On the martingale problem for pseudo-differential operators of variable order
Consider parabolic pseudo-differential operators L = ∂t − p(x,Dx) of variable order α(x) ≤ 2. The function α(x) is assumed to be smooth, but the symbol p(x, ξ) is not always differentiable with respect to x. We will show the uniqueness of Markov processes with the generator L. The essential point in...
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irk-123456789-45512009-12-07T12:00:31Z On the martingale problem for pseudo-differential operators of variable order Komatsu, T. Consider parabolic pseudo-differential operators L = ∂t − p(x,Dx) of variable order α(x) ≤ 2. The function α(x) is assumed to be smooth, but the symbol p(x, ξ) is not always differentiable with respect to x. We will show the uniqueness of Markov processes with the generator L. The essential point in our study is to obtain the Lp-estimate for resolvent operators associated with solutions to the martingale problem for L. We will show that, by making use of the theory of pseudo-differential operators and a generalized Calderon–Zygmund inequality for singular integrals. As a consequence of our study, the Markov process with the generator L is constructed and characterized. The Markov process may be called a stable-like process with perturbation. 2008 Article On the martingale problem for pseudo-differential operators of variable order / T. Komatsu // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 42–51. — Бібліогр.: 10 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4551 519.21 en Інститут математики НАН України |
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Consider parabolic pseudo-differential operators L = ∂t − p(x,Dx) of variable order α(x) ≤ 2. The function α(x) is assumed to be smooth, but the symbol p(x, ξ) is not always differentiable with respect to x. We will show the uniqueness of Markov processes with the generator L. The essential point in our study is to obtain the Lp-estimate for resolvent operators associated with solutions to the martingale problem for L. We will show that, by making use of the theory of pseudo-differential operators and a generalized Calderon–Zygmund inequality for singular integrals. As a consequence of our study, the Markov process with the generator L is constructed and characterized. The Markov process may be called a stable-like process with perturbation. |
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Article |
| author |
Komatsu, T. |
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Komatsu, T. On the martingale problem for pseudo-differential operators of variable order |
| author_facet |
Komatsu, T. |
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Komatsu, T. |
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On the martingale problem for pseudo-differential operators of variable order |
| title_short |
On the martingale problem for pseudo-differential operators of variable order |
| title_full |
On the martingale problem for pseudo-differential operators of variable order |
| title_fullStr |
On the martingale problem for pseudo-differential operators of variable order |
| title_full_unstemmed |
On the martingale problem for pseudo-differential operators of variable order |
| title_sort |
on the martingale problem for pseudo-differential operators of variable order |
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Інститут математики НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/4551 |
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On the martingale problem for pseudo-differential operators of variable order / T. Komatsu // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 42–51. — Бібліогр.: 10 назв.— англ. |
| work_keys_str_mv |
AT komatsut onthemartingaleproblemforpseudodifferentialoperatorsofvariableorder |
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2025-07-02T07:46:03Z |
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2025-07-02T07:46:03Z |
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1836520428413124608 |
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Theory of Stochastic Processes
Vol. 14 (30), no. 2, 2008, pp. 42–51
UDC 519.21
TAKASHI KOMATSU
ON THE MARTINGALE PROBLEM FOR
PSEUDO-DIFFERENTIAL OPERATORS OF VARIABLE ORDER
Consider parabolic pseudo-differential operators L = ∂t − p(x, Dx) of variable order
α(x) ≤ 2. The function α(x) is assumed to be smooth, but the symbol p(x, ξ) is
not always differentiable with respect to x. We will show the uniqueness of Markov
processes with the generator L. The essential point in our study is to obtain the
Lp-estimate for resolvent operators associated with solutions to the martingale prob-
lem for L. We will show that, by making use of the theory of pseudo-differential
operators and a generalized Calderon–Zygmund inequality for singular integrals. As
a consequence of our study, the Markov process with the generator L is constructed
and characterized. The Markov process may be called a stable-like process with
perturbation.
1. Introduction and Notation
Set Dx = −i∂x, where x = (xj) ∈ Rd and ∂ = ∂x = (∂/∂xj )). Then a symbol p(x, ξ) is
associated with the pseudo-differential operator p(x,Dx) by the relation p(x,Dx)eix·ξ =
eix·ξp(x, ξ). We consider a symbol
−p(x, ξ) ≡ ψ(x, α(x), ξ) + ϕ(x, ξ)
which is a negative definite function of ξ, where α, ψ, ϕ are functions satisfying the
following condition.
