2 semigroup theory - springer

2

Semigroup Theory

This chapter is devoted to a review of standard topics from the theory ofsemigroups which forms a functional analytic background for the proof ofTheorems 1.2, 1.3, 1.4 and 1.5.

2.1 Analytic Semigroups

This section provides a brief description of the basic results of the theoryof analytic semigroups which forms a functional analytic background for theproof of Theorems 1.2 and 1.3. Moreover, Subsection 2.1.3 is devoted to thesemigroup approach to a class of initial-boundary value problems for semi-linear parabolic differential equations (Theorem 2.8). Theorem 1.5 follows byverifying all the conditions of Theorem 2.8. For more leisurely treatmentsof analytic semigroups, the reader is referred to Friedman [Fr1], Pazy [Pa],Tanabe [Tn], Yosida [Yo] and also Taira [Ta4].

2.1.1 Generation of Analytic Semigroups

Let E be a Banach space over the real or complex number field, and letA : E → E be a densely defined, closed linear operator with domain D(A).

Assume that the operator A satisfies the following two conditions (seeFigure 2.1 below):

(1) The resolvent set of A contains the region

Σω = {λ ∈ C : λ �= 0, | argλ| < π/2 + ω}, 0 < ω < π/2.

(2) For each ε > 0, there exists a positive constant Mε such that the resolventR(λ) = (A− λI)−1 satisfies the estimate

‖R(λ)‖ ≤ Mε

|λ| for all λ ∈ Σεω = {λ ∈ C : λ �= 0, | argλ| ≤ π/2 + ω − ε}.

(2.1)

K. Taira, Boundary Value Problems and Markov Processes, 13Lecture Notes in Mathematics 1499, DOI 10.1007/978-3-642-01677-6 2,c© Springer-Verlag Berlin Heidelberg 2009

14 2 Semigroup Theory

0

Σω

Fig. 2.1.

Then we letU(t) = − 1

2πi

∫

Γ

eλtR(λ) dλ. (2.2)

Here Γ is a path in the set Σεω consisting of the following three curves (see

Figure 2.2):

Γ (1) ={re−i(π/2+ω−ε) : 1 ≤ r <∞

},

Γ (2) ={eiθ : −(π/2 + ω − ε) ≤ θ ≤ π/2 + ω − ε

},

Γ (3) ={rei(π/2+ω−ε) : 1 ≤ r <∞

}.

0

Γ (1)

Γ (2)

Γ (3)

Fig. 2.2.

It is easy to see that the integral

U(t) = − 12πi

3∑

k=1

∫

Γ (k)eλtR(λ) dλ

2.1 Analytic Semigroups 15

converges in the uniform operator topology of the Banach space L(E,E) forall t > 0, and thus defines a bounded linear operator on E. Here L(E,E)denotes the space of bounded linear operators on E.

Furthermore, we have the following:

Proposition 2.1. The operators U(t), defined by formula (2.2), form a semi-group on E, that is, they enjoy the semigroup property

U(t+ s) = U(t) · U(s) for all t, s > 0.

Proof. By Cauchy’s theorem, we may assume that

U(s) = − 12πi

∫

Γ ′eμsR(μ) dμ, s > 0.

Here Γ ′ is a path obtained from the path Γ by translating each point of Γ tothe right by a fixed small positive distance (see Figure 2.3).

′Γ Γ

0

Fig. 2.3.

Then we have, by Fubini’s theorem,

U(t) · U(s) =1

(2πi)2

∫

Γ

∫

Γ ′eλteμsR(λ)R(μ) dλ dμ

=1

(2πi)2

∫

Γ

∫

Γ ′eλteμsR(λ)−R(μ)

λ− μ dλdμ

=1

2πi

∫

Γ

eλtR(λ)[

12πi

∫

Γ ′

eμs

λ− μ dμ]dλ

− 12πi

∫

Γ ′eμsR(μ)

[1

2πi

∫

Γ

eλt

λ− μ dλ]dμ.

We calculate the two terms in the last part.(a) We let

f(μ) =eμs

λ− μ, μ ∈ C.


|¹|= r

¸

0

Fig. 2.4.

Then, by applying the residue theorem we obtain that (see Figure 2.4)∫

Γ ′(1)∩{|μ|≤r}f(μ) dμ+

∫

Γ ′(2)f(μ) dμ+

∫

Γ ′(3)∩{|μ|≤r}f(μ) dμ

+∫ π/2+ω−ε

−(π/2+ω−ε)

f(reiθ)rieiθ dθ

= 2πiRes [f(μ)]μ=λ

= −2πieλs.

However, we have, as r →∞,∫

Γ ′(1)∩{|μ|≤r}f(μ) dμ −→

∫

Γ ′(1)f(μ) dμ,

∫

Γ ′(3)∩{|μ|≤r}f(μ) dμ −→

∫

Γ ′(3)f(μ) dμ,

and∣∣∣∣∣

∫ π/2+ω−ε

−(π/2+ω−ε)

f(reiθ)rieiθ dθ

∣∣∣∣∣ ≤ e−rs·sin(ω−ε)

∫ π/2+ω−ε

−(π/2+ω−ε)

dθ∣∣λr − eiθ

∣∣ −→ 0.

Therefore, we find that

12πi

∫

Γ ′

eμs

λ− μ dμ = −eλs.

(b) Similarly, since the path Γ lies to the left of the path Γ ′, we find that

12πi

∫

Γ

eλt

λ− μ dλ = 0.


Summing up, we obtain that

U(t) · U(s) = − 12πi

∫

Γ

eλ(t+s)R(λ) dλ = U(t+ s) for all t, s > 0.

The proof of Proposition 2.1 is complete. �

The next theorem states that the semigroup U(t) can be extended to ananalytic semigroup in some sector containing the positive real axis.

Theorem 2.2. The semigroup U(t), defined by formula (2.2), can be extendedto a semigroup U(z) which is analytic in the sector

Δω = {z = t+ is : z �= 0, | arg z| < ω} ,

and enjoys the following properties:

(a) The operators AU(z) and dUdz (z) are bounded operators on E for each z ∈

Δω, and satisfy the relation

dU

dz(z) = AU(z) for all z ∈ Δω. (2.3)

(b) For each 0 < ε < ω/2, there exist positive constants M0(ε) and M1(ε)such that

‖U(z)‖ ≤ M0(ε) for all z ∈ Δ2εω , (2.4)

‖AU(z)‖ ≤ M1(ε)|z| for all z ∈ Δ2ε

ω , (2.5)

where (see Figure 2.5)

Δ2εω = {z ∈ C : z �= 0, | arg z| ≤ ω − 2ε} .

