Construction of time inhomogeneous Markov
processes via evolution equations using
pseudo-differential operators
Björn Böttcher
January, 2007
Abstract
For a pseudo-differential operator with symbol which is time and space
dependent, elliptic and continuous negative definite the corresponding evolution equation is solved. Further it is shown that the solution defines a
Markov process. In general this will be a time and space inhomogeneous
jump process.
To solve the evolution equation we combine a fixed-point method with
the symbolic calculus for negative definite symbols developed by Hoh. The
properties of the fundamental solution which ensure the existence of a corresponding Markov process are proved along the lines of Eidelman, Ivasyshen
and Kochubei. But instead of hyper-singular integral representations we use
the pseudo-differential operator representation together with the positive
maximum principle to get the required properties.
Key words: Pseudo-differential operator, generator, time inhomogeneous Markov
process, jump process, evolution equation, fundamental solution, parametrix.
MSC (2000): 60J35, 35S10, 47G30, 60J75, 35K90.
Motivation and introduction
For a time inhomogeneous Markov process Xt on Rn the corresponding evolution
system
U (s, t)f (x) := E(f (Xt )|Xs = x), s ≤ t, s, t ∈ R
is well defined on the continuous and bounded functions Cb on Rn . The linear
operators U (s, t) are positivity preserving and satisfy U (s, t)1 = 1, U (s, s) = id
and the evolution property U (s, t) = U (s, τ )U (τ, t) for s ≤ τ ≤ t.
1
Consider the corresponding family of left derivatives
d−
U (s − h, s)f − f
As f :=
U (s, t) := lim
for s > 0
h↓0
ds
h
which is defined for all f ∈ C∞ , i.e. f is continuous and lim|x|→∞ f (x) = 0, such
that the limit exists in a strong sense (i.e. w.r.t. k · k∞ ). In this case we write
f ∈ D(As ).
The evolution property leads to the evolution equation corresponding to the
process
d−
U (s, t) = −As U (s, t).
(0.1)
ds
Equation (0.1) is called backward equation since the initial value problem would
be solved on (−∞, t]. This equation is equivalent to the backward Kolmogorov
equation if the process has transition densities.
The operators As satisfy the positive maximum principle, i.e. for f ∈ D(As ) the
following implication holds: If there exists x0 ∈ Rn with f (x0 ) = supx∈Rn f (x) ≥ 0
then
As f (x)x=x0 ≤ 0.
This property of As is a consequence of its definition and the fact, that for
f ∈ D(As ) attaining its positive maximum at some point x0 the following inequality
holds:
U (s, t)f (x0 ) ≤ U (s, t)f + (x0 ) ≤ kf + k∞ = f (x0 ), where f + := f 1{f ≥0} .
Therefore, if the set C0∞ of arbitrarily often differentiable functions with compact support is a subset of D(As ), we know by Courrège [4] (see Jacob [8] section
4.5) that As on C0∞ is a pseudo-differential operator with continuous negative definite symbol, i.e. an operator of the form
Z
−n
2
eixξ q(s, x, ξ)fˆ(ξ) dξ,
As f (x) = −(2π)
Rn
n R
where fˆ(ξ) = (2π)− 2 Rn e−ixξ f (x) dx denotes the Fourier transform of f and
q(s, x, ·) is for fixed (s, x) a continuous negative definite function as defined in
Berg and Forst [1] (see our Definition 1.1).
Thus it is natural to ask, for which continuous negative definite symbols q(s, x, ·)
we can construct a Markov process by solving the evolution equation (0.1), where
As is a pseudo-differential operator with symbol q.
This question has been answered for a class of symbols which are homogeneous
of degree γ > 1 and have finite smoothness at 0 (see [5]). The approach is based
on hyper-singular integrals and the order up to which the decay at infinity of the
2
derivatives of the symbols must improve is 2n + 2[γ] + 1, where n is the space
dimension. But in general for a continuous negative definite function only an
improvement of the decay of the derivatives up to order 2 can be expected, see [7].
For a class of symbols which are arbitrarily often differentiable, the corresponding evolution equation was solved by Tsutsumi [12] and others (see also [11]), but
their approaches rely on estimates of the derivatives of the symbols with decay
improving up to an arbitrary order.
In [6, 7] Hoh developed a calculus for a class of arbitrarily often differentiable
negative definite symbols, for which the estimates of the decay of the derivatives
only improve up to order 2. For time independent symbols in Hoh’s symbol class
we constructed in [2, 3] the fundamental solution to the corresponding evolution
equation, where our construction was based on [11]. Now we extend this approach
to time dependent symbols and show that the constructed fundamental solution
gives rise to a Markov process. To prove this result it is necessary to use time
dependent symbols, since the usual way of transforming a time inhomogeneous