(1) Rd � x −→ α(x) ∈ (0, 2] : smooth,
(2) ψ(x, γ, λξ) = λγ ψ(x, γ, ξ) (λ > 0) ,
(3) (0, 2] � γ −→ ψ(x, γ, ξ) (|ξ| = 1) : smooth,
(4) x −→ ∂ξ
νψ(x, γ, ξ) (|ν| ≤ d+ 1) : continuous and bounded,
(5) ∃ε > 0, ϕ(x, ξ) = o(|ξ|α(x)−ε) (|ξ| → ∞) .
Let W = D(R+ → Rd) be the càd-làg path space, and Xt(w) := w(t) for w = (w(t)) ∈
W . Set Wt =
⋂
ε>0 σ(Xs; s ≤ t + ε), W = σ(Xs; s < ∞). We consider a parabolic
pseudo-differential operator L = ∂t − p(x,Dx). A probability measure P on (W,W) is
called a solution to the martingale problem for the operator L if the process(
exp[ iXt · ξ +
∫ t
0
p(Xs, ξ) ds ]
)
is a martingale w.r.t. (Wt, P ) for any ξ ∈ Rd. It is usually expected that the process
(W, (Wt), P ;Xt) is a Markov process with the generator L.
Bass [1] and Negoro [10] studied on a Markov process with the generator −(−Δ)α(x)/2
of variable order 0 < α(x) < 2. The Markov processes associated with pseudo-differential
operators with smooth symbols were studied in several articles (Hoh [3], Jacob-Leopold
[4], Jacob [5], etc). There are two typical cases where the martingale problem for L is
well-posed.
2000 AMS Mathematics Subject Classification. Primary 60H20, 60J75.
Key words and phrases. Martingale problem, pseudo-differential operator, variable order.
42
MARTINGALE PROBLEM 43
Case 1: p(x, ξ) is a smooth symbol. Applying the theory of pseudo-differential opera-
tors, under the non-degenerate condition
sup {�e ψ(x, γ, ξ) | x ∈ Rd, 0 < γ ≤ 2, |ξ| = 1} < 0,
we can show the existence of the smooth transition function of the Markov process with
the generator L (Komatsu [8]).
Case 2: α(x) is a constant function. Using a generalized Hörmander inequality for
singular integrals, under the non-degenerate condition, it is proved that the existence
and the uniqueness of solutions to the martingale problem for the operator L hold good
(Komatsu [6], Komatsu [7]).
One of the key points of this article is the unusual but well-devised definition of the
pseudo-differential operator ψ(x, γ,Dx), where the analytic distribution λ −→ [rλ
+] is
used. Though the general notion of the analytic distribution can be found in [2], it
might be better to give here a short sketch of the analytic distribution.
Let D = C∞
0 (Rd) be a space of test functions on Rd, and let D′ denote the space
of distributions. Consider a distribution fλ = 〈fλ, · 〉 ∈ D′ with parameter λ ∈ Λ,
where Λ is an open domain in C. We say that fλ is an analytic distribution if the
function Λ � λ −→ 〈fλ, φ〉 is analytic for any φ ∈ D. Define derivatives (d/dλ)nfλ by
〈(d/dλ)nfλ, φ〉 = (d/dλ)n〈fλ, φ〉. From the sequential completeness of the space D′, we
have (d/dλ)nfλ ∈ D′, and the Taylor expansion
fλ+h =
∞∑
n=0
hn
n!
(
d
dλ
)nfλ
holds in the sense of D′. Then it is possible to consider the analytic continuation of the
analytic distribition in the following way. Let fλ (λ ∈ Λ) be an analytic distribution,
and Λ ⊂ Λ1 ⊂ C. Assume that the function Λ � λ −→ 〈fλ, φ〉 can be extended to the
analytic function Λ1 � λ −→ gλ(φ) for any φ ∈ D, and set 〈fλ, φ〉 := gλ(φ). Then the
distribution Λ1 � λ −→ fλ is an analytic extension of the distribution Λ � λ −→ fλ.
Let d = 1, and let [xλ
±] denote the analytic distribution defined for �e λ > −1
associated with the function xλ± on R1. The largest extension of the analytic distribution
[xλ±] is the analytic distribution defined on Λ = {λ ∈ C | λ �= −1,−2, · · · }. If −n− 1 <
�e λ < −n, the equality
〈[xλ
±], φ(x)〉 =
∫ ∞
0
xλ
(
φ(±x) −
n−1∑
k=0
(±x)k
k!
φ(k)(0)
)
dx
holds for any φ ∈ D. Note that the analytic distribution λ −→ [xλ
±] has poles of order 1
at negative integers, but its modification λ −→ [xλ±]/Γ(λ + 1) is an entire distribution.
We see that
lim
λ→−n
〈
[xλ
±]
Γ(λ+ 1)
, φ(x)〉 = (∓)n−1φ(n−1)(0) = 〈δ(n−1)(x), φ(x)〉.