(c) For each x ∈ E, we have, as z → 0, z ∈ Δ2εω ,

U(z)x −→ x in E.

Proof. (i) The analyticity of U(z): If λ ∈ Γ (3) and z ∈ Δ2εω , that is, if we have

the formulas

λ = |λ|eiθ, θ = π/2 + ω − ε,z = |z|eiϕ, |ϕ| ≤ ω − 2ε,

then it follows thatλz = |λ| |z|ei(θ+ϕ),

withπ/2 + ε ≤ θ + ϕ ≤ π/2 + 2ω − 3ε < 3π/2− 3ε.


Δ2εω

0

Fig. 2.5.

Note thatcos(θ + ϕ) ≤ cos(π/2 + ε) = − sin ε.

Hence we have the inequality

|eλz| ≤ e−|λ| |z| sin ε for all λ ∈ Γ (3) and z ∈ Δ2εω . (2.6)

Similarly, we have the inequality

|eλz| ≤ e−|λ| |z| sin ε for all λ ∈ Γ (1) and z ∈ Δ2εω . (2.7)

For each small ε > 0, we let

Kεω = Δ2ε

ω

⋂{z ∈ C : |z| ≥ ε} = {z ∈ C : |z| ≥ ε, | arg z| ≤ ω − 2ε} .

Then, by combining estimates (2.1), (2.6) and (2.7) we obtain that

∥∥eλzR(λ)∥∥ ≤ Mε

|λ| e−ε sin ε·|λ| for all λ ∈ Γ (1)

⋃Γ (3) and z ∈ Kε

ω. (2.8)

On the other hand, we have the estimate∥∥eλzR(λ)

∥∥ ≤Mεe|z| for all λ ∈ Γ (2) and z ∈ Kε

ω. (2.9)

Therefore, we find that the integral

U(z) = − 12πi

∫

Γ

eλzR(λ) dλ = − 12πi

3∑

k=1

∫

Γ (k)eλzR(λ) dλ (2.10)

converges in the Banach space L(E,E), uniformly in z ∈ Kεω, for every ε > 0.

This proves that the operator U(z) is analytic in the domain Δω =⋃

ε>0Kεω.

By the analyticity of U(z), it follows that the operators U(z) also enjoythe semigroup property


U(z + w) = U(z) · U(w) for all z, w ∈ Δω.

(ii) We prove that the operators U(z) enjoy properties (a), (b) and (c).(b) First, by using Cauchy’s theorem we obtain that

U(z) = − 12πi

∫

Γ

eλzR(λ) dλ = − 12πi

∫

Γ|z|eλzR(λ) dλ,

where Γ|z| is a path consisting of the following three curves (see Figure 2.6):

Γ(1)|z| =

{re−i(π/2+ω−ε) :

1|z| ≤ r <∞

},

Γ(2)|z| =

{1|z|e

iθ : −(π/2 + ω − ε) ≤ θ ≤ π/2 + ω − ε},

Γ(3)|z| =

{rei(π/2+ω−ε) :

1|z| ≤ r <∞

}.

0

Γ(1)

Γ(2)

Γ(3)

|z|

|z|

|z|

Fig. 2.6.

However, by estimates (2.1), (2.6) and (2.7), it follows that

∥∥eλzR(λ)∥∥ ≤ Mε

|λ| e−|λ ||z| sin ε for all λ ∈ Γ (1)

|z|⋃Γ

(3)|z| and z ∈ Δ2ε

ω .

Hence we have, for k = 1, 3,∫

Γ(k)|z|

∥∥eλzR(λ)∥∥ | dλ| ≤ Mε

∫ ∞

1|z|

e−ρ|z| sin ερ−1 dρ

= Mε

∫ ∞

1

e− sin ε·ss−1 ds.


We have also, for k = 2,∫

Γ(2)|z|

∥∥eλzR(λ)∥∥ | dλ| ≤ Mε

∫ π/2+ω−ε

−(π/2+ω−ε)

e dθ

= 2eMε(π/2 + ω − ε)≤ 2πeMε.

Summing up, we obtain the following estimate:

‖U(z)‖ ≤ 12π

3∑

k=1

∫

Γ(k)|z|

∥∥eλzR(λ)∥∥ | dλ|

≤ 12π

(2Mε

∫ ∞

1

s−1e− sin ε·s ds+ 2πeMε

)

=Mε

π

(∫ ∞

1

s−1e− sin ε·s ds+ πe

).

This proves the desired estimate (2.4), with

M0(ε) =Mε

π

(∫ ∞

1

s−1e− sin ε·s ds+ πe

).

To prove estimate (2.5), note that

AR(λ) = (A− λI + λI)R(λ) = I + λR(λ),

so that‖AR(λ)‖ ≤ 1 +Mε for all λ ∈ Σε

ω.

Hence, by arguing just as in the proof of estimate (2.4) we obtain that∥∥∥∥∫

Γ

eλzAR(λ) dλ∥∥∥∥ ≤ 2

∫ ∞

1|z|

e−ρ|z| sin ε (1 +Mε) dρ

+∫ π/2+ω−ε

−(π/2+ω−ε)

(1 +Mε) e1|z| dθ

≤ 2 (1 +Mε)(∫ ∞

1

e− sin ε·s ds+ πe

)1|z|

for all z ∈ Δ2εω . (2.11)

This proves that the integral∫

Γ

eλzAR(λ) dλ

is convergent in the Banach space L(E,E), for every z ∈ Δ2εω . By the closed-

ness of A, it follows that

U(z) ∈ D(A) for all z ∈ Δ2εω ,


andAU(z) = − 1

2πi

∫

Γ

eλzAR(λ) dλ for all z ∈ Δ2εω . (2.12)

Therefore, the desired estimate (2.5) follows from estimate (2.11), with

M1(ε) =1 +Mε

π

(∫ ∞

1

e− sin ε·s ds+ πe

).

We remark that formula (2.12) remains valid for all z ∈ Δω, since Δω =⋃ε>0 Δ2ε

ω .(a) By estimates (2.8) and (2.9), we can differentiate formula (2.10) under

the integral sign to obtain that

dU

dz(z) = − 1

2πi

∫

Γ

eλzλR(λ) dλ for all z ∈ Δω. (2.13)

On the other hand, it follows from formula (2.12) that

AU(z) = − 12πi

∫

Γ

eλzAR(λ) dλ

= − 12πi

∫

Γ

eλz(I + λR(λ)) dλ

= − 12πi

∫

Γ

eλzλR(λ) dλ for all z ∈ Δω, (2.14)

since we have, by Cauchy’s theorem,∫

Γ

eλz dλ = 0.

Therefore, the desired formula (2.3) follows immediately from formulas(2.13) and (2.14).