evolution to time homogeneous evolution leads to a degenerated symbol to which
our previous construction cannot be applied, compare Remark 2.1.
The paper is structured as follows: In the first section the symbol classes are
introduced and and the corresponding calculus is summarized. In section 2 some
general properties of the solution to the evolution equation are discussed. Afterwards in section 3 the solution to the evolution equation is constructed and in
section 4 we conclude that this fundamental solution defines a Markov process.
The final section just contains the proof of Lemma 3.3 which we have postponed
for readability.
1
The symbols and the corresponding calculus
The main definitions and results needed in the following are given in this section.
The presented symbolic calculus was developed by Hoh in [6, 7].
A central notion is that of a continuous negative definite function.
Definition 1.1. A function ψ : Rn → C is continuous negative definite, if it
is continuous and if for any choice of k ∈ N and vectors ξ 1 , . . . , ξ k ∈ Rn the matrix
ψ(ξ j ) + ψ(ξ l ) − ψ(ξ j − ξ l )
j,l=1,...,k
is positive Hermitian.
An equivalent definition is that ψ is the characteristic exponent of a Lévy
process, i.e. it has a Lévy-Khinchin representation, compare section 3.7 in [8].
3
Some typical examples of continuous negative definite functions are: |ξ|α for α ∈
(0, 2], 1 − e−isξ for s ≥ 0 and log(1 + ξ 2 ) + i arctan ξ.
Whenever a derivative is used in the following definitions, the assumption of
existence of the derivative is also implicitly part of the definition.
Definition 1.2. A continuous negative definite function ψ : Rn → R belongs to
the class Λ if for all α ∈ Nn0 there exist constants cα ≥ 0 such that
|∂ξα (1 + ψ(ξ))| ≤ cα (1 + ψ(ξ))
2−(|α|∧2)
2
where |α| ∧ 2 := min(|α|, 2).
Now the symbol classes can be defined.
Definition 1.3. Let m ∈ R, j ∈ {0, 1, 2} and ψ ∈ Λ. A function q : R×Rn ×Rn →
C is in the class Sjψ,m if for all α, β ∈ Nn0 and for any compact K ⊂ R there are
constants cα,β,K ≥ 0 such that
|∂ξα ∂xβ q(t, x, ξ)| ≤ cα,β,K (1 + ψ(ξ))
m−(|α|∧j)
2
(1.1)
holds for all t ∈ K, x ∈ Rn and ξ ∈ Rn . m ∈ R ist called the order of the symbol.
Sometimes we denote such a symbol by q(t, x, ξ) ∈ Sjψ,m , which is a slight abuse of
notation.
For a symbol q depending on two parameters s, t ∈ R we also write q ∈ Sjψ,m if
(1.1) holds for all s, t ∈ K. Furthermore the notation q ∈ (t − s)Sjψ,m is used if
for s ≤ t
m−(|α|∧j)
|∂ξα ∂xβ q(s, t, x, ξ)| ≤ (t − s)cα,β (1 + ψ(ξ)) 2
holds, where the constants cα,β are independent of s and t.
The difference between the classical symbol classes in the theory of pseudodifferential operators and the symbol classes used by Hoh, is that the estimates for
the decay of the derivatives of the symbol only improve up to order 2. Furthermore
in the classical setting the derivatives
p are compared to powers of |.|, while in the
above setting they are compared to ψ(.) for ψ ∈ Λ.
For a given symbol the corresponding pseudo-differential operator can now be
defined.
Definition 1.4. Let m ∈ R, j ∈ {0, 1, 2} and ψ ∈ Λ. For q ∈ Sjψ,m the pseudodifferential operator q(t, x, D) is defined on the Schwartz space S(Rn ) by
Z
−n
eixξ q(t, x, ξ)fˆ(ξ) dξ.
q(t, x, D)f (x) := (2π) 2
Rn
4
The symbol of a pseudo-differential operator q(t, x, D) is formally denoted by
σ(q(t, x, D))(x, ξ) but if it is clear by the context just q(t, x, ξ) or q is used.
The main theorem for the symbolic calculus is the following.
Theorem 1.5. Let ψ ∈ Λ.
A) If q1 ∈ S0ψ,m1 and q2 ∈ S0ψ,m2 , then the composition of the corresponding
operators is a pseudo-differential operator and
σ (q1 (t, x, D) ◦ q2 (t, x, D)) ∈ S0ψ,m1 +m2 .
B) For q1 ∈ S2ψ,m1 and q2 ∈ S0ψ,m2 the symbol q of the operator q(t, x, D) :=
q1 (t, x, D) ◦ q2 (t, x, D) is given by
q(t, x, ξ) = q1 (t, x, ξ) · q2 (t, x, ξ) +
n
X
∂ξj q1 (t, x, ξ)Dxj q2 (t, x, ξ) + rq1 q2 (t, x, ξ)
j=1
with rq1 q2 ∈
S0ψ,m1 +m2 −2 ,
where Dxj denotes −i∂xj .
C) For q ∈ S0ψ,0 , K ⊂ R compact and k ∈ N there exist constants Cα,β,K > 0
(independent of k) such that for all t ∈ K and k ∈ N
1
|∂ξα ∂xβ σ q(t, x, D)k (x, ξ)| k ≤ Cα,β,K ,
where q(t, x, D)k denotes the k-fold composition of the operator q(t, x, D).
Remark 1.6. i) For a proof of part A) and B) of the theorem see Corollary 3.5
and 3.11 in [7] and Remark 1.6 in [3], where the time dependence does not
affect the result. Part C) can be proved analogous to chapter 7 section 2 in
[11].
P
ii) In B) it is also known that nj=1 ∂ξj q1 (t, x, ξ)Dxj q2 (t, x, ξ) ∈ S1ψ,m1 +m2 −1 . Furthermore the remainder satisfies rq1 q2 ∈ (t − s)S0ψ,m1 +m3 −2 , if q2 depends on
parameters t, s and additionally satisfies (m3 ≥ m2 ):
|∂ξα ∂xβ q2 (s, t, x, ξ)| ≤ cα,β (t − s)(1 + ψ(ξ))
m3 −(|α|∧2)
2
for |α + β| ≥ 1.
