )h(t )−(d+|β|)+ l . d
(2.5)
For r > 0 and ξ ∈ R , define d
Ψr (ξ ) := Φ (ξ ) − Φr (ξ ) =
1 − ei⟨ξ ,z ⟩ ν(dz ) − i ξ ,
|z |>r
1<|z |≤r
z ν(dz ) − b.
Since Ψr is given by a Lévy–Khintchine formula, it is the characteristic exponent of some d-dimensional infinitely divisible random variable. Let {πtr , t ≥ 0} be the family of infinitely divisible measures, whose Fourier transforms are of the form
πtr (ξ ) = exp(−t Ψr (ξ )) for ξ ∈ Rd . Clearly, µt = µrt ∗ πtr for all t , r > 0. Therefore, for all f ∈ Bb (Rd ), we have Z Pt f (x) = f x + z µt (dz ) = f x + z µrt ∗ πtr (dz ) = f x + z1 + z2 πtr (dz1 ) µrt (dz2 ). Taking r = h(t ), we get, using the conclusions above, that for all t ≤ t1 , PtZ f (x) =
h(t )
pt
(z2 ) dz2
h(t )
f x + z1 + z2 πt
(dz1 ) =
h(t )
pt
z2 − x dz2
h(t )
f (z1 + z2 ) πt
(dz1 ).
According to (2.4) and the dominated convergence theorem, for any β ∈ Nd0 and x ∈ Rd ,
∂ β PtZ f (x) =
∂xβ pht (t ) z2 − x dz2
h(t )
f (z1 + z2 ) πt
(dz1 ).
(2.6)
Step 2: We begin with the proof of the required assertion for all s ∈ N0 . For any p ∈ [1, ∞] and f ∈ Lp (Rd ), set
f (x) :=
h(t )
(dz1 ). Then, for any p ∈ [1, ∞), by the Hölder inequality, p 1/p 1/p h(t ) h(t ) p ∥f ∥p = ≤ |f (z1 + x)| πt (dz1 ) dx f (z1 + x) πt (dz1 ) dx
f (z1 + x) πt
=
h(t ) t
|f (z1 + x)| dx π p
(dz1 )
1/p
= ∥f ∥p . It is easy to check that the inequality above also holds for p = ∞. Applying the Young inequality, we get from (2.5) and (2.6) that for any p ∈ [1, ∞], t ≤ t1 and α ∈ Nd0 with |α| ≤ s,
α Z ∂ P f (x) = ∂ α pht (t ) z2 − x f˜ (z2 ) dz2 x t ≤ ∥∂ α pht (t ) ∥q ∥f ∥p
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J. Wang / Statistics and Probability Letters 89 (2014) 23–30
≤
C (q, d, |α|, Φ ) h(t )
d+|α|− dq
C
≤
s+ dp
h(t )
∥f ∥p
∥f ∥p ,
where in the first inequality q ∈ [1, ∞] such that p−1 + q−1 = 1, and the last inequality we have used the fact that h(t ) → 0 as t → 0. Next, we treat any non-integer s > 0. It suffices to consider the term
|∂ β PtZ f (x) − ∂ β PtZ f (y)| , |x − y|{s}
It ,s,β (x, y) :=
where β ∈ Nd0 with |β| = [s]. Let the constants C below vary from line to line. According to (2.5), (2.6), the Hölder inequality and the Cr -inequality, we get that for any t ∈ (0, t1 ] and p ∈ (1, ∞],
∂ β pht (t ) z2 − x − ∂ β pht (t ) z2 − y ˜f (z2 ) dz2 |x − y|{s} β h(t ) ∂ pt z2 − x − ∂ β pht (t ) z2 − y q{s} ≤ ∥f ∥p |x − y| 1/q q(1−{s}) × ∂ β pht (t ) z2 − x − ∂ β pht (t ) z2 − y dz2 β h(t ) {s}/q ∂ pt z2 − x − ∂ β pht (t ) z2 − y q dz2 ≤ C ∥f ∥p |x − y| (1−{s})/q β h(t ) q ∂ pt z2 − x − ∂ β pht (t ) z2 − y dz2 ×