On the other hand, the analytic distribution λ −→ [(x± i0)λ] is defined by
[(x± i0)λ] =
{ [xλ
+] + e±iπλ[xλ
−] (−λ /∈ N),
[x−n] ± iπ(−1)n/(n− 1)! × δ(n−1)(x) (−λ = n ∈ N),
44 TAKASHI KOMATSU
where the distribution [x−n] (n ∈ N) is defined by the formula
〈[x−n], φ(x)〉
=
⎧⎨⎩
∫∞
0
x−2m
(
φ(x) + φ(−x) − 2
∑m−1
k=0
x2k
(2k)!φ
(2k)(0)
)
dx, n = 2m,∫∞
0
x−2m−1
(
φ(x) − φ(−x) − 2
∑m−1
k=0
x2k+1
(2k+1)!φ
(2k+1)(0)
)
dx, n = 2m+ 1.
We have the equality
F
[
x−γ−1
±
Γ(−γ)
]
(ξ) = exp[± iπγ
2
] (ξ ∓ i0)γ (ξ ∈ R1)
where F [ · ](ξ) denotes the Fourier transform in the distribution sense. Since the Fourier
transform of an entire distribution is also an entire distribution, γ −→ (ξ ∓ i0)γ is an
entire distribution.
Consider the general case d ≥ 1. Let σ(dω) be the area element on Sd−1. It is natural
to define the analytic distribition λ −→ [|x|λ] associated with the function |x|λ on Rd by
〈[|x|λ], φ(x)〉 = 〈[rλ+d−1
+ ],
∫
|ω|=1
φ(rω) σ(dω)〉 (λ+ d �= 0,−2,−4, · · · ),
because these equalities hold in the usual sense for �e λ > −d. This suggests a natural
way to define the pseudo-differential operator ψ(x, γ,Dx). Consider a function m(x, γ, ω)
on Rd × (0, 2] × Sd−1 such that
(1) ∀(x, γ), m(x, γ, ω) ∈ Cd(Sd−1),
(2) (0, 2] � γ −→ m(x, γ, ω) ≥ 0 : smooth,
(3) m(x, 2, ω) = 0,
∫
|ω|=1
ω m(x, 1, ω) σ(dω) = 0.
We define a pseudo-differential operator ψ(x, γ,Dx) of order 0 < γ ≤ 2 by
ψ(x, γ,Dx)f(x) = 〈[r−γ−1
+ ],
∫
|ω|=1
f(x+ rω) m(x, γ, ω)σ(dω)〉.
Note that the analytic distribution γ −→ [r−γ−1
+ ] has poles of order 1 at non-negative
integers. For γ = 1, 2, we have
ψ(x, 1, Dx)f(x) =
∫
(f(x+ y) − f(x) − I{|y|≤1} y · ∂f(x)) m(x, 1, y/|y|)|y|−d−1 dy
−
(∫
|ω|=1
ω [∂γm(x, γ, ω)]γ=1 σ(dω)
)
· ∂f(x),
ψ(x, 2, Dx)f(x) =
1
2
tr[
(∫
|ω|=1
ωω∗ [∂γm(x, γ, ω)]γ=2 σ(dω)
)
(∂∂∗f(x)) ].
Consider an operator ϕ(x,Dx) defined by
ϕ(x,Dx)f(x) =
∫
[f(x+ y) − f(x) − I{α(x)>1+ε,|y|≤1} y · ∂f(x)]N(x, dy)
+I{α(x)>1+ε} b(x) · ∂f(x).
We assume that the following condition is satisfied:
sup
x
|b(x)| +
∫
sup
x
(1 ∧ |y|α(x)−ε)|N(x, dy)| <∞.
Then we see that ϕ(x, ξ) = o(|ξ|α(x)−ε). It is not necessary that b(·) and N(·, dy) be
continuous.
MARTINGALE PROBLEM 45
Theorem. Under the non-degenerate condition that ψ(x, γ, ξ) < 0 for (x, γ, ξ) ∈ Rd ×
(0, 2] × Sd−1, the martingale problem for the operator
L = ∂t + ψ(x, α(x), Dx) + ϕ(x,Dx)
is well-posed, that is, the existence and the uniqueness of solutions holds good.