These pseudo-differential operators have nice mapping properties on the following family of anisotropic Sobolev spaces (for details see chapter 4 in [6]).
Definition 1.7. Let r ∈ R and ψ : Rn → R be a continuous negative definite
function. The space H ψ,r is defined by
H ψ,r (Rn ) := {f ∈ S 0 (Rn ); kf kψ,r < ∞}
where the norm is
kf kψ,r := k(1 + ψ(D))r/2 f (.)k0 ,
k.k0 denotes the L2 norm and S 0 (Rn ) is the space of tempered distributions.
5
By section 3.10 in [8] we know C0∞ is a dense subset of H ψ,r for all r ∈ R and
all continuous negative definite ψ. Furthermore if there exist c0 > 0, r0 > 0 such
that
(1.2)
ψ(ξ) ≥ c0 |ξ|r0 for all ξ large
and r is chosen such that
n
(1.3)
r0
then, by Sobolev’s embedding theorem, H ψ,r ⊂ C∞ and there exists c > 0 such
that for all f ∈ H ψ,r :
kf k∞ ≤ ckf kψ,r .
(1.4)
r>
Theorem 1.8. Let ψ ∈ Λ, m ∈ R and q ∈ S0ψ,m . Then for all r ∈ R the corresponding pseudo-differential operator q(t, x, D) satisfies
kq(t, x, D)f kψ,r ≤ ckf kψ,m+r .
Note that for ψ ∈ Λ, m ∈ R and q ∈ (t − s)S0ψ,m analogously
kq(s, t, x, D)f kψ,r ≤ c · (t − s)kf kψ,m+r
(1.5)
holds.
The following assumptions on the time dependent symbol q will be used throughout the paper.
q(., x, ξ) is a continuous function for all x ∈ Rn , ξ ∈ Rn .
(1.6)
q(t, x, .) is continuous negative definite for all t ∈ R, x ∈ Rn .
(1.7)
lim sup |q(t, x, ξ)| = 0 holds uniformly in t on compact sets.
(1.8)
ξ→0 x∈Rn
q ∈ Sjψ,m is elliptic, i.e. uniformly in t on compact sets holds:
m
∃R, c > 0 ∀x, |ξ| ≥ R : Re q(t, x, ξ) ≥ c(1 + ψ(ξ)) 2 .
(1.9)
If (1.2) and (1.3) are satisfied and q(t, x, D) has the symbol q ∈ S0ψ,m satisfing
(1.7) then the operator −q(t, x, D) satisfies the positive maximum principle on
H ψ,r+m . I.e. for real-valued f ∈ H ψ,r+m the following implication holds: If there
exists x0 ∈ Rn with f (x0 ) = supx∈Rn f (x) ≥ 0 then
−q(t, x, D)f (x)x=x0 ≤ 0.
This is a consequence of Theorem 1.8, Theorem 4.5.6 in [8] and Theorem 2.6.1 in
[9]. Note that q(t, x, D) always maps real-valued functions to real-valued functions,
if the symbol satisfies (1.7).
We end this section with a simple lemma which will be used later.
6
Lemma 1.9. Let ψ satisfy (1.2), r satisfy (1.3) and p ∈ S0ψ,0 . Suppose the corresponding pseudo-differential operator is positivity preserving, i.e. p(t, x, D)f ≥ 0
for all (real-valued) positive f ∈ H ψ,r . Then for any compact K ⊂ R there exists a
c > 0 such that
kp(t, x, D)f k∞ ≤ ckf k∞
for all real-valued f ∈ H ψ,r and t ∈ K.
Proof. Note that |p(t, x, ξ)| < c0,0 uniformly in x and ξ, since p ∈ S0ψ,0 . Let k ∈ N
x2
(ξk)2
and ϕk (x) := e− 2k2 . Thus ϕk ∈ S, ϕ̂k (ξ) = k n e− 2 and
Z
ixξ
−n
e p(t, x, ξ)ϕ̂k (ξ) dξ 0 ≤ p(t, x, D)ϕk (x) = (2π) 2 n
ZR
(ξk)2
n
≤ (2π)− 2
c0,0 k n e− 2 dξ = c0,0 .
Rn
±g
≤ 2ϕk . Now the above calculation
Let g ∈ C0∞ . Then there exists a k such that kgk
∞
together with the positivity preserving property implies
p(t, x, D)
±g
≤ 2p(t, x, D)ϕk ≤ 2c0,0 .
kgk∞
Thus by linearity
kp(t, x, D)gk∞ ≤ ckgk∞
holds. It remains to extend the inequality form C0∞ to H ψ,r . For f ∈ H ψ,r there
exists a sequence gk ∈ C0∞ with gk → f in H ψ,r . Thus with Theorem 1.8 and (1.4)
kp(t, x, D)f k∞ ≤ kp(t, x, D)(f − gk )k∞ + kp(t, x, D)gk k∞
≤ c̃kf − gk kψ,r + ckgk k∞
≤ c̃kf − gk kψ,r + ckgk − f k∞ + ckf k∞
implies the result.
2
(1.10)
The evolution equation
Let ψ ∈ Λ, m ∈ R and q(t, x, D) be a pseudo-differential operator with symbol
q ∈ S2ψ,m . Assume for this section (1.7) and (1.8).
A solution to the initial value problem of the backward equation is a function
u(s, x), such that for fixed t ∈ R
d
− q(s, x, D) u(s, x) = h(s, x) for s ∈ (−∞, t]
ds
(2.1)
u(t, x) = ϕ(x),
7
where ϕ and h are in suitable spaces.
An operator U (s, t), s < t is called fundamental solution to the backward
equation if
d
U (s, t) − q(s, x, D)U (s, t) = 0 and lim U (s, t) = id.
s→t
ds
(2.2)
Note that this is a short notation for
d
(U (s, t)u(x)) − q(s, x, D)U (s, t)u(x) = 0 and lim U (s, t)u(x) = u(x)
s→t
ds
(2.3)
for all u from a suitable domain and for all x. The domain of the operator will be
discussed later.
Remark 2.1. It is well known that the time inhomogeneous initial value problem
d
− q(s, x, D) u(s, x) = 0, u(0, x) = ϕ(x)
ds
can be transformed into a time homogeneous problem by treating time as a further
dimension of the state space. Now one would like to apply the results of [3] to this
case. But the operator of the new time homogeneous problem has the symbol
a(r, x, τ, ξ) = −iτ − q(r, x, ξ),
where (r, x) is an element of the new state space with covariable (τ, ξ).
This symbol is not elliptic in the sense of [3] and thus the method we developed
therein in is not applicable.
First the positivity of certain solutions to the initial value problem is shown.
Lemma 2.2. Let u be a solution to (2.1) with h ≡ 0 and (real-valued) initial value