(c) Now let x0 be an arbitrary element of D(A). By the residue theorem,it follows that

x0 =1

2πi

∫

Γ

eλz

λx0 dλ.

Hence we have the formula

U(z)x0 − x0 = − 12πi

∫

Γ

eλz

(R(λ) +

1λ

)x0 dλ

= − 12πi

∫

Γ

eλz

λR(λ)Ax0 dλ.

Here we remark that∥∥∥∥

1λR(λ)

∥∥∥∥ ≤Mε

|λ|2 for all λ ∈ Γ,∣∣eλz

∣∣ ≤ 2e−|λ| |z| sin ε + e|z| for all z ∈ Δ2εω and λ ∈ Γ.


Thus it follows from an application of the Lebesgue dominated convergencetheorem that, as z → 0, z ∈ Δ2ε

ω ,

U(z)x0 − x0 −→ −1

2πi

∫

Γ

1λR(λ)Ax0 dλ.

However, we have the assertion∫

Γ

1λR(λ)Ax0 dλ = 0.

Indeed, by Cauchy’s theorem it follows that∫

Γ

1λR(λ)Ax0 dλ = lim

r→∞

∫

Γ∩{|λ|≤r}

1λR(λ)Ax0 dλ

= − limr→∞

∫

Cr

1λR(λ)Ax0 dλ

= 0,

where Cr is a closed path shown in Figure 2.7.

0

ΓCr

Fig. 2.7.

Summing up, we have proved that

U(z)x0 −→ x0 as z → 0, z ∈ Δ2εω ,

for each x0 ∈ D(A).Since the domain D(A) is dense in E and ‖U(z)‖ ≤ M0(ε) for all z ∈ Δ2ε

ω ,it follows that, for each x ∈ E,

U(z)x −→ x as z → 0, z ∈ Δ2εω .

The proof of Theorem 2.2 is now complete. �


Remark 2.1. Assume that the operator A satisfies a stronger condition thancondition (2.1):

‖R(λ)‖ ≤ Mε

|λ|+ 1for all λ ∈ Σε

ω. (2.15)

Then we have the estimates

‖U(z)‖ ≤ M0(ε)e−δ·Re z for all z ∈ Δ2εω , (2.16)

‖AU(z)‖ ≤ M1(ε)|z| e−δ·Re z for all z ∈ Δ2ε

ω , (2.17)

with some positive constant δ.

Proof. Take a real number δ such that

0 < δ <1Mε

.

Then we have, by estimate (2.15),

δ∥∥(A− λI)−1

∥∥ ≤ δMε

|λ|+ 1≤ δMε < 1 for all λ ∈ Σε

ω.

Hence it follows that the operator (A+ δI)− λI has the inverse

((A+ δI)− λI)−1 = (I + δ(A− λI)−1)−1(A− λI)−1,

and

‖((A+ δI)− λI)−1‖ ≤ ‖(I + δ(A− λI)−1)−1‖ · ‖(A− λI)−1‖

≤ Mε

|λ|+ 11

1− ‖δ(A− λI)−1‖

≤ Mε

|λ|+ 11

1− δMε

≤(

Mε

1− δMε

)1|λ| .

This proves that the operator A+δI satisfies condition (2.1), so that estimates(2.4) and (2.5) remain valid for the operator A+ δI:

‖V (z)‖ ≤ M0(ε) for all z ∈ Δ2εω , (2.18)

‖(A+ δI)V (z)‖ ≤ M1(ε)|z| for all z ∈ Δ2ε

ω , (2.19)

whereV (z) = − 1

2πi

∫

Γ

eλz(A+ δI − λI)−1 dλ.


However, we have, by Cauchy’s theorem,

V (z) = − 12πi

∫

Γ

eλz(A+ δI − λI)−1 dλ

= − 12πi

∫

Γ+δ

eλz(A+ δI − λI)−1 dλ

= − 12πi

∫

Γ

eμzeδz(A− μI)−1 dμ = eδzU(z)

for all z ∈ Δ2εω . (2.20)

In view of formula (2.16), the desired estimates (2.16) and (2.17) follow fromestimates (2.18) and (2.19). �

2.1.2 Fractional Powers

Assume that the operator A satisfies a stronger condition than condition (2.1):

(1) The resolvent set of A contains the region Σ as in Figure 2.8.

0

Σ

Fig. 2.8.

(2) There exists a positive constant M such that the resolvent R(λ) = (A −λI)−1 satisfies the estimate

‖R(λ)‖ ≤ M

(1 + |λ|) for all λ ∈ Σ. (2.21)

If α > 0, we can define the fractional power (−A)−α of −A by the followingformula:

(−A)−α = − 12πi

∫

Γ

(−λ)−αR(λ) dλ. (2.22)

Here the path Γ runs in the set Σ from∞e−iω to ∞eiω, avoiding the positivereal axis and the origin (see Figure 2.9), and for the function (−λ)−α =


e−α log(−λ), we choose the branch whose argument lies between −απ and απ;it is analytic in the region obtained by omitting the positive real axis.

The integral (2.22) converges in the uniform operator topology of the Ba-nach space L(E,E) for all α > 0, and thus defines a bounded linear operatoron E.

Γ

0

Fig. 2.9.

Some basic properties of the fractional power (−A)−α are summarized inthe following:

Proposition 2.3. (i) We have, for all α, β > 0,

(−A)−α(−A)−β = (−A)−(α+β).

(ii) If α is a positive integer n, then we have the formula

(−A)−α =((−A)−1

)n.

(iii) The fractional power (−A)−α is invertible for all α > 0.

If 0 < α < 1, we have the following useful formula for the fractional power(−A)−α:

Theorem 2.4. We have, for all 0 < α < 1,

(−A)−α = − sinαππ

∫ ∞

0

s−αR(s)ds. (2.23)

By Remark 2.1, we may assume that there exist positive constants M0,M1 and a such that

‖U(t)‖ ≤M0e−at for all t > 0.

‖AU(t)‖ ≤M1e−at 1

tfor all t > 0.

Then we can prove still another useful formula for the fractional power (−A)−α

for all 0 < α < 1.



(−A)−α =1

Γ (α)

∫ ∞

0

tα−1U(t) dt.

In view of part (iii) of Proposition 2.3, we can define the fractional power(−A)α for all α > 0 as follows:

(−A)α = the inverse of (−A)−α, α > 0.

The next theorem states that the domain D((−A)α) of (−A)α is biggerthan the domain D(A) of A when 0 < α < 1.


D(A) ⊂ D ((−A)α) .

We can give an explicit formula for the fractional power (−A)α (0 < α < 1)on the domain D(A):

Theorem 2.7. Let 0 < α < 1. Then we have, for any x ∈ D(A),

(−A)αx =sinαππ

∫ ∞

0

sα−1R(s)Axds.