It ,s,β (x, y) ≤
{s}/q β h(t ) q ∥∂ β pht (t ) ∥1q−{s} ≤ C ∥f ∥p ∇∂ pt z2 − x − θ (y − x) dz2 ≤ C ∥f ∥p ∥∇∂ β pht (t ) ∥{qs} ∥∂ β pht (t ) ∥1q−{s} ≤ C ∥f ∥p h(t )−(1−{s})(d+|β|−d/q)−{s}(d+|β|+1−d/q) ≤ C ∥f ∥p h(t )−(1−{s})([s]+d/p)−{s}([s]+1+d/p) = C ∥f ∥p h(t )−s−d/p , where in the second inequality the constant q ∈ [1, ∞) is such that p−1 + q−1 = 1, and in the fourth inequality we have used the mean value theorem and θ ∈ [0, 1]. Note that, according to (2.5) and the Hölder inequality, one can easily verify that the estimate above also holds for p = 1. Therefore, the required assertion for t ∈ (0, t1 ] follows from all the conclusions above and the definition of ∥f ∥C s (Rd ) . b
For any t > t1 , one can write PtZ f = PtZ1 PtZ−t1 f . Then, we have
∥PtZ f ∥C s (Rd ) ≤ C ∥PtZ−t1 f ∥p h(t1 )−s−d/p b
≤ C ∥f ∥p h(t1 )−s−d/p . The proof is completed.
Proof of Example 1.3. The symbol of the subordinate Brownian motion Z here satisfies Re Φ (ξ ) = |ξ |α log(1 + |ξ |2 )
β/2
.
1/α For r > 0, set g (r ) = r α (log(1 + r 2 ))β/2 and h(r ) = r (log(1 + r ))−β/2 . Then, for r → ∞, we have g h(r ) = r log(1 + r 2 )
≍ r (log r )−β/2
−β/2
log 1 + r (log(1 + r ))−β/2
2 log r − β log log r
α
β/2
2/α β/2
J. Wang / Statistics and Probability Letters 89 (2014) 23–30
=r
2 log r − β log log r
α log r
≍ r.
This shows that g −1 (r ) ≍ h(r ) for r → ∞, and now Corollary 1.2 applies. Proof of Example 1.4. Let Z1
ν (A) :=
r0
27
β/2
0
Zt1
and
1A (sθ )s
Zt2
−1−α
be Lévy processes whose Lévy measures are given by ds µ(dθ ) +
∞
r0
S
1A (sθ )s−1−β ds µ(dθ ) S
and 2
1
ν Z (dz ) := ν(dz ) − ν Z (dz ) ≥ 0, respectively. After some elementary calculations, we see that the symbol Φ Z of Zt1 satisfies Re Φ Z (ξ ) ≍ |ξ |α as |ξ | → ∞. 1
1
Z1
Let Pt denote the semigroup of Zt1 . According to Theorem 1.1 or Corollary 1.2, we can prove the claim first for the Lévy process Zt1 . To come back to the original semigroup PtZ we can now use the following facts that for any p ∈ [1, ∞] and s > 0, 1
2
2
∥PtZ f ∥C s (Rd ) = ∥PtZ PtZ f ∥C s (Rd ) and ∥PtZ f ∥p ≤ ∥f ∥p . b
b
Combining with all the conclusions above, we prove the desired assertion.
3. Extension and application 3.1. Extension: smooth properties for semigroups of Ornstein–Uhlenbeck processes Let (Xt (x))t ≥0 be a d-dimensional Ornstein–Uhlenbeck process, which is defined as the unique strong solution of the following stochastic differential equation dXt = AXt dt + dZt ,
X0 = x ∈ R d .