|x|→∞
ϕ(x) ≥ 0. If u is jointly continuous and u(s, x) −−−−→ 0 for all s ∈ (−∞, t], then
u is real-valued, unique and
u(s, x) ≥ 0.
Proof. The proof is partly modeled after Lemma 4.7 in [5]. First suppose u is
real. Fix S < t and select χ(·) ∈ C0∞ such that 0 ≤ χ ≤ 1, χ(x) = 1 for |x| < 1
and χ(x) = 0 for |x| > 2. Now suppose
α :=
inf
(s,x)∈[S,t]×Rn
8
u(s, x) < 0.
|x|→∞
Note that the infimum is actually a minimum, since ϕ ≥ 0 and u(s, x) −−−−→ 0.
Then define
t − s |α| x
vk (s, x) := u(s, x) +
χ( ).
t−S 2
k
Clearly
α
inf n vk (s, x) ≤
(s,x)∈[S,t]×R
2
holds and this infimum is again a minimum attained at some point (s0 , x0 ), since
|x|→∞
vk (s, x) −−−−→ 0 uniformly in s ∈ [S, t] for |x| large. Furthermore for k large
d
u(s, x) exists since u is
enough the minimum is also independent of k. Moreover ds
a solution to (2.1). Thus the property of the minimum and the fact that s0 ∈ [S, t)
imply
d
vk (s, x)(s,x)=(s0 ,x0 ) ≥ 0.
ds
Therefore on the one hand
d
− q(s, x, D) vk (s, x)(s,x)=(s0 ,x0 ) ≥ −q(s0 , x0 , D)vk (s0 , x0 ) ≥ 0
ds
holds, where the last inequality is due to the positive maximum principle. On the
other hand for k large
|α|
x
|α|(t − s)
x
d
− q(s, x, D) vk (s, x) = −
χ( ) −
q(s, x, D)χ( ) < 0
ds
2(t − S) k
2(t − S)
k
·
[
holds, since u is a solution of (2.1) and χ(
)(ξ) = k n χ̂(ξk) with (1.8) implies
k
Z
η
x
η
−n
|q(s, x, D)χ( )| =(2π) 2
eix k q(s, x, )χ̂(η) dη
k
k
Z
(2.4)
−→ q(s, x, 0)χ̂(η) dη = 0, as k −→ ∞.
But this is a contradiction to the previous inequality.
|x|→∞
To show the uniqueness suppose v is another real solution such that v(s, x) −−−−→
0. Then u(s, x) ≥ 0 by the previous considerations and
|x|→∞
v(s, x) − u(s, x) −−−−→ 0 and v(0, x) − u(0, x) = 0.
Thus v − u and u − v are both positive real solutions and therefore u = v.
Now suppose u is a complex valued solution to (2.1) with real initial value. Then
by linearity Im u is the solution to (2.1) with initial value 0. Therefore Im u ≡ 0. 9
Theorem 2.3. Let U = U (s, t) be the fundamental solution as in (2.2). If U
is a pseudo-differential operator with symbol in S0ψ,0 and ψ satisfying (1.2) and r
satisfying (1.3) then
i) U (s, t)f ≥ 0 for all (real-valued) positive f ∈ H ψ,r ,
ii) the initial value problem (2.1) with initial value ϕ ∈ H ψ,r for a fixed t ∈ R
and h ∈ H ψ,r+m has the unique solution
Z t
U (s, τ )h(τ, x) dτ.
(2.5)
u(s, x) := U (s, t)ϕ(x) −
s
Proof. i) The operator U maps H ψ,r into itself and this space is for r > rn0 a subset
of C∞ . Since U is a fundamental solution it is clear that U (s, t)f solves (2.1) with
initial value f . Thus the positivity preserving property follows by Lemma 2.2.
ii) Define u by (2.5). Then it satisfies
Z t
U (t, τ )h(τ, x) dτ = ϕ(x)
u(t, x) = U (t, t)ϕ(x) −
t
and
Z t
d
d
U (s, τ )h(τ, x) dτ
− q(s, x,D) u(s, x) = 0 −
− q(s, x, D)
ds
ds
s
Z t
d
= U (s, s)h(s, x) +
q(s, x, D) −
U (s, τ )h(τ, x) dτ
ds
s
= h(s, x),
since the operator is continuous on H ψ,r+m .
To see the uniqueness, note that u ∈ H ψ,r and suppose w ∈ H ψ,r is another
solution. Then w(s, x) − u(s, x) solve (2.1) with ϕ ≡ 0 and h ≡ 0. But now Lemma
2.2 implies w(s, x) − u(s, x) ≡ 0.
Theorem 2.4. Suppose the assumptions of Theorem 2.3 hold. Then U can be
extended to the constants such that
U 1 := sup U uk = 1
(2.6)
k
for uk ∈ H ψ,r , uk real-valued and uk ↑ 1, i.e. uk is point wise monotone increasing
to 1. Furthermore
kU f k∞ ≤ kf k∞
(2.7)
holds for real-valued f ∈ H ψ,r .
10
Proof. Let χ ∈ C0∞ such that χ(x) = 1 for |x| < 1, 0 ≤ χ(x) ≤ 1 for 1 ≤
|x| ≤ 2 and χ(x) = 0 for |x| > 2. Now, (2.1) with ϕ(x) := χ( xk ) and h(s, x) :=
−q(s, x, D)χ( xk ) is obviously solved by u(s, x) := χ( xk ). Therefore by Theorem 2.3
ii)
Z t
x
x
x
U (s, τ )q(τ, x, D)χ( ) dτ.
χ( ) = U (s, t)χ( ) +
k
k
k
s
Using Lemma 1.9
Z t
. U (s, τ )q(τ, x, D)χ( . ) dτ ≤ (t − s) U
(s,
τ
)q(τ,
x,
D)χ(
)
k
k ∞
s
∞
.
≤ (t − s) · c q(τ, x, D)χ( ) −→ 0
k ∞
holds as k → ∞, by the same argument as in (2.4). Thus it follows that
x
lim U (s, t)χ( ) = 1.
k→∞
k
For uk ∈ H ψ,r with uk ↑ 1 it holds that uk · χ( l. ) ∈ H ψ,r and thus
x
x
sup U (s, t)uk (x) = sup U (s, t)(uk (x) · χ( )) = sup U (s, t)χ( ) = 1.
l
l
k
l,k
l
Finally for each g ∈ C0∞ there exists a k such that
constant c in Lemma 1.9 is in this case equal to 1.
3
±g
kgk∞
≤ χ( xk ) and thus the
Construction of the fundamental solution
First the main theorem of this section and the ideas of the proof are presented,
then the precise construction is given.
To improve the readability the arguments (x, ξ) respectively (x, D) are omitted
when it is clear by the context that the symbol respectively the operator is meant.
Furthermore equations with operators will be written in short notation, as (2.2)
is the short notation
for (2.3). In particular differentiation and integration as in