2. Estimates for Fundamental Solutions
One of the bases of our reasoning is the theory of pseudo-differential operators. For a
bdd function ζ(x) and 0 < δ < 1, we define
|∼p|ζk := sup
|β+γ|≤k
sup
x,ξ
{|∂β
ξ D
γ
x
∼
p|〈ξ〉|β|−ζ(x)−δ|γ|},
Sζ
1,δ = {∼p(x, ξ) ∈ C∞(R2d) | |∼p|ζk <∞ (∀k)},
where 〈ξ〉 =
√
2 + |ξ|2. Each pseudo-differential operator in the class
Pζ
1,δ = {∼p(x,Dx) | ∼p(x, ξ) ∈ Sζ
1,δ}
is called an operator of variable order ζ(x). If pj(x, ξ) ∈ Sζj
1,δ (j = 1, 2), the symbol
(p1 ◦p2)(x, ξ) of the iterated operator p1(x,Dx)p2(x,Dx) belongs to the class Sζ1+ζ2
1,δ , and
the asymptotic expansion formula
p1 ◦ p2 −
∑
|�|<N
1
�!
∂�
ξp1D
�
xp2 ∈ Sζ1+ζ2−N(1−δ)
holds for any N ∈ Z+ (see Kumano-go [9]).
Let ρ(r) be a smooth function on R+ such that ρ(r) = 1 for r ≤ 1, ρ(r) = 0 for r ≥ 2
and 0 < ρ(r) < 1 for 1 < r < 2. We fix a point x0 ∈ Rd and set
q(x, ξ) = − (ψ(x0, α(x), ·) ∗ ρ̂)(ξ) + (ψ(x0, α(x), ·) ∗ ρ̂)(0),
where ρ̂(ξ) := F−1[ρ(| · |)](ξ). Note that we consider the symbol not ψ(x0, α(x0), ξ)
but ψ(x0, α(x), ξ). The symbol q(x, ξ) belongs to the class Sα
1,δ. Let u(s, x,Dx) be the
fundamental solution to the Cauchy problem for the operator ∂s + q(x,Dx).
We now survey how to construct the fundamental solution. Set q ≡ q(x, ξ) and u0(s) ≡
u0(s, x, ξ) = exp(−sq). We may assume that there exists a constant c > 0 such that
|u0(s, x, ξ)| ≤ exp(−cs〈ξ〉α(x)).
Define symbols {uj(s)}j≥1 by uj(0) = 0 and
−(∂s + q)uj(s) =
∑
|�|+k=j,|�|
=0
1
�!
∂�
ξq D
�
xuk(s).
The following estimates hold good (see [8], Lemma 3).
Proposition 1. Fix 0 < δ < 1. There exist constants Cβγj such that∣∣∣∣∣∂
β
ξ D
γ
xuj(s, x, ξ)
u0(s, x, ξ)
∣∣∣∣∣ ≤ Cβγj 〈ξ〉−|β|+δ|γ|−j(1−δ)
|β|+|γ|+2j∑
k=1
(s〈ξ〉α(x))k.
For sufficiently large N , we define the symbol
∼
uN (s) ≡ ∼
uN (s, x, ξ) :=
N−1∑
j=0
uj(s).
46 TAKASHI KOMATSU
From the asymptotic expansion formula, we have
∼
rN (s) := −(∂s + q) ◦ ∼
uN (s) ∈ Sα(·)−(1−δ)N
1,δ .
The symbol u(s) = u(s, x, ξ) of the fundamental solution can be constructed by
w0(t) := δ(t), wj(s) =
∫ s
0
∼
rN (τ) ◦ wj−1(s− τ)dτ (j ≥ 1),
u(s, x, ξ) :=
∼
uN (s, x, ξ) +
∫ s
0
∼
uN (τ) ◦ (
∞∑
j=1
wj(s− τ)) dτ.
Hereafter, we assume that infx α(x) > 0. We define the resolvent operator Gλ (λ > 0) by
Gλf(x) =
∫ ∞
0
e−λsu(s, x,Dx)f(x) ds.
We use the convention of letting c.’s to stand for positive absolute constants. Each c. may
denote a constant different from other c.’ s. From the next proposition and the Young
inequality, we have the estimate
λ‖Gλf‖Lp ≤ c. ‖f‖Lp.
Proposition 2. For any β ∈ Rd, there exists a constant cβ such that
|F−1[∂β
ξ u(s, z, ξ)](y)| ≤ cβ (s > 0, y, z ∈ Rd),
and there is a constant C such that∫ ∞
0
e−sλ‖ sup
z
|F−1[u(s, z, ξ)](·)| ‖L1 ds ≤ C
1
λ
.
Proof. From Proposition 1, for 0 < s < 1,
|F−1[∂β
ξ uj(s, z, ξ)](y)| = |F−1[∂β
η uj(s, z, η)|η=s−1/αξ](s1/αy)|
≤ c.
|β|+2j∑
k=1
∫
(〈s−1/αξ〉α)k e−sq(z,s−1/αξ) dξ
≤ c.+ c.