(3.7)
Here A is a real d × d matrix, and Z is a Lévy process in R with symbol Φ given by (1.1). Let Pt be the semigroup of the ddimensional Ornstein–Uhlenbeck process (Xt (x))t ≥0 given by (3.7). Then, we have the following statement, which extends Theorem 1.1 to Ornstein–Uhlenbeck processes driven by Lévy processes. d
Proposition 3.1. Assume that for every t > 0,
t 0
lim inf
⊤
Re Φ esA ξ ds
log(1 + |ξ |)
|ξ |→∞
= ∞,
and there exists a constant c > 0 such that for any t ∈ (0, 1] and all n ∈ N0 ,
t
⊤
n+d
Re Φ esA ξ ds |ξ |n dξ ≤ c ϕt−1 (1)
exp −
,
0
where A⊤ is the transpose of A, and
ϕt (ρ) := sup
t
|ξ |≤ρ
⊤
Re Φ esA ξ ds.
0
Then, for all p ∈ [1, ∞] and s > 0, there is a constant C > 0 such that for any t ∈ [0, 1],
s+d/p ∥Pt f ∥C s (Rd ) ≤ C ∥f ∥p ϕt−∧11 (1) . b
Sketch of the proof of Proposition 3.1. The Ornstein–Uhlenbeck process (Xt (x))t ≥0 determined by (3.7) is of the following explicit form Xtx
t
tA
=e x+
e(t −s)A dZs ,
t ≥ 0.
0
For t > 0, denote by µt the law of Xt0 := characteristic exponent of µt is given by
Φt (ξ ) :=
0
t
⊤ Φ esA ξ ds,
ξ ∈ Rd .
t 0
e(t −s)A dZs . Then, µt is an infinitely divisible probability distribution, and the
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J. Wang / Statistics and Probability Letters 89 (2014) 23–30
Thus, for all f ∈ Bb (Rd ), Pt f (x) =
f etA x + z µt (dz ).
Now, the required assertion essentially follows from the arguments of Theorem 1.1 and Wang (2013a, Theorem 1.1).
3.2. Application: strong Feller property of stochastic differential equations driven by additive Lévy processes We consider the following stochastic differential equation (SDE) dXt = b(Xt ) dt + dZt ,
X 0 = x,
(3.8)
where (Zt )t ≥0 is a Lévy process with symbol Φ satisfying (1.2) and (1.3), and b : R → R is a continuous function such that for any x, y ∈ Rd , d
⟨b(x) − b(y), x − y⟩ ≤ K |x − y|2
d
(3.9)
holds for some constant K > 0. It is known that (e.g. see the proofs of Applebaum (2004, Theorems 6.2.3, 6.2.9 and 6.2.11)) there exists a unique strong solution to the SDE (3.8), which will be denoted by (Xt (x))t ≥0 . Let (PtX )t ≥0 be the semigroup corresponding to (Xt (x))t ≥0 . As an application of Theorem 1.1, we have the following statement for the strong Feller property of (PtX )t ≥0 , which means that for any t > 0 and f ∈ Bb (Rd ), PtX f ∈ Cb (Rd ). Proposition 3.2. Let ν be the Lévy measure corresponding to the symbol Φ given by (1.1), and h(t ) be the positive measurable function defined in Theorem 1.1. If 1
1 h(t )
0
dt < ∞
and there exists a constant α ∈ (0, 2) such that {|z |≥1} |z |α ν(dz ) < ∞, then the semigroup (PtX )t ≥0 associated with the process (Xt (x))t ≥0 above is strong Feller.
Two equivalent characterizations for the strong Feller property of Markov semigroups have been studied in Schilling and Wang (2012). In particular, motivated by Schilling and Wang (2012, Theorem 2.8), we have obtained in Wang (2013b, Proposition 3.2) some conditions for the strong Feller property of stochastic differential equations driven by additive Lévy processes. The proof of Wang (2013b, Proposition 3.2) is based on the positive maximum principle and lower-boundedness of the associated generator, while the approach of Proposition 3.2 depends on Duhamel’s formula. Note that Proposition 3.2 strengthens Wang (2013b, Proposition 3.2) and Kusuoka and Marinelli (in press, Theorem 4.5), where the drift term is required to be differentiable with bounded derivatives or to be bounded, respectively. The point is important, because stochastic differential equations with unbounded drift are more interesting in applications; see e.g. Wang and Wang (2014) and Wang (2013c). Proof of Proposition 3.2. (1) We first assume that b is a bounded continuous function. According to Wang (2010, Theorem 2.1), the semigroup (PtX )t ≥0 is Cb -Feller, i.e., for any f ∈ Cb (Rd ) and t ≥ 0, PtX f ∈ Cb (Rd ). Let f ∈ Bb (Rd ) and t > 0. It is obvious that PtX f is a bounded measurable function, and, by Duhamel’s formula (see e.g., Chen et al. (2012, (1.1))), PtX f = PtZ f +
t
PtX−s ⟨b, ∂ PsZ f ⟩ ds.