Rt
d
d
U (s, t) and s VB (s, τ )B(τ, t) dτ are understood in the sense of ds
(U (s, t)u(x))
ds
Rt
and s VB (s, τ )B(τ, t)u(x) dτ for suitable u.
Theorem 3.1. Let ψ ∈ Λ and m ≤ 2. For q ∈ S2ψ,m satisfying (1.6) and (1.9) there
exists a fundamental solution U (s, t) to (2.2). This solution is a pseudo-differential
operator whose symbol satisfies
σ(U (s, t)) ∈ S0ψ,0
and
11
σ(
d
U (s, t)) ∈ S0ψ,m
ds
and
σ(U )(s, t, x, ξ) = e−
Rt
s
q(τ,x,ξ) dτ
+ r0 (s, t, x, ξ)
where
ψ,max{−1,m−2}
r 0 ∈ S0
∩ (t − s)S0ψ,m−1 .
This result generalizes the time homogeneous result from [3] to the time inhomogeneous case.
Remark 3.2. i) m ≤ 2 is no restriction, since a continuous negative definite
function grows at most as fast as a second order polynomial.
ii) If q ∈ S2ψ,m satisfies (1.9) there exists a λ ≥ 0 such that qλ := λ + q is also an
element of S2ψ,m which satisfies (1.9) (with the same constant) for all ξ. If now
Uλ is a fundamental solution to
d
Uλ (s, t) − qλ (s, x, D)Uλ (s, t) = 0
ds
then
U := eλ(t−s) Uλ
is a fundamental solution to (2.2) and the factor eλ(t−s) does not affect the
class to which the corresponding symbol belongs.
By Remark 3.2 ii) it can be assumed for the proof of the theorem without
loss of generality that q satisfies (1.9) not only for |ξ| large but for all ξ. Thus
the inequality
∂ β ∂ α q(s, x, ξ) −(|α|∧j)
x ξ
(3.1)
≤ cα,β (1 + ψ(ξ)) 2
Re q(s, x, ξ) holds for all ξ.
To construct the fundamental solution the method of Levi-Mizohata as presented by Kumano-go in [11] is applied. For the backward equation this means:
Starting with an approximation VB to the solution of (2.2) an operator B satisfying
Z t
d
d
− q(s, x, D) VB (s, t) +
− q(s, x, D) VB (s, τ )B(τ, t) dτ
B(s, t) =
ds
ds
s
(3.2)
will be constructed. Then the fundamental solution to (2.2) is given by
Z t
U (s, t) := VB (s, t) +
VB (s, τ )B(τ, t) dτ.
s
The main problem is to show that the constructed operators have symbols in an
appropriate symbol class.
12
Now the above will be made precise. The first approximation to the solution is
VB (s, t, x, ξ) := e(s, t, x, ξ) + eB (s, t, x, ξ),
(3.3)
where
Rt
e(s, t, x, ξ) := e− s q(τ,x,ξ) dτ ,
Z t
qB (τ, t, x, ξ)
eB (s, t, x, ξ) := −
dτ e(s, t, x, ξ)
s e(τ, t, x, ξ)
(3.4)
(3.5)
with
qB (s, t, x, ξ) :=
X
∂ξα q(s, x, ξ)Dxα e(s, t, x, ξ).
|α|=1
Lemma 3.3. Let ψ ∈ Λ, m ∈ R and q ∈ S2ψ,m satisfy (1.6) and (1.9) for all ξ,
then the previously defined symbols satisfy
i) e ∈ S2ψ,0 ,
ii) |∂ξα ∂xβ e(s, t, x, ξ)| ≤ cα,β (t − s)(1 + ψ(ξ))
m−(|α|∧2)
2
for |α + β| ≥ 1,
iii) eB ∈ S1ψ,−1 ,
iv) eB ∈ (t − s)S1ψ,m−1 .
Since the proof is very technical and not enlightening for the main topic, it is
postponed to section 5 at the end of the paper. The operator B satisfying (3.2) is
B(s, t) :=
∞
X
Bν (s, t, x, D)
(3.6)
ν=1
where the operators Bν are defined recursively by
d
B1 (s, t, x, D) :=
− q(s, x, D) VB (s, t),
ds
Z t
Bν (s, t, x, D) :=
B1 (s, τ )Bν−1 (τ, t) dτ, ν = 2, 3, . . . .
(3.7)
s
The convergence of
sum in (3.6) is meant on the level of the symbols, i.e.
Pthe
∞
it will
ν (s, t, x, ξ) is in an appropriate symbol class and thus
P be shown that ν=1
PB
∞
σ( ∞
B
(s,
t,
x,
D))
=
ν=1 ν
ν=1 Bν (s, t, x, ξ).
13
Lemma 3.4. The symbols of the operators defined in (3.6) and (3.7) satisfy
B1 (s, t, x, ξ) ∈ S0ψ,m−2 ∩ (t − s)S0ψ,2m−2
(3.8)
and if m ≤ 2
B(s, t, x, ξ) = σ
∞
X
!
Bν (s, t, x, D) (s, t, x, ξ) ∈ S0ψ,m−2 ∩ (t − s)S0ψ,2m−2 . (3.9)
ν=1
Furthermore B solves (3.2), i.e.
Z
B(s, t) = B1 (s, t) +
t
B1 (s, τ )B(τ, t) dτ
(3.10)
s
holds.
Proof. By Theorem 1.5 B) and Lemma 3.3 it follows that
σ (q(s, x, D)e(s, t, x, D)) = q(s, x, ξ)e(s, t, x, ξ) + qB (s, t, x, ξ) + rqe (s, t, x, ξ)
and
σ (q(s, x, D)eB (s, t, x, D)) = q(s, x, ξ)eB (s, t, x, ξ) + rqeB (s, t, x, ξ)
where rqe , rqeB ∈ S0ψ,m−2 , rqeB ∈ (t − s)S0ψ,2m−2 and due to Remark 1.6 and Lemma
3.3 ii) also rqe ∈ (t − s)S0ψ,2m−2 holds. This implies
d
d
e(s, t, x, ξ) + eB (s, t, x, ξ)
ds
ds
− σ (q(s, x, D)e(s, t, x, D)) − σ (q(s, x, D)eB (s, t, x, D))
d
=
− q(s, x, ξ) e(s, t, x, ξ) − qB (s, t, x, ξ)
ds
d
+
− q(s, x, ξ) eB (s, t, x, ξ) − rqe (s, t, x, ξ) − rqeB (s, t, x, ξ)
ds
= −rqe (s, t, x, ξ) − rqeB (s, t, x, ξ),
σ (B1 (s, t)) =
where the last step is due to
d
− q(s, x, ξ) e(s, t, x, ξ) = 0 since q(., x, ξ) is continuous
ds
and
d
− q(s, x, ξ) eB (s, t, x, ξ) = qB (s, t, x, ξ).
ds
14
The last equality holds, since eB is by definition the solution to this ordinary
inhomogeneous differential equation. Thus (3.8) is proved.
By definition is
Z t
Z tZ t
B1 (s, τ1 )B1 (τ1 , τ2 ) · · · B1 (τν−1 , t) dτν−1 · · · dτ2 dτ1
···
Bν (s, t, x, D) =
s
τ1
τν−1
and since m ≤ 2 Theorem 1.5 C) implies
(
(Cα,β )ν (1 + ψ(ξ))m−2
|∂ξα ∂xβ σ(B1 (s, τ1 )B1 (τ1 , τ2 ) · · · B1 (τν−1 , t))| ≤
0
C(Cα,β
)ν−1 (t − s)(1 + ψ(ξ))2m−2
.
Thus evaluating the integrals yields