2.1.3 The Semilinear Cauchy Problem

This subsection is devoted to the semigroup approach to a class of initial-boundary value problems for semilinear parabolic differential equations. Bymaking good use of fractional powers of analytic semigroups, we formulate alocal existence and uniqueness theorem for semilinear initial-boundary valueproblems (Theorem 2.8). Our semigroup approach can be traced back to thepioneering work of Fujita–Kato [FK] for the Navier–Stokes equation in fluiddynamics.

Assume that the operator A satisfies condition (2.21). Then we can de-fine the fractional power (−A)α for all 0 < α < 1 (see Subsection 2.1.2).The operator (−A)α is a closed linear, invertible operator with domainD((−A)α) ⊃ D(A). We let

Eα = the space D((−A)α) endowed with the graph norm ‖ · ‖αof (−A)α,

where‖x‖α =

(‖x‖2 + ‖(−A)αx‖2

)1/2, x ∈ D((−A)α).

Then we have the following three assertions:

(i) The space Eα is a Banach space.

2.2 Markov Processes and Feller Semigroups 27

(ii) The graph norm ‖x‖α is equivalent to the norm ‖(−A)αx‖.(iii) If 0 < α < β < 1, then we have Eβ ⊂ Eα with continuous injection.

Now we consider the following semilinear Cauchy problem:{

dudt = Au(t) + f(t, u(t)), t0 < t < t1,u(t0) = x0.

(2.24)

Here f(t, x) is a function defined on an open subset U of [0,∞)×Eα (0 < α <1), taking values in E. We assume that f(t, x) is locally Holder continuous int and locally Lipschitz continuous in x. That is, for each point (t, x) of U thereexist a neighborhood V ⊂ U , constants L = L(t, x, V ) > 0 and 0 < γ ≤ 1such that

‖f(s1, y1)− f(s2, y2)‖ ≤ L (|s1 − s2|γ + ‖y1 − y2‖α) ,(s1, y1), (s2, y2) ∈ V.

A function u(t) : [t0, t1) → E is called a solution of problem (2.24) if itsatisfies the following three conditions:

(1) u(t) ∈ C([t0, t1);E) ∩ C1((t0, t1);E) and u(t0) = x0.(2) u(t) ∈ D(A) and (t, u(t)) ∈ U for all t0 < t < t1.(3) du

dt = Au(t) + f(t, u(t)) for all t0 < t < t1.

Here C([t0, t1);E) denotes the space of continuous functions on [t0, t1) takingvalues in E, and C1((t0, t1);E) denotes the space of continuously differentiablefunctions on (t0, t1) taking values in E, respectively.

Our main result is the following local existence and uniqueness theoremfor problem (2.24):

Theorem 2.8. Let f(t, x) be a function defined on an open subset U of[0,∞) × Eα (0 < α < 1), taking values in E. Assume that f(t, x) is lo-cally Holder continuous in t and locally Lipschitz continuous in x. Then,for every (t0, x0) ∈ U , problem (2.24) has a unique local solution u(t) ∈C([t0, t1];E) ∩C1((t0, t1);E) where t1 = t1(t0, x0) > t0.

For a proof of Theorem 2.8, the reader is referred to [He, Theorem 3.3.3],Pazy [Pa, Chapter 6, Theorem 3.1] and also Taira [Ta4, Theorem 1.18].

2.2 Markov Processes and Feller Semigroups

This section provides a brief description of basic definitions and results aboutMarkov processes and a class of semigroups (Feller semigroups) associatedwith Markov processes. In Subsection 2.2.6 we prove various generation the-orems of Feller semigroups by using the Hille–Yosida theory of semigroups(Theorems 2.16 and 2.18) which form a functional analytic background forthe proof of Theorem 1.4. The results discussed here are adapted from


Blumenthal–Getoor [BG], Dynkin [Dy2], Lamperti [La], Revuz–Yor [RY] andalso Taira [Ta2, Chapter 9]. The semigroup approach to Markov processes canbe traced back to the pioneering work of Feller [Fe1] and [Fe2] in early 1950s(cf. [BCP], [SU], [Ta3]).

2.2.1 Markov Processes

In 1828 the English botanist R. Brown observed that pollen grains suspendedin water move chaotically, incessantly changing their direction of motion. Thephysical explanation of this phenomenon is that a single grain suffers innumer-able collisions with the randomly moving molecules of the surrounding water.A mathematical theory for Brownian motion was put forward by A. Einsteinin 1905 (cf. [Ei]). Let p(t, x, y) be the probability density function that a one-dimensional Brownian particle starting at position x will be found at positiony at time t. Einstein derived the following formula from statistical mechanicalconsiderations:

p(t, x, y) =1√

2πDtexp

[− (y − x)2

2Dt

].

Here D is a positive constant determined by the radius of the particle, theinteraction of the particle with surrounding molecules, temperature and theBoltzmann constant. This gives an accurate method of measuring Avogadro’snumber by observing particles. Einstein’s theory was experimentally testedby J. Perrin between 1906 and 1909.

Brownian motion was put on a firm mathematical foundation for the firsttime by N. Wiener in 1923 ([Wi]). Let Ω be the space of continuous functionsω : [0,∞) �→ R with coordinates xt(ω) = ω(t) and let F be the smallestσ-algebra in Ω which contains all sets of the form {ω ∈ Ω : a ≤ xt(ω) < b},t ≥ 0, a < b. Wiener constructed probability measures Px, x ∈ R, on F forwhich the following formula holds:

Px{ω ∈ Ω : a1 ≤ xt1(ω) < b1, a2 ≤ xt2(ω) < b2, . . . , an ≤ xtn(ω) < bn}=∫ b1

a1

∫ b2a2. . .∫ bn

anp(t1, x, y1)p(t2 − t1, y1, y2) . . .

p(tn − tn−1, yn−1, yn) dy1 dy2 . . . dyn,

0 < t1 < t2 < . . . < tn <∞.

This formula expresses the “starting afresh” property of Brownian motionthat if a Brownian particle reaches a position, then it behaves subsequentlyas though that position had been its initial position. The measure Px is calledthe Wiener measure starting at x.

P. Levy found another construction of Brownian motion, and gave a pro-found description of qualitative properties of the individual Brownian path inhis book: Processus stochastiques et mouvement brownien (1948).

Markov processes are an abstraction of the idea of Brownian motion. LetK be a locally compact, separable metric space and let B be the σ-algebra


of all Borel sets in K, that is, the smallest σ-algebra containing all open setsin K. Let (Ω,F , P ) be a probability space. A function X(ω) defined on Ωtaking values in K is called a random variable if it satisfies the condition

X−1(E) = {ω ∈ Ω : X(ω) ∈ E} ∈ F for all E ∈ B.