(3.10)
0
Since the symbol of Lévy process (Zt )t ≥0 satisfies (1.2) and (1.3), Theorem 1.1 yields that for any t > 0, PtZ f ∈ Cb1 (Rd ) and ∥∂ PtZ f ∥∞ ≤ Ch(t ∧ 1)−1 ∥f ∥∞ holds with some constant C > 0. Note that, b ∈ Cb (Rd ) implies that ⟨b, ∂ PtZ f ⟩ ∈ Cb (Rd ); since
1
h(t )−1 dt < ∞ and ∥PtX f ∥∞ ≤ ∥f ∥∞ , the sup-norm of the integral is finite. This, along with the dominated convergence theorem, gives us that PtX f is a bounded continuous function. (2) Next, we consider the general drift term b. For any l ≥ 1, let Bl = {x ∈ Rd : |x| ≤ l}, and hl : Rd → [0, 1] be a smooth function with compact support such that hl = 1 on Bl . Define bl = bhl . Consider the following stochastic differential equation (SDE) 0
(l)
dXt
= bl (Xt(l) ) dt + dZt ,
(l)
X0 = x.
(3.11)
It is easy to check that under (3.9) for any x ∈ Rd ,
⟨bl (x), x⟩ = hl (x)⟨b(x), x⟩ ≤ ⟨b(x), x⟩ ∨ 0 ≤ K |x|2 + |b(0)||x| ≤ K1 (1 + |x|2 ), and any x, y ∈ Rd with |x − y| ≤ 1,
⟨bl (x) − bl (y), x − y⟩ ≤ hl (x)⟨b(x) − b(y), x − y⟩ + |b(y)||hl (x) − hl (y)||x − y| ≤ K + ∥∇ h∥∞ sup |b(y)| |x − y|2 , |y|≤l+1
J. Wang / Statistics and Probability Letters 89 (2014) 23–30
29
where in the last inequality we have used the fact that for any x, y ∈ Rd with |y| ≤ l + 1 and |x − y| ≤ 1, |hl (x) − hl (y)| = 0. Therefore, according to the proofs of Applebaum (2004, Theorems 6.2.3, 6.2.9 and 6.2.11) and Wang (2010, Theorem 2.1) (l) again, there exists a unique strong solution to the SDE (3.11), which is denoted by X (l) (x) = (Xt (x))t ≥0 , and the semigroup (l) associated with X (x) is Cb -Feller. On the other hand, since the function bl is bounded and continuous, by (1) we know that the semigroup associated with X (l) (x) is also strong Feller. For any l ≥ 1, define the stopping time
τl (x) := inf{s ≥ 0 : Xs (x) ̸∈ Bl }. (l)
It is clear that for any x ∈ Bl and t ≤ τl (x), Xt (x) = Xt (x), and for any x ∈ Rd , liml→∞ τl (x) = ∞. Now, for any f ∈ Bb (Rd ) d and x0 ∈ Rd , we choose a sequence {xk }∞ k=1 ⊆ R such that xk → x0 as k → ∞. Thus, for n large enough such that x0 ∈ Bn−1 and for k large enough,
|Ef (Xt (xk )) − Ef (Xt (x0 ))| ≤ |Ef (Xt(n) (xk )) − Ef (Xt(n) (x0 ))| + |Ef (Xt(n) (xk )) − Ef (Xt (xk ))| + |Ef (Xt(n) (x0 )) − Ef (Xt (x0 ))| ≤ |Ef (Xt(n) (xk )) − Ef (Xt(n) (x0 ))| + ∥f ∥∞ P(τn (xk ) ≤ t ) + P(τn (x0 ) ≤β t ) supk≥0 E sups∈[0,t ] |Xs (xk )| ≤ |Ef (Xt(n) (xk )) − Ef (Xt(n) (x0 ))| + 2∥f ∥∞ β
(3.12)
n
where β ∈ (0, α) and α is the constant in the assumptions of Proposition 3.2. On the other hand, let φ ∈ C 2 (Rd ) such that φ(x) = |x|β for |x| ≥ 1. Let L be the generator of the process (Xt (x))t ≥0 . For any f ∈ Cc∞ (Rd ), Lf (x) = ⟨b(x), ∇ f (x)⟩ +
f (x + z ) − f (x) − ∇ f (x) · z 1{|z |≤1} ν(dz ).