ν−1

(Cα,β )ν (t − s) (1 + ψ(ξ))m−2

(ν − 1)!
|∂ξα ∂xβ Bν (s, t, x, ξ)| ≤
(t − s)ν−1

0
ν−1

(1 + ψ(ξ))2m−2
C(Cα,β ) (t − s)
(ν − 1)!
and therefore
(
Cα,β eCα,β (t−s) (1 + ψ(ξ))m−2
|∂ξα ∂xβ B(s, t, x, ξ)| ≤
0
(t − s)CeCα,β (t−s) (1 + ψ(ξ))2m−2
.
By induction
l
X
Z
t
B1 (s, τ )
Bν (s, t, x, D) = B1 (s, t) +
s
ν=1
l−1
X
Bν (τ, t) dτ
ν=1
holds since the integrand is bounded on [s, t] for suitable functions. Finally looking
at the above equation on the level of the symbols and letting l tend to ∞ yields
Z t
B1 (s, τ )B(τ, t) dτ.
B(s, t) = B1 (s, t) +
s
Proof of Theorem 3.1. Lemma 3.3 implies that VB ∈ S2ψ,0 ⊂ S0ψ,0 and by
Lemma 3.4 also B ∈ S0ψ,m−2 holds. Thus by Theorem 1.5 A) is σ (VB (s, τ )B(τ, t)) ∈
S0ψ,m−2 and for
Z
t
U (s, t) := VB (s, t) +
VB (s, τ )B(τ, t) dτ
s
15
(3.11)
it follows that U ∈ S0ψ,0 and
−
σ(U (s, t))(x, ξ) = e
Rt
s
q(τ,x,ξ) dτ
Z
t
VB (s, τ )B(τ, t) dτ
+ eB (s, t, x, ξ) + σ
(x, ξ).
s
For
Z
r0 (s, t, x, ξ) := eB (s, t, x, ξ) + σ
t
VB (s, τ )B(τ, t) dτ
(x, ξ)
s
ψ,max{−1,m−2}
it follows that r0 ∈ S0
∩ (t − s)S0ψ,m−1 since eB ∈ S1ψ,−1 ∩ (t − s)S1ψ,m−1 .
Applying Lemma 3.3 and Theorem 1.5 we get
d
e(s, t, x, ξ) = q(s, x, ξ)e(s, t, x, ξ) ∈ S0ψ,m ,
ds
d
eB (s, t, x, ξ) = qB (s, t, x, ξ) + eB (s, t, x, ξ)q(s, x, ξ) ∈ S0ψ,m−1
ds
since q ∈ S2ψ,m , qB ∈ S1ψ,m−1 (see Remark 1.6) and eB ∈ S1ψ,−1 . Thus we have
Z t
d
d
d
σ
U (s, t) = σ
VB (s, t) − B(s, t) +
VB (s, τ )B(τ, t) dτ ∈ S0ψ,m ,
ds
ds
ds
s
d
d
U (s, t) u(x) = ds
(U (s, t)u(x)) , i.e. it
where the derivative is meant such that ds
can be understood on the level of the symbols.
Finally by the definition of B1 and (3.10) we find
d
d
− q(s, x, D) U (s, t) =
− q(s, x, D) VB (s, t) − B(s, t)
ds
ds
Z t
d
+
− q(s, x, D) VB (s, τ )B(τ, t) dτ
ds
s
= 0.
Corollary 3.5. Let U be the operator constructed in Theorem 3.1. Then
i) σ (U (s, t) − id) ∈ (t − s)S0ψ,m ,
ii) σ (U (s, t) − U (r, t)) ∈ (r − s)S0ψ,m for s ≤ r ≤ t.
Proof. As in the proof of Theorem 3.1 the symbol of U has the representation
Z t
σ(U (s, t))(x, ξ) = e(s, t, x, ξ) + eB (s, t, x, ξ) + σ
VB (s, τ )B(τ, t) dτ (x, ξ)
s
= e(s, t, x, ξ) + r0 (s, t, x, ξ).
16
Note that r0 (s, t, x, ξ) ∈ (t − s)S0ψ,m−1 ⊂ (t − s)S0ψ,m and
e(s, t, x, ξ) − 1 = e−
Rt
s
q(τ,x,ξ) dτ
− 1 ∈ (t − s)S0ψ,m
(3.12)
where the elementary estimate |e−z − 1| ≤ |z| for Re z ≥ 0 was used. Thus i)
holds, i.e.
σ(U (s, t) − id)(x, ξ) = e(s, t, x, ξ) − 1 + r0 (s, t, x, ξ) ∈ (t − s)S0ψ,m .
To prove ii) note
e(s, t, x, ξ) = e−
Rt
s
q(τ,x,ξ) dτ
= e−
Rr
s
q(τ,x,ξ) dτ −
e
Rt
r
q(τ,x,ξ) dτ
= e(s, r, x, ξ)e(r, t, x, ξ).
Therefore
e(s, t, x, ξ) − e(r, t, x, ξ) = (e(s, r, x, ξ) − 1) e(r, t, x, ξ) ∈ (r − s)S0ψ,m ,
holds, where (3.12) and e(r, t, x, ξ) ∈ S0ψ,0 was used. Using the same argument
with eB (s, r) ∈ S0ψ,0 ∩ (r − s)S0ψ,m also the symbol
Z t
qB (τ, t)
eB (s, t) − eB (r, t) = −
dτ e(s, t) − eB (r, t)
s e(τ, t)
Z r
Z t
qB (τ, t)
qB (τ, t)
dτ +
dτ e(s, r)e(r, t) − eB (r, t)
=−
e(τ, t)
s
r e(τ, t)
Z r
qB (τ, t)
dτ e(s, r)e(r, t) + (e(s, r) − 1)eB (r, t)
=−
e(τ, t)
s
= eB (s, r)e(r, t) + (e(s, r) − 1)eB (r, t)
is an element of (r − s)S0ψ,m . Using VB (s, t) = e(s, t) + eB (s, t) ∈ S0ψ,0 with the
above yields
VB (s, t) − VB (r, t) = (e(s, r) − 1)VB (r, t) + eB (s, r)e(r, t) ∈ (r − s)S0ψ,m .
Finally consider
σ(U (s, t) − U (r, t))
Z
t
Z
= VB (s, t) − VB (r, t) + σ
VB (s, τ )B(τ, t) dτ −
Z r
= VB (s, t) − VB (r, t) + σ
VB (s, τ )B(τ, t) dτ
s
Z t
+σ
(VB (s, τ ) − VB (r, τ )) B(τ, t) dτ
s
r
17
t
VB (r, τ )B(τ, t) dτ
r
and note that
Z
σ
r
VB (s, τ )B(τ, t) dτ
∈ (r − s)S0ψ,0 ,
s
Z
t
(VB (s, τ ) − VB (r, τ )) B(τ, t) dτ
σ
∈ (r − s)S0ψ,m ,
r
since VB (s, t), σ(B(τ, t)) ∈ S0ψ,0 and VB (s, t) − VB (r, t) ∈ (r − s)S0ψ,m . Thus we have
shown
σ(U (s, t) − U (r, t)) ∈ (r − s)S0ψ,m .
4
The associated Markov process
In this section it is shown that operator constructed in Theorem 3.1 defines a
Markov process.
Lemma 4.1. Let U be the operator constructed in Theorem 3.1 and assume that
ψ satisfies (1.2) and r satisfies (1.3).
A) U is strongly continuous in s and t, i.e. for f ∈ H ψ,r+m
lim kU (s, t)f − f k∞ = 0
|t−s|→0
holds.
B) The function (s, x) 7→ U (s, t)f (x) is jointly continuous for f ∈ H ψ,r+m .
Proof. By Corollary 3.5 i) we have
σ (U (s, t) − id) ∈ (t − s)S0ψ,m
and thus for f ∈ H ψ,r+m
kU (s, t)f − f k∞ ≤ ck(U (s, t) − id)f kψ,r ≤ c̃(t − s)kf kψ,r+m
implies part A).
Note that part B) is a simple consequence of A) once we know that U (s, t) has
the evolution property. But this property is a consequence of the uniqueness of
the solution to the evolution equation. And for the uniqueness (i.e. Theorem 2.3)
the joint continuity is needed. Therefore we proceed differently.
18
By the same argument as in part A) together with statement ii) of Corollary
3.5 also
lim kU (s, t)f − U (τ, t)f k∞ = 0
|s−τ |→0
holds for s, τ ≤ t.
To show B) fix s, x and note that U (s, t)f ∈ H ψ,r . Thus x 7→ U (s, t)f (x) is
continuous. Then
|U (s, t)f (x) − U (τ, t)f (y)| ≤ |U (s, t)f (x) − U (s, t)f (y)| + |U (s, t)f (y) − U (τ, t)f (y)|
≤ |U (s, t)f (x) − U (s, t)f (y)| + kU (s, t)f − U (τ, t)f k∞
implies the joint continuity.
Lemma 4.2. Let ψ ∈ Λ satisfy (1.2) and m ≤ 2. A pseudo-differential operator
with symbol in S2ψ,m satisfying (1.6),(1.7),(1.8) and (1.9) defines an operator on
C∞ such that
a) U (s, t) is a linear operator,
d) U (t, t) = id,
b) U (s, t) is a contraction,
e) U (s, t) = U (s, τ )U (τ, t) t ≥ τ ≥ s,
c) U (s, t) is positivity preserving,
f ) U (s, t)1 = 1.
Where in f ) is understood in the sense of monotone limits, i.e. limk→∞ U (s, t)uk =
1 holds for uk ∈ C∞ with uk ↑ 1.
Proof. Let U be the operator constructed in Theorem 3.1 and let r satisfy (1.3).
The operator U is obviously linear and by Lemma 4.1, Theorem 2.3 and 2.4 a
positivity preserving contraction with respect to k.k∞ for real-valued functions
of H ψ,r . Property d) follows from (3.11). The evolution property e) holds for
ϕ ∈ H ψ,r since U (s, t)ϕ and U (s, τ )U (τ, t)ϕ solve (2.1) with initial value ϕ and the
solution is unique by Theorem 2.3. For the uniqueness the joint continuity shown
in part B) of Lemma 4.1 was needed. Property f) is just Theorem 2.4.
Since C0∞ ⊂ H ψ,r and C0∞ is dense in C∞ the operator U can be extended onto
C∞ .
Now in the usual (non-trivial) way a Markov process can be constructed starting
with the operator from Lemma 4.2, see for example section 4.8 in [8] and chapter 3
in [10]. The construction is the same as for a Feller semigroup, just the semigroup
property is replaced by the evolution property. I.e. due to property c) of Lemma 4.2
a family of positive measures can be associated to the operators. These measures
are probability measures due to f) and they define a projective limit, due to d) and
e). Thus by Kolmogorov’s canonical construction a corresponding process exists.
19
Corollary 4.3. The operator given in Lemma 4.2 defines Markov process.
Note that in most cases this Markov process will be time and space inhomogeneous.
5
Proof of Lemma 3.3
Repeatedly the following formulas and notations will be used:
l
Y
∂ ( fj )
γ
=
cγ 1 ,...,γ l
Ql
γ 1 +...+γ l =γ
j=1
γ
P
f
∂ (e )
f
=e
P
γj
j=1
∂ fj
=:
prod l
X
Y
j
∂ γ fj
(γ,l) j=1
cγ 1 ,...,γ l
γ 1 +...+γ l =γ
l=1,...,|γ|
Ql
j=1
γj
∂ f
f
=: e
exp l
X
Y
j
∂ γ f (5.2)
(γ;l) j=1
sk e−s ≤ k k e−k for s ≥ 0, k ∈ N ∪ {0}
Note that the symbol
exp
X
(5.1)
(5.3)
in (5.2) is set to equal 1 if the sum is empty, i.e. γ = 0.
(γ;l)
Using (3.1) resp. (1.1) we find
Z t