We express this by saying that X is F/B-measurable. A family {xt}t≥0 ofrandom variables is called a stochastic process, and it may be thought of asthe motion in time of a physical particle. The space K is called the state spaceand Ω the sample space. For a fixed ω ∈ Ω, the function xt(ω), t ≥ 0, definesin the state space K a trajectory or path of the process corresponding to thesample point ω.

In this generality the notion of a stochastic process is of course not sointeresting. The most important class of stochastic processes is the class ofMarkov processes which is characterized by the Markov property. Intuitively,this is the principle of the lack of any “memory” in the system. More precisely,(temporally homogeneous) Markov property is that the prediction of subse-quent motion of a particle, knowing its position at time t, depends neither onthe value of t nor on what has been observed during the time interval [0, t);that is, a particle “starts afresh”.

Now we introduce a class of Markov processes which we will deal with inthis book (cf. [Dy2], [BG], [RY]).

Assume that we are given the following:

(1) A locally compact, separable metric space K and the σ-algebra B of allBorel sets in K. A point ∂ is adjoined to K as the point at infinity if K isnot compact, and as an isolated point if K is compact (see Figure 2.10).We let

K∂ = K ∪ {∂},B∂ = the σ-algebra in K∂ generated by B.

K∂K

∂

Fig. 2.10.

(2) The space Ω of all mappings ω : [0,∞] → K∂ such that ω(∞) = ∂ andthat if ω(t) = ∂ then ω(s) = ∂ for all s ≥ t. Let ω∂ be the constant mapω∂(t) = ∂ for all t ∈ [0,∞].


(3) For each t ∈ [0,∞], the coordinate map xt defined by xt(ω) = ω(t), ω ∈ Ω.(4) For each t ∈ [0,∞], a mapping ϕt : Ω → Ω defined by (ϕtω)(s) = ω(t+s),

ω ∈ Ω. Note that ϕ∞ω = ω∂ and xt ◦ ϕs = xt+s for all t, s ∈ [0,∞].(5) A σ-algebra F in Ω and an increasing family {Ft}0≤t≤∞ of sub-σ-algebras

of F .(6) For each x ∈ K∂, a probability measure Px on (Ω,F).

We say that these elements define a (temporally homogeneous) Markov processX = (xt,F ,Ft, Px) if the following four conditions are satisfied:

(i) For each 0 ≤ t <∞, the function xt is Ft/B∂-measurable, that is,

{xt ∈ E} = {ω ∈ Ω : xt(ω) ∈ E} ∈ Ft for all E ∈ B∂ .

(ii) For all 0 ≤ t <∞ and E ∈ B, the function

pt(x,E) = Px{xt ∈ E} (2.25)

is a Borel measurable function of x ∈ K.(iii) Px{ω ∈ Ω : x0(ω) = x} = 1 for each x ∈ K∂.(iv) For all t, h ∈ [0,∞], x ∈ K∂ and E ∈ B∂ , we have the formula

Px{xt+h ∈ E | Ft} = ph(xt, E) a. e.,

or equivalently,

Px(A ∩ {xt+h ∈ E}) =∫

A

ph(xt(ω), E) dPx(ω) for all A ∈ Ft.

Here is an intuitive way of thinking about the above definition of a Markovprocess. The sub-σ-algebra Ft may be interpreted as the collection of eventswhich are observed during the time interval [0, t]. The value Px(A), A ∈ F ,may be interpreted as the probability of the event A under the condition thata particle starts at position x; hence the value pt(x,E) expresses the transitionprobability that a particle starting at position x will be found in the set E attime t (see Figure 2.11). The function pt(x, ·) is called the transition functionof the process X . The transition function pt(x, ·) specifies the probabilitystructure of the process. The intuitive meaning of the crucial condition (iv) isthat the future behavior of a particle, knowing its history up to time t, is thesame as the behavior of a particle starting at xt(ω), that is, a particle startsafresh. A particle moves in the space K until it “dies” at the time when itreaches the point ∂; hence the point ∂ is called the terminal point.

With this interpretation in mind, we let

ζ(ω) = inf{t ∈ [0,∞] : xt(ω) = ∂}.

The random variable ζ is called the lifetime of the process X .


E

x

t

Fig. 2.11.

2.2.2 Markov Transition Functions

In the first works devoted to Markov processes, the most fundamental wasA. N. Kolmogorov’s work ([Ko]) where the general concept of a Markov tran-sition function was introduced for the first time and an analytic method ofdescribing Markov transition functions was proposed. From the point of viewof analysis, the transition function is something more convenient than theMarkov process itself. In fact, it can be shown that the transition functionsof Markov processes generate solutions of certain parabolic partial differentialequations such as the classical diffusion equation; and, conversely, these differ-ential equations can be used to construct and study the transition functionsand the Markov processes themselves.

In the 1950s, the theory of Markov processes entered a new period of inten-sive development. We can associate with each transition function in a naturalway a family of bounded linear operators acting on the space of continuousfunctions on the state space, and the Markov property implies that this fam-ily forms a semigroup. The Hille–Yosida theory of semigroups in functionalanalysis made possible further progress in the study of Markov processes, aswill be shown in Subsection 2.2.5.

Our first job is thus to give the precise definition of a transition functionadapted to the theory of semigroups:

Definition 2.1. Let (K, ρ) be a locally compact, separable metric space andlet B be the σ-algebra of all Borel sets in K. A function pt(x,E), defined forall t ≥ 0, x ∈ K and E ∈ B, is called a (temporally homogeneous) Markovtransition function on K if it satisfies the following four conditions:

(a) pt(x, ·) is a non-negative measure on B and pt(x,K) ≤ 1 for all t ≥ 0 andx ∈ K.

(b) pt(·, E) is a Borel measurable function for all t ≥ 0 and E ∈ B.(c) p0(x, {x}) = 1 for all x ∈ K.


(d) (The Chapman–Kolmogorov equation) For all t, s ≥ 0, x ∈ K and E ∈ B,we have the equation

pt+s(x,E) =∫

K

pt(x, dy)ps(y,E). (2.26)

K

0

t

t+s

y

E

x

Fig. 2.12.

Here is an intuitive way of thinking about the above definition of a Markovtransition function. The value pt(x,E) expresses the transition probabilitythat a physical particle starting at position x will be found in the set Eat time t. The Chapman–Kolmogorov equation (2.26) expresses the idea thata transition from the position x to the set E in time t + s is composed of atransition from x to some position y in time t, followed by a transition fromy to the set E in the remaining time s; the latter transition has probabilityps(y,E) which depends only on y (see Figure 2.12). Thus a particle “startsafresh”; this property is called the Markov property.