It is easy to see that under (3.9) and the assumptions of Proposition 3.2, there is a constant c > 0 such that Lφ ≤ c φ, which together with the Ito’s formula gives us sup E sup |Xs (xk )|β < ∞.
k≥0
s∈[0,t ]
Therefore, first letting k → ∞ and then n → ∞ in (3.12), we can show that Pt f is continuous. The proof is complete.
Remark 3.3. There are a few papers that deal with stochastic differential equations from the point-of-view of the symbol, see e.g. Schilling and Schnurr (2010) and Schnurr (2013). Unfortunately it seems that one cannot adapt the argument of Proposition 3.2 to study the strong Feller property of the following stochastic differential equation with multiplicative noise dXt = b(Xt ) dt + σ (Xt ) dZt , where σ : Rd → Rd ⊗ Rd and b : Rd → Rd are continuous. The reason is essentially due to that in this case the relation such like (3.10) between the semigroup of Xt and that of Zt is not available. Acknowledgments Financial support through the National Natural Science Foundation of China (No. 11201073), and the Program for New Century Excellent Talents in Universities of Fujian (No. JA12053) and the Program for Nonlinear Analysis and Its Applications (No. IRTL1206) is gratefully acknowledged. References Applebaum, D., 2004. Lévy Processes and Stochastic Calculus. Cambridge Univ. Press, Cambridge. Chen, Z.-Q., Kim, P., Song, R., 2012. Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation. Ann. Probab. 40, 2483–2538. Hartman, P., Wintner, A., 1942. On the infinitesimal generators of integral convolutions. Am. J. Math. 64, 273–298. Jacob, N., 2001. Pseudo differential operators and Markov processes. In: Fourier Analysis and Semigroups, vol. 1. Imperial College Press, London. Knopova, V., Schilling, R.L., 2013. A note on the existence of transition probability densities for Lévy processes. Forum Math. 25, 125–149. Kusuoka, S., Marinelli, C., 2013. On smoothing properties of transition semigroups associated to a class of SDEs with jumps. Ann. Inst. H. Poincaré Probab. Statist. see arXiv:1208.2860 (in press). Sato, K., 1999. Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge. Schilling, R.L., Schnurr, A., 2010. The symbol associated with the solution of a stochastic differential equation. Electr. J. Probab. 15, 1369–1393. Schilling, R.L., Sztonyk, P., Wang, J., 2012. Coupling property and gradient estimates of Lévy processes via the symbol. Bernoulli 18, 1128–1149. Schilling, R.L., Wang, J., 2012. Strong Feller continuity of Feller processes and semigroups. Infin. Dimens. Anal. Quandum Probab. 15, 1250010-1–125001028. Schnurr, A., 2013. Generalization of the Blumenthal–Getoor index to the class of homogeneous diffusions with jumps and some applications. Bernoulli 19, 2010–2032. Wang, J., 2010. Regularity of semigroups generated by Lévy type operators via coupling. Stoc. Proc. Appl. 120, 1680–1700. Wang, J., 2013a. Regularity for semigroups of Ornstein–Uhlenbeck processes. Positivity 17, 205–221.
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