−(2∧|α|)
Z t

α β
cα,β
Re q(τ, x, ξ) dτ (1 + ψ(ξ)) 2 ,
∂ξ ∂x −
q(τ, x, ξ) dτ ≤
s
m−(2∧|α|)

s
cα,β (t − s)(1 + ψ(ξ)) 2 .
For α1 + . . . + αl = α we have
−(2 ∧ |α1 |) − . . . − (2 ∧ |αl |) ≤ −(2 ∧ |α|)
and thus get
Z t
l
Y
j
q(τ, x, ξ) dτ ∂ξα ∂xβ −
s
j=1

R
l
−(2∧|α|)

t
cα1 ,...,αl ,β
Re
q(τ,
x,
ξ)
dτ
(1 + ψ(ξ)) 2 ,
s
R
l−1
≤
m−(2∧|α|)

cα1 ,...,αl ,β (t − s) st Re q(τ, x, ξ) dτ
(1 + ψ(ξ)) 2 .
20
(5.4)
Now using (5.2) leads to
Z t
exp k
X
Y
j
∂ξα −
q(τ, x, ξ) dτ |∂ξα e(s, t, x, ξ)| = e(s, t, x, ξ)
s
(α;k) j=1

k
Z t
exp
X
Rt
−(2∧|α|)


−
Re
q(τ,x,ξ)
dτ
s
e
Re q(τ, x, ξ) dτ (1 + ψ(ξ)) 2 ,
cα1 ,...,αk ,β



s

(α;k)

k−1
Z
exp
t
X
R
≤
− st Re q(τ,x,ξ) dτ

cα1 ,...,αk ,β
Re q(τ, x, ξ) dτ
·
e



s

(α;k)


m−(2∧|α|)

· (t − s)(1 + ψ(ξ)) 2 .
(5.5)
But note that for the second inequality it is necessary that |α| > 0. Putting these
together we get (where starting from the third line we write γ for α1 and δ for α2
to increase the readability)