The Chapman–Kolmogorov equation (2.26) asserts that pt(x,K) is mono-tonically increasing as t ↓ 0, so that the limit

p+0(x,K) = limt↓0

pt(x,K)

exists.A transition function pt(x, ·) is said to be normal if it satisfies the condition

p+0(x,K) = 1 for all x ∈ K.

The next theorem, due to Dynkin [Dy1, Chapter 4, Section 2], justifies thedefinition of a transition function, and hence it will be fundamental for ourfurther study of Markov processes:


Theorem 2.9. For every Markov process, the function pt(x, ·), defined byformula (2.25), is a Markov transition function. Conversely, every normalMarkov transition function corresponds to some Markov process.

Here are some important examples of normal transition functions on theline R = (−∞,∞):

Example 2.1 (Uniform motion). If t ≥ 0, x ∈ R and E ∈ B, we let

pt(x,E) = χE(x+ vt),

where v is a constant, and χE(y) = 1 if y ∈ E and = 0 if y �∈ E.

This process, starting at x, moves deterministically with constant veloc-ity v.

Example 2.2 (Poisson process). If t ≥ 0, x ∈ R and E ∈ B, we let

pt(x,E) = e−λt∞∑

n=0

(λt)n

n!χE(x+ n),

where λ is a positive constant.

This process, starting at x, advances one unit by jumps, and the probabilityof n jumps during the time 0 and t is equal to e−λt(λt)n/n!.

Example 2.3 (Brownian motion). If t > 0, x ∈ R and E ∈ B, we let

pt(x,E) =1√2πt

∫

E

exp[− (y − x)2

2t

]dy,

andp0(x,E) = χE(x).

This is a mathematical model of one-dimensional Brownian motion. Itscharacter is quite different from that of the Poisson process; the transitionfunction pt(x,E) satisfies the condition

pt(x, [x− ε, x+ ε]) = 1− o(t) as t ↓ 0,

for all ε > 0 and x ∈ R. This means that the process never stands still, as doesthe Poisson process. Indeed, this process changes state not by jumps but bycontinuous motion. A Markov process with this property is called a diffusionprocess.

Example 2.4 (Brownian motion with constant drift). If t > 0, x ∈ R andE ∈ B, we let

pt(x,E) =1√2πt

∫

E

exp[− (y −mt− x)2

2t

]dy,

andp0(x,E) = χE(x),

where m is a constant.


This represents Brownian motion with a constant drift of magnitude msuperimposed; the process can be represented as {xt + mt}, where {xt} isBrownian motion on R.

Example 2.5 (Cauchy process). If t > 0, x ∈ R and E ∈ B, we let

pt(x,E) =1π

∫

E

t

t2 + (y − x)2 dy,


This process can be thought of as the “trace” on the real line of trajec-tories of two-dimensional Brownian motion, and it moves by jumps (see [Kn,Lemma 2.12]). More precisely, if B1(t) and B2(t) are two independent Brow-nian motions and if T is the first passage time of B1(t) to x, then B2(T ) hasthe Cauchy density

1π

|x|x2 + y2

, −∞ < y <∞.

Here are two more examples of diffusion processes on the half line R+ =[0,∞) in which we must take account of the effect of the boundary point 0:

Example 2.6 (Reflecting barrier Brownian motion). If t > 0, x ∈ R+ andE ∈ B, we let

pt(x,E) =1√2πt

(∫

E

exp[− (y − x)2

2t

]dy +

∫

E

exp[− (y + x)2

2t

]dy

),


This represents Brownian motion with a reflecting barrier at x = 0; theprocess may be represented as {|xt|}, where {xt} is Brownian motion on R.Indeed, since {|xt|} goes from x to y if {xt} goes from x to ±y due to thesymmetry of the transition function in Example 2.3 about x = 0, it followsthat

pt(x,E) = Px{|xt| ∈ E}

=1√2πt

(∫

E

exp[− (y − x)2

2t

]dy +

∫

E

exp[− (y + x)2

2t

]dy

).

Example 2.7 (Sticking barrier Brownian motion). If t > 0, x ∈ R+ and E ∈ B,we let

pt(x,E) =1√2πt

(∫

E

exp[− (y − x)2

2t

]dy −

∫

E

exp[− (y + x)2

2t

]dy

)

+(

1− 1√2πt

∫ x

−x

exp[−z

2

2t

]dz

)χE(0),



This represents Brownian motion with a sticking barrier at x = 0. Whena Brownian particle reaches the boundary point 0 for the first time, insteadof reflecting it sticks there forever; in this case the state 0 is called a trap.

2.2.3 Path Functions of Markov Processes

It is naturally interesting and important to ask the following problem:

Problem. Given a Markov transition function pt(x, ·), under which condi-tions on pt(x, ·) does there exist a Markov process with transition functionpt(x, ·) whose paths are almost surely continuous?

A Markov process X = (xt,F ,Ft, Px) is said to be right continuous pro-vided that we have, for each x ∈ K,

Px{ω ∈ Ω : the mapping t �→ xt(ω) is a right continuous functionfrom [0,∞) into K∂} = 1.

Furthermore, we say that X is continuous provided that we have, for eachx ∈ K,

Px{ω ∈ Ω : the mapping t �→ xt(ω) is a continuous functionfrom [0, ζ(ω)) into K∂} = 1,

where ζ is the lifetime of the process X .Now we give some useful criteria for path continuity in terms of

Markov transition functions (see Dynkin [Dy1, Chapter 6], [Dy2, Chapter 3,Section 2]):

Theorem 2.10. Let (K, ρ) be a locally compact, separable metric space andlet pt(x, ·) be a normal Markov transition function on K.(i) Assume that the following two conditions are satisfied:

(L) For each s > 0 and each compact E ⊂ K, we have the condition

limx→∂

sup0≤t≤s

pt(x,E) = 0.

(M) For each ε > 0 and each compact E ⊂ K, we have the condition

limt↓0

supx∈E

pt(x,K \ Uε(x)) = 0,

where Uε(x) = {y ∈ K : ρ(y, x) < ε} is an ε-neighborhood of x.


Then there exists a Markov process X with transition function pt(x, ·)whose paths are right continuous on [0,∞) and have left-hand limits on [0, ζ)almost surely.(ii) Assume that condition (L) and the following condition (N) (replacingcondition (M)) are satisfied:

(N) For each ε > 0 and each compact E ⊂ K, we have the condition

limt↓0

1t

supx∈E

pt(x,K \ Uε(x)) = 0,

or equivalently

supx∈E

pt(x,K \ Uε(x)) = o(t) as t ↓ 0.