∫
|ξ|>1
e−c.|ξ|α dξ ≤ c. ,
where α = α(z). It is much more easy to show that
sup
s≥1,y,z
|F−1[∂β
ξ uj(s, z, ξ)](y)| < ∞.
These prove the first claim. The second claim is proved by the inequality
〈y〉d+1|F−1[u(s, z, ·)](y)| ≤ c.
∑
|β|≤d+1
|F−1[∂β
ξ u(s, z, ξ)](y)|.
Similarly to the above proof, we can prove that there exist constants c. such that
〈y〉d|F−1[u(s, z, ·)](s1/α(z)y)| ≤ c. s−d/α(z),
〈y〉d+1|F−1[ξju(s, z, ξ)](s1/α(z)y)| ≤ c. s−(d+1)/α(z).
Moreover, the following proposition can be proved with the use of Proposition 1 (see [7],
Lemma 2.3).
MARTINGALE PROBLEM 47
Proposition 3. Let 0 < η < γ ∧ 1. There is a constant Cηγ such that
〈y〉d+η |F−1[φ(·)u(s, z, ·)](s1/α(z)y)| ≤ Cηγ
⎛⎝ sup
|ξ|=1
∑
|β|≤d+1
|∂β
ξ φ(ξ)|
⎞⎠ s−(d+γ)/α(z)
for any homogeneous function φ(ξ) with index γ.
From the above-presented estimates and the Hölder inequality, we obtain the following
proposition.
Proposition 4. Let α0 = infx α(x) > 0.
(1) If pα0 > d, then ‖Gλf‖∞ ≤ c. λ−1+d/pα0 ‖f‖Lp,
(2) If p(α0 − 1) > d, then‖DxGλf‖∞ ≤ c. λ−1+(1+d/p)/α0 ‖f‖Lp ,
(3) If 0 < η < α0 ∧ 1 and (α0 − η)p > d, then
‖ |Dx|ηGλf ‖∞ ≤ c. λ−1+(η+d/p)/α0 ‖f‖Lp.
3. Estimates for Singular Integrals
Though the order function α(x) is smooth, the symbols ψ(x, α(x), ξ) and p(x, ξ) are not
smooth. Then we need the theory of singular integrals, as well as the theory of pseudo-
differential operators, on which we will base the analysis for the operator p(x,Dx). Let
φ(ξ) be a homogeneous function with index 0, and let μ(φ) be the average of φ(·) over
Sd−1. Then kφ(x) := F−1[φ](x) − μ(φ)δ(x) is a homogeneous function with index −d.
Define the singular integral operator [f −→ kφ ∗ f ] by
(kφ ∗ f)(x) = lim
η↓0
∫
|y|>η
kφ(y)f(x− y) dy.
Then we have φ(Dx)f(x) = (kφ∗f)(x)+μ(φ)f(x). The estimate in the following theorem
(Komatsu [7], Theorem 2.1) is a key in this theory.
Lemma 1 (generalized Hörmander inequality).
‖ sup
z
|φz(Dx)f | ‖Lp ≤ Cp
⎛⎝ sup
z,|ξ|=1
∑
|β|≤d
|∂β
ξ φz(ξ)|
⎞⎠ ‖f‖Lp
for any system {φz(ξ)} of homogeneous functions with index 0.
Define a pseudo-differential operator H by
Hf(x) = h(x,Dx)f(x) = ψ(x0, α(x), Dx)f(x) + q(x,Dx)f(x).
We have
h(x, ξ) = 〈[r−α(x)−1
+ ](1−ρ(r)),
∫
|ω|=1
eirξ·ω m(x0, α(x), ω)σ(dω)〉+(ψ(x0 , α(x), ·)∗ ρ̂)(0).
We see that the symbol (1 − ρ(|ξ|))(h(x, ξ) − h(x, 0)) belongs to the class Sα(x)−1
1,δ .
Proposition 5. Let {ψz(γ, ξ)} be a system of functions on (0, 2] × Rd such that
(1) ψz(γ, λξ) = λγ ψz(γ, ξ) (λ > 0),
(2) (0, 2] × Sd−1 � (γ, ξ) −→ ψz(γ, ξ) is a smooth mapping.
Then there exists a constant Cp such that
‖ sup
z
|(ψz(α(x), Dx)Gλf)(x)| ‖Lp ≤ Cp
⎛⎝ sup
z,γ,|ξ|=1
∑
|β|≤d+1
|∂β
ξ ψz(γ, ξ)|
⎞⎠ ‖f‖Lp.