Z t
exp l
X
Y
α
α β
βj


|∂ξ ∂x e(s, t, x, ξ)| = ∂ξ e(s, t, x, ξ)
∂x (−
q(τ, x, ξ) dτ ) s
(β;l) j=1
!
prod
Z t
exp
l
X
Y
X α1
2
j
= ∂ξ e(s, t, x, ξ)
∂ξα
∂xβ (−
q(τ, x, ξ) dτ ) s
(α,2)
j=1
(β;l)
prod
exp prod l
X
XY j j Z t
X γ
δ
β
= ∂ξ e(s, t, x, ξ)
∂ξ ∂x (−
q(τ, x, ξ) dτ )
s
(α,2)
(β;l) (δ,l) j=1


k
Z t
prod
exp
X
X
Rt
−(2∧|γ|)
≤
e− s Re q(τ,x,ξ) dτ 
cγ 1 ,...,γ k
Re q(τ, x, ξ) dτ (1 + ψ(ξ)) 2 
(α,2)
·
(γ;k)
exp prod
X
X
Z
cδ1 ,...,δl ,β
s
l
t
Re q(τ, x, ξ) dτ
(1 + ψ(ξ))
−(2∧|δ|)
2
.
s
(β;l) (δ,l)
(5.6)
Finally (5.3) and (5.4) imply
|∂ξα ∂xβ e(s, t, x, ξ)| ≤ cα,β (1 + ψ(ξ))
−(2∧|α|)
2
and if |α + β| ≥ 1, we can apply the second inequality from (5.5) in one of the two
exp
P
terms to get
(...)
|∂ξα ∂xβ e(s, t, x, ξ)| ≤ cα,β (t − s)(1 + ψ(ξ))
21
−(2∧|α|)
2
.
It remains to show iii) and iv). First note that we can simplify the first factor of
eB , since |γ| = 1,
Z t
Z tP
γ
γ
qB (τ, t, x, ξ)
|γ|=1 ∂ξ q(τ, x, ξ)(−i∂x ) e(τ, t, x, ξ)
dτ =
dτ
e(τ, t, x, ξ)
s e(τ, t, x, ξ)
s
R
t
γ
γ
Z tP
−
q(r,
x,
ξ)
dr
∂
q(τ,
x,
ξ)(−i)e(τ,
t,
x,
ξ)∂
x
|γ|=1 ξ
s
=
dτ
e(τ, t, x, ξ)
s
Z t
Z tX
γ
γ
=i
q(r, x, ξ) dr dτ.
∂ξ q(τ, x, ξ)∂x
s |γ|=1
s
Using the above and 2 ∧ |α + γ| = 1 + (1 ∧ |α|) for |γ| = 1 we get
Z
α β t qB (τ, t, x, ξ) ∂ξ ∂x
dτ e(τ,
t,
x,
ξ)
s
X
Z t
Z t
prod prod
X
X
α1 +γ β 1
α2 β 2 +γ
=
∂ξ
∂x q(τ, x, ξ)∂ξ ∂x
q(r, x, ξ) dr dτ s
(α,2) (β,2) |γ|=1 s
prod prod
X
XXZ t
−(2∧|α1 +γ|)
2
Re q(τ, x, ξ)
≤
cα1 ,α2 ,β 1 ,β 2 ,γ (1 + ψ(ξ))
(α,2) (β,2) |γ|=1
s
· (1 + ψ(ξ))
−(2∧|α2 |)
2
Z
t
Re q(r, x, ξ) dr
dτ
s
≤ cα,β (1 + ψ(ξ))
−1−(1∧|α|)
2
Z
2
t
Re q(τ, x, ξ) dτ
s
and
Z
α β
∂ξ ∂x
s
t
Z t
m−1−(1∧|α|)
qB (τ, t, x, ξ) 2
dτ ≤ cα,β (t − s)(1 + ψ(ξ))
Re q(τ, x, ξ) dτ .
e(τ, t, x, ξ)
s
And finally with (5.6) we get
prod prod
X X 1 1 Z t qB (τ, t, x, ξ)
α β
α β
α2 β 2
|∂ξ ∂x eB (s, t, x, ξ)| = ∂ξ ∂x
dτ ∂ξ ∂x e(s, t, x, ξ)
s e(τ, t, x, ξ)
(α,2) (β,2)
(
−1−(1∧|α|)
2
cα,β (1 + ψ(ξ))
≤
m−1−(1∧|α|)
2
cα,β (t − s)(1 + ψ(ξ))
.
22
References
[1] Berg, C. and Forst, G. Potential Theory on Locally Compact Abelian Groups,
Vol. 87 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag,
1975.
[2] Böttcher, B. Some investigations on Feller processes generated by pseudodifferential operators. Ph.D. Thesis, University of Wales, Swansea, 2004.
[3] Böttcher, B. A parametrix construction for the fundamental solution of the
evolution equation associated with a pseudo-differential operator generating
a Markov process. Math. Nachr., 278(11):1235–1241, 2005.
[4] Courrège, P. Sur la forme intégro-différentielle des opérateurs de Ck∞ dans C
satisfaisant au principe du maximum. Sém. Théorie du potentiel. Exposé 2,
38pp, 1965/66.
[5] Eidelman, S., Ivasyshen, S., and Kochubei, A. Analytic Methods in the Theory
of Differential and Pseudo-Differential Equations of Parabolic Type, Vol. 152
of Operator Theory, Advances and Applications. Birkhäuser, 2004.
[6] Hoh, W. Pseudo differential operators generating Markov processes. Habilitationsschrift. Universität Bielefeld, Bielefeld, 1998.
[7] Hoh, W. A symbolic calculus for pseudo differential operators generating
Feller semigroups. Osaka Math. J., 35(4):789–820, 1998.
[8] Jacob, N. Pseudo-Differential Operators and Markov Processes I. Fourier
Analysis and Semigroups. Imperial College Press, 2001.
[9] Jacob, N. Pseudo-Differential Operators and Markov Processes II. Generators
and Their Potential Theory. Imperial College Press, 2002.
[10] Jacob, N. Pseudo-Differential Operators and Markov Processes III. Markov
Processes and Applications. Imperial College Press, 2005.
[11] Kumano-go, H. Pseudo-differential Operators. MIT Press, 1974.
[12] Tsutsumi, C. The fundamental solution for a degenerate parabolic pseudodifferential operator. Proc. Japan Acad., 50(1):11–15, 1974.
Björn Böttcher
[email protected]
Philipps-Universität Marburg
FB12 Mathematik und Informatik
Hans-Meerwein-Straße
35032 Marburg
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