Then there exists a Markov process X with transition function pt(x, ·)whose paths are almost surely continuous on [0, ζ).

Remark 2.2. It is known (see Dynkin [Dy1, Lemma 6.2]) that if the paths ofa Markov process are right continuous, then the transition function pt(x, ·)satisfies the condition

limt↓0

pt(x, Uε(x)) = 1 for all x ∈ K.

2.2.4 Strong Markov Processes and Transition Functions

A Markov process is called a strong Markov process if the “starting afresh”property holds not only for every fixed moment but also for suitable randomtimes.

We shall formulate precisely this “strong” Markov property. Let X =(xt,F ,Ft, Px) be a Markov process. A mapping τ :Ω → [0,∞] is called astopping time or Markov time with respect to {Ft} if it satisfies the condition

{τ ≤ t} = {ω ∈ Ω : τ(ω) ≤ t} ∈ Ft for all t ∈ [0,∞).

Intuitively, this means that the events {τ ≤ t} depend on the process only upto time t, but not on the “future” after time t. It should be noticed that anynon-negative constant mapping is a stopping time.

If τ is a stopping time with respect to {Ft}, we let

Fτ = {A ∈ F : A ∩ {τ ≤ t} ∈ Ft for all t ∈ [0,∞)}.

Intuitively, we may think of Fτ as the “past” up to the random time τ . It iseasy to verify that Fτ is a σ-algebra. If τ ≡ t0 for some constant t0 ≥ 0, thenFτ reduces to Ft0 .

For each t ∈ [0,∞], we define a mapping

Φt: [0, t]×Ω −→ K∂


by the formulaΦt(s, ω) = xs(ω).

A Markov process X = (xt,F ,Ft, Px) is said to be progressively measurablewith respect to {Ft} if the mapping Φt is B[0,t] ×Ft/B∂-measurable for eacht ∈ [0,∞], that is, if we have the condition

Φ−1t (E) = {Φt ∈ E} ∈ B[0,t] ×Ft for all E ∈ B∂ .

Here B[0,t] is the σ-algebra of all Borel sets in the interval [0, t] and B∂ is theσ-algebra in K∂ generated by B. It should be noticed that if X is progressivelymeasurable and if τ is a stopping time, then the mapping xτ :ω �→ xτ(ω)(ω) isFτ/B∂-measurable.

The next definition expresses the idea of “starting afresh” at random times:

Definition 2.2. We say that a progressively measurable Markov process X =(xt,F ,Ft, Px) has the strong Markov property with respect to {Ft} if thefollowing condition is satisfied: For all h ≥ 0, x ∈ K∂ , E ∈ B∂ and all stoppingtimes τ , we have the formula

Px{xτ+h ∈ E | Fτ} = ph(xτ , E),

or equivalently,

Px(A ∩ {xτ+h ∈ E}) =∫

A

ph(xτ(ω)(ω), E) dPx(ω) for all A ∈ Fτ .

We shall state a simple criterion for the strong Markov property in termsof transition functions.

Let (K, ρ) be a locally compact, separable metric space. We add a point∂ to the metric space K as the point at infinity if K is not compact, and asan isolated point if K is compact; so the space K∂ = K ∪ {∂} is compact(see Figure 2.10). Let C(K) be the space of real-valued, bounded continuousfunctions f(x) on K; the space C(K) is a Banach space with the supremumnorm

‖f‖∞ = supx∈K|f(x)|.

We say that a function f ∈ C(K) converges to zero as x → ∂ if, for eachε > 0, there exists a compact subset E of K such that

|f(x)| < ε for all x ∈ K \ E,

and we then write limx→∂ f(x) = 0. We let

C0(K) ={f ∈ C(K) : lim

x→∂f(x) = 0

}.

The space C0(K) is a closed subspace of C(K); hence it is a Banach space.Note that C0(K) may be identified with C(K) if K is compact.


Now we introduce a useful convention as follows:

Any real-valued function f(x) on K is extended tothe space K∂ = K ∪ {∂}by setting f(∂) = 0.

From this point of view, the space C0(K) is identified with the subspace ofC(K∂) which consists of all functions f(x) satisfying the condition f(∂) = 0:

C0(K) = {f ∈ C(K∂) : f(∂) = 0} .

Furthermore, we can extend a Markov transition function pt(x, ·) on K to aMarkov transition function p′t(x, ·) on K∂ by the formulas:

⎧⎨

⎩

p′t(x,E) = pt(x,E) for all x ∈ K and E ∈ B,p′t(x, {∂}) = 1− pt(x,K) for all x ∈ K,p′t(∂,K) = 0, p′t(∂, {∂}) = 1.

Intuitively this means that a Markovian particle moves in the space K untilit “dies” at the time when it reaches the point ∂; hence the point ∂ is calledthe terminal point.

Now we introduce some conditions on the measures pt(x, ·) related to con-tinuity in x ∈ K, for fixed t ≥ 0:

Definition 2.3. (i) A Markov transition function pt(x, ·) is called a Fellerfunction if the function

Ttf(x) =∫

K

pt(x, dy)f(y)

is a continuous function of x ∈ K whenever f is in C(K), that is, if we havethe condition

f ∈ C(K) =⇒ Ttf ∈ C(K).

(ii) We say that pt(x, ·) is a C0-function if the space C0(K) is an invariantsubspace of C(K) for the operators Tt:

f ∈ C0(K) =⇒ Ttf ∈ C0(K).

Remark 2.3. The Feller property is equivalent to saying that the measurespt(x, ·) depend continuously on x ∈ K in the usual weak topology, for everyfixed t ≥ 0.

The next theorem gives a useful criterion for the strong Markov property(see [Dy1, Theorem 5.10]):

Theorem 2.11. If the transition function pt(x, ·) of a right continuous Mar-kov process X has the C0-property, then X is a strong Markov process.


Furthermore, we state a simple criterion for the strong Markov property interms of Markov transition functions. To do this, we introduce the followingdefinition:

Definition 2.4. A Markov transition function pt(x, ·) on K is said to be uni-formly stochastically continuous on K if the following condition is satisfied:For each ε > 0 and each compact E ⊂ K, we have the condition

limt↓0

supx∈E

[1− pt(x, Uε(x))] = 0, (2.27)

where Uε(x) = {y ∈ K : ρ(y, x) < ε} is an ε-neighborhood of x.

It should be emphasized that every uniformly stochastically continuoustransition function is normal and satisfies condition (M) in Theorem 2.10. Bycombining part (i) of Theorem 2.10 and Theorem 2.11, we obtain the followingresult (see [Dy1, Theorem 6.3]):

Theorem 2.12. Assume that a uniformly stochastically continuous, C0-transition function pt(x, ·) satisfies condition (L). Then it is the transitionfunction of some strong Markov process X whose paths are right continuousand have no discontinuities other than jumps.