48 TAKASHI KOMATSU
Proof. Set φ̃z(ξ) := ψz(γ, ξ)/ψ(x0, γ, ξ) which is independent of γ, and ρ̃0(ξ) :=
ρ(|ξ|), ρ̃1(ξ) := 1 − ρ(|ξ|). Let gλ(x, ξ) be the symbol of a pseudo-differential opera-
tor Gλ:
gλ(x, ξ) =
∫ ∞
0
e−λsu(s, x, ξ) ds.
Then we have
ψz(α(x), ξ) ◦ gλ(x, ξ)
= (ψz(α(x), ξ)ρ̃0(ξ)) ◦ gλ(x, ξ) + (ψz(α(x), ξ)ρ̃1(ξ)) ◦ gλ(x, ξ)
= (ψz(α(x), ξ)ρ̃0(ξ)) ◦ gλ(x, ξ) + (φ̃z(ξ)ρ̃1(ξ)ψ(x0, α(x), ξ)) ◦ gλ(x, ξ)
= (ψz ρ̃0) ◦ gλ + (φ̃z ρ̃1h) ◦ gλ − (φ̃z ρ̃1q) ◦ gλ
= (ψz ρ̃0) ◦ gλ + (φ̃z ρ̃1h) ◦ gλ + [(φ̃z ρ̃1) ◦ q − (φ̃z ρ̃1)q] ◦ gλ − (φ̃z ρ̃1) ◦ q ◦ gλ
= (ψz ρ̃0) ◦ gλ + (φ̃z ρ̃1h) ◦ gλ + [(φ̃z ρ̃1) ◦ q − (φ̃z ρ̃1)q] ◦ gλ + (φ̃z ρ̃1) ◦ (λgλ − 1).
Set η = (infx α(x) ∧ 1)/2 and
C∗ = sup
z,γ,|ξ|=1
∑
|β|≤d+1
|∂β
ξ ψz(γ, ξ)|.
It can be proved that
sup
x,y,z
〈y〉d+η|F−1[ψz(α(x), ·)ρ̃0](y)| ≤ c. C∗.
We observe that the symbol p̃z(x, ξ) := [(ψzρ̃1) ◦ q − ψz ρ̃1q)](x, ξ) belongs also to the
class Sα(x)−1
1,δ . We have estimates
sup
y
〈y〉d+1|F−1[(ρ̃1h)(x, ξ) ◦ gλ(x, ξ)](y)| ≤ c.
(
1
λ
)(1/α(x))∧1
,
sup
y,z
〈y〉d+1|F−1[(p̃z(x, ξ) ◦ gλ(x, ξ)](y)| ≤ c. C∗
(
1
λ
)(1/α(x))∧1
.
It may not be a routine work to show these estimates, but these can be proved in a
similar way to the proof of Proposition 2. Since
ψz(α(x), Dx)Gλf(x) = ψz(α(x), Dx)ρ̃0(Dx)Gλf(x) + φ̃z(x,Dx)Gλf(x)
+ φ̃z(Dx)ρ̃1(Dx)((h(x,Dx) + λ)Gλf(x) − f(x)),
from the generalized Hörmander inequality and the Young inequality, the proof is com-
pleted.
Define the operators
Uλ = (q(x,Dx) − p(x,Dx))Gλ
= (ψ(x, α(x), Dx) − ψ(x0, α(x), Dx) + h(x,Dx) + ϕ(x,Dx))Gλ.
Since ϕ(x, ξ) = o(|ξ|α(x)−ε), it can be proved that
‖ϕ(x,Dx)Gλ‖Lp −→ 0
as λ→ ∞ (see [6], Theorem 2). From Proposition 5, we see that Uλ is a bounded operator
on Lp. Here, we assume that the value
‖α(·) − α(x0)‖∞ + sup
γ,|ξ|=1
∑
|β|≤d+1
‖∂β
ξ ψ(·, γ, ξ) − ∂β
ξ ψ(x0, γ, ξ)‖∞
MARTINGALE PROBLEM 49
is sufficiently small. Then there exists λ0 such that ‖Uλ‖Lp < 1 for λ ≥ λ0. Let p >
d/(infx α(x)), and let us define bounded operators on Lp by
Rλ = Gλ[I − Uλ]−1 (λ > λ0).
We see from Proposition 4 that Rλ is a bounded operator from Lp to C(Rd)
⋂
Lp in
both Lp and L∞ norms. We have
(λ+ p(x,Dx))Rλf = f (f ∈ Lp),
and if f ∈ [I − Uλ]C∞
0 (Rd), then Rλf ∈ Gλ C
∞
0 (Rd) ⊂ C∞(Rd).
4. Lp−Estimate and Proof of the Theorem
The proof of the uniqueness of solutions to the martingale problem is based on the
following lemma (see [6], Lemma 3.1).
Lemma 2. Let P 1 and P 2 be two probability measures on (W,W) with P 1[X0 ∈ dx] =
P 2[X0 ∈ dx]. Let E�[ · | Ws] denote the conditional expectation by P �. The property
∀s ≥ 0, ∀λ ≥ λ0, ∀f ∈ C(Rd)∩Lp, ∃g ∈ C(Rd),
E�
[∫ ∞
0
e−tλf(Xs+t)dt | Ws
]
= g(Xs) (� = 1, 2)
implies that P 1 = P 2 on W.
Let P be a solution to the martingale problem for L = ∂t−p(x,Dx). Then the process
Mλ
t := e−tλGλφ(Xt) +
∫ t
0
e−sλ(λ+ p(x,Dx))Gλφ(Xs) ds
is a martingale w.r.t. (Wt, P ) for any φ ∈ C∞
0 (Rd). This implies that
E
[∫ ∞
0
e−tλ[I − Uλ]φ(Xs+t)dt | Ws
]
= Gλφ(Xs),
for (λ+ p(x,Dx))Gλφ(x) = [I − Uλ]φ(x). Then the equality
E
[∫ ∞
0
e−tλf(Xs+t)dt | Ws
]
= Rλf(Xs)
is satisfied for any function f ∈ [I − Uλ]C∞
0 (Rd). It holds that, for sufficiently large
p, ‖Rλf‖∞ ≤ cλ ‖f‖Lp (λ ≥ λ0). Since the space [I − Uλ]C∞
0 (Rd) is dense in Lp, the
property in the above Lemma holds good if the “Lp−estimate ”∣∣∣∣E [∫ ∞
0
e−tλf(Xs+t)dt | Ws
]∣∣∣∣ ≤ cλ ‖f‖Lp
holds good.
To prove the “Lp-estimate”, we define a sequence of stable-like processes( ∼
W, (
∼
Wt),
∼
P ;
∼
Xn
t
)
with perturbations, whose laws approximate the law of the solution (Xt, P ) to the mar-
tingale problem for L. Let JX denote the counting measure of jumps of X :
JX(dt, dy) = #{τ | τ ∈ dt, 0 �= Xτ −Xτ− ∈ dy}.
Set γt = α(Xt) and
M(x, γ, dy) = m(x, γ, y/|y|)|y|−d−γ dy.
50 TAKASHI KOMATSU
We see that the measure JX(dt, dy) − (M(Xt, γt, dy) + N(Xt, dy))dt is a martingale
random measure w.r.t. (Wt, P ). Let a(x) be the Rd ⊗ Rd−valued continuous function
such that
ψ(x, 2, Dx)f(x) = (1/2) tr[aa∗(x) (∂∂∗f(x))],
that is,
a(x)a∗(x) =
∫
|ω|=1
ωω∗[∂γm(x, γ, ω)]γ=2 σ(dω).
Set
b0(x) := ψ(x, 1, Dx)x = −
∫
|ω|=1
ω[∂γm(x, γ, ω)]γ=1 σ(dω).
Then there exists a continuous martingale BX(t) such that
dXt = a(Xt) dBX(t) + (I(γt=1)b0(Xt) + I(γt>1+ε)b(Xt)) dt
+
∫
y [JX(dt, dy) − I(γt>1)M(Xt, γt, dy)dt− I(γt>1+ε, |y|≤1)N(Xt, dy)dt,
〈dBi
X(t), dBj
X(t)〉 = δij I(γt=2) dt.
Let Z = (Zt) be a Cauchy process which is independent of X = (Xt), and let JZ denote
the counting measure of jumps of Z. Set
π(n, t) = [nt]/n, ωy = y/|y|, Θn(y) = y I(|y|≤1/n), Θc
n(y) = y I(|y|>1/n).
Define processes (
∼
Xn
t ) by the formula
d
∼
Xn
t = a(Xπ(n,t)) dBX(t) + (I(γt=1)b0(Xπ(n,t)) + I(γt>1+ε)b(Xt)) dt
+
∫
Θc
n
([
m(Xπ(n,t), γt, ωy)
m(Xt, γt, ωy)
]1/γt
y
)
× [JX(dt, dy) − I(γt>1)M(Xt, γt, dy)dt− I(γt>1+ε, |y|≤1)N(Xt, dy)dt ]
+
∫
Θn( [m(Xπ(n,t), γt, ωy)|y| ]1/γt ωy ) [JZ(dt, dy) − I(|y|≤1)|y|−d−1dydt].
Since m(x, γ, ω), a(x), b(x) are continuous in x, it is a routine work to show that
∼
Xn
t −→
Xt in probability. We observe that, for g(x) ∈ C∞
0 (Rd), there exists a martingale (
∼
Mn
t [g])
such that
d g(
∼
Xn
t ) = d
∼
Mn
t [g] + (ψ(Xπ(n,t), γt, Dx)g)(
∼
Xn
t ) dt
+
∫
[g(
∼
Xn
t + Θc
n) − g(
∼
Xn
t ) − I(γt>1+ε,|y|≤1)Θc
n · ∂xg(
∼
Xn
t )]N(Xt, dy) dt
+ I(γt>1+ε)b(Xt) · ∂xg(
∼
Xn
t ) dt,
where Θc
n = Θc
n([m(Xπ(n,t), γt, ωy)/m(Xt, γt, ωy)]1/γty).
We have the following estimate (see [6], Lemma 1.1):
sup
γ,|ξ|=1
∑
|β|≤d+1
‖∂β
ξ (ψ(·, γ, ξ) − ψ(x0, γ, ξ))‖∞
≤ c. sup
γ,|ω|=1
[
∑
|β|≤d
‖∂β
ω(m(·, γ, ω) −m(x0, γ, ω))‖∞ + ‖∂γ(m(·, γ, ω) −m(x0, γ, ω))‖∞ ].
MARTINGALE PROBLEM 51
Under the assumption that the value of ‖α(·) − α(x0)‖∞ and the right-hand side of
the above inequality are sufficiently small, from estimates for the operator Gλ : Lp −→
C(Rd)
⋂
Lp, it can be proved that each process (
∼
Xn
t ,
∼
P ) admits the Lp−estimate∣∣∣∣∼E [∫ ∞
0
e−tλf(
∼
Xn
s+t)dt
∣∣ ∼
Ws
]∣∣∣∣ ≤ cn,λ ‖f‖Lp,
and that cλ := supn cn,λ < ∞ (see Komatsu [7], Lemma 4.5). Then we obtain the
Lp−estimate for the resolvent operators associated with solutions to the martingale
problem for L, which implies the uniqueness of solutions to the martingale problem.
The existence of solutions to the martingale problem for L can be proved under the same
assumption (see [7], Theorem 3.1).
In the general case, to prove the existence and uniqueness of solutions to the martingale
problem, we make use of “ the localization methods ”. Let ρr(x) = ρ(|x − x0|2/r2) and
define αr, ψr, Lr by
αr(x) = α(x0) + ρr(x)(α(x) − α(x0)),
ψr(x, γ, ξ) = ψ(x0, γ, ξ) + ρr(x)(ψ(x, γ, ξ) − ψ(x0, γ, ξ)),
Lr = ∂t + ψr(x, αr(x), Dx) + ρr(x)ϕ(x,Dx).
We see that, for sufficiently small r > 0, the existence and uniqueness of solutions to the
martingale problem for Lr holds good. Set
Tr = inf{t ≥ 0 | |Xt − x0| > r}.
Any local solution on [0, Tr] to the martingale problem for L can be extended to a solution
to the martingale problem for Lr. From the uniqueness of solutions to the martingale
problem for Lr, we see that the local solution on [0, Tr] to the martingale problem for L
is uniquely determined. Repeating such localization methods, we see that the solution
to the martingale problem for L exists and is uniquely determined.
Bibliography
1. R.F.Bass, Uniqueness in law for pure jump type Markov processes, Probab. Theory Related
Fields 79 (1988), 271-278.
2. I.M.Gel’fand, G.E.Shilov, Generalized Functions, I, Academic Press, New York, London, 1964.
3. W.Hoh, Pseudo differential operators with negative definite symbols of variable order, Rev.
Mat. Iberoamericana 16 (2000), 219-241.
4. N.Jacob, L.G.Leopold, Pseudo differential operators with variable order of differentiation gen-
erating Feller semigroups, Integral Equations and Operator Theory 17 (1993), 544-553.
5. N.Jacob, A class of Feller semigroups generated by pseudo differential operators, Math. Z. 215
(1994), 151-166.
6. T.Komatsu, On the martingale problem for generators of stable processes with perturbations,
Osaka J. Math. 21 (1984), 113-132.
7. T. Komatsu, Pseudo-differential operators and Markov processes, J. Math. Soc. Japan 36
(1984), 387-418.
8. T.Komatsu, On stable-like processes, Probability Theory and Mathematical Statistics, World
Scientific, Singapore, 1996, pp. 681-691.
9. H.Kumano-go, Pseudo-Differential Operators, The MIT Press, Cambridge, Massachusetts, Lon-
don, 1982.
10. A.Negoro, Stable-like processes; construction of the transition density and the behavior of sam-
ple paths near t = 0, Osaka J. Math. 31 (1994), 189-214.
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