A continuous strong Markov process is called a diffusion process.The next theorem states a sufficient condition for the existence of a diffu-

sion process with a prescribed Markov transition function:

Theorem 2.13. Assume that a uniformly stochastically continuous, C0-transition function pt(x, ·) satisfies conditions (L) and (N). Then it is thetransition function of some diffusion process X .

This is an immediate consequence of part (ii) of Theorem 2.10 andTheorem 2.12.

2.2.5 Markov Transition Functions and Feller Semigroups

The Feller or C0-property deals with continuity of a Markov transition func-tion pt(x,E) in x, and does not, by itself, have no concern with continuityin t. We give a necessary and sufficient condition on pt(x,E) in order that itsassociated operators {Tt}t≥0, defined by the formula

Ttf(x) =∫

K

pt(x, dy)f(y), f ∈ C0(K), (2.28)

is strongly continuous in t on the space C0(K):

lims↓0‖Tt+sf − Ttf‖∞ = 0, f ∈ C0(K). (2.29)


Then we have the following (cf. [Ta2, Theorem 9.2.3]):

Theorem 2.14. Let pt(x, ·) be a C0-transition function on K. Then the as-sociated operators {Tt}t≥0, defined by formula (2.28) is strongly continuousin t on C0(K) if and only if pt(x, ·) is uniformly stochastically continuous onK and satisfies condition (L).

Proof. (i) The “if” part: Since continuous functions with compact support aredense in C0(K), it suffices to prove the strong continuity of {Tt} at t = 0:

limt↓0‖Ttf − f‖∞ = 0 (2.30)

for all such functions f .For any compact subset E of K containing the support supp f of f , we

have the inequality

‖Ttf − f‖∞ ≤ supx∈E|Ttf(x)− f(x)| + sup

x∈K\E

|Ttf(x)|

≤ supx∈E|Ttf(x)− f(x)| + ‖f‖∞ · sup

x∈K\E

pt(x, supp f). (2.31)

However, condition (L) implies that, for each ε > 0, we can find a compactsubset E of K such that, for all sufficiently small t > 0,

supx∈K\E

pt(x, supp f) < ε. (2.32)

On the other hand, we have, for each δ > 0,

Ttf(x)− f(x) =∫

Uδ(x)

pt(x, dy)(f(y)− f(x))

+∫

K\Uδ(x)

pt(x, dy)(f(y)− f(x))− f(x)(1 − pt(x,K)),

and hence

supx∈E|Ttf(x)− f(x)|

≤ supρ(x,y)<δ

|f(y)− f(x)|+ 3‖f‖∞ · supx∈E

[1− pt(x, Uδ(x))] .

Since the function f(x) is uniformly continuous, we can choose a positiveconstant δ such that

supρ(x,y)<δ

|f(y)− f(x)| < ε.

Furthermore, it follows from condition (2.27) with ε := δ (the uniform stochas-tic continuity of pt(x, ·)) that, for all sufficiently small t > 0,


supx∈E

[1− pt(x, Uδ(x))] < ε.

Hence we have, for all sufficiently small t > 0,

supx∈E|Ttf(x)− f(x)| < ε(1 + 3‖f‖∞). (2.33)

Therefore, by carrying inequalities (2.32) and (2.33) into inequality (2.31)we obtain that, for all sufficiently small t > 0,

‖Ttf − f‖∞ < ε(1 + 4‖f‖∞).

This proves the desired formula (2.30), that is, the strong continuity (2.29) of{Tt}.

(ii) The “only if” part: For any x ∈ K and ε > 0, we define a continuousfunction fx(y) by the formula (see Figure 2.13)

fx(y) ={

1− 1ερ(x, y) if ρ(x, y) ≤ ε,

0 if ρ(x, y) > ε.(2.34)

fx

xε ε

Fig. 2.13.

Let E be an arbitrary compact subset of K. Then, for all sufficiently smallε > 0, the functions fx, x ∈ E, are in C0(K) and satisfy the condition

‖fx − fz‖∞ ≤1ερ(x, z) for all x, z ∈ E. (2.35)

However, for any δ > 0, by the compactness of E we can find a finite numberof points x1, x2, . . ., xn of E such that

E =n⋃

k=1

Uδε/4(xk),

and hencemin

1≤k≤nρ(x, xk) ≤ δε

4for all x ∈ E.

Thus, by combining this inequality with inequality (2.35) with z := xk weobtain that


min1≤k≤n

‖fx − fxk‖∞ ≤

δ

4for all x ∈ E. (2.36)

Now we have, by formula (2.34),

0 ≤ 1− pt(x, Uε(x)) ≤ 1−∫

K∂

pt(x, dy)fx(y)

= fx(x)− Ttfx(x)≤ ‖fx − Ttfx‖∞≤ ‖fx − fxk

‖∞ + ‖fxk− Ttfxk

‖∞+‖Ttfxk

− Ttfx‖∞≤ 2‖fx − fxk

‖∞ + ‖fxk− Ttfxk

‖∞ for all x ∈ E.

In view of inequality (2.36), the first term on the last inequality is bounded byδ/2 for the right choice of k. Furthermore, it follows from the strong continuity(2.30) of {Tt} that the second term tends to zero as t ↓ 0 for each k = 1, 2,. . ., n.

Consequently, we have, for all sufficiently small t > 0,

supx∈E

[1− pt(x, Uε(x))] ≤ δ.

This proves the desired condition (2.27), that is, the uniform stochastic con-tinuity of pt(x, ·).

Finally, it remains to verify condition (L). Assume, to the contrary, that:For some s > 0 and some compact E ⊂ K, there exist a positive constant

ε0, a sequence {tk}, tk ↓ t (0 ≤ t ≤ s) and a sequence {xk}, xk → ∂, suchthat

ptk(xk, E) ≥ ε0. (2.37)

Now we take a relatively compact subset U of K containing E, and let(see Figure 2.14)

f(x) =ρ(x,K \ U)

ρ(x,E) + ρ(x,K \ U).

Then it follows that the function f(x) is in C0(K) and satisfies the condi-tion

Ttf(x) =∫

K

pt(x, dy)f(y) ≥ pt(x,E) ≥ 0.

Therefore, by combining this inequality with inequality (2.37) we obtain that

Ttkf(xk) ≥ ptk

(xk, E) ≥ ε0. (2.38)

However, we have the inequality

Ttkf(xk) ≤ ‖Ttk

f − Ttf‖∞ + Ttf(xk). (2.39)


E

U

f

Fig. 2.14.

Since the semigroup {Tt} is strongly continuous and since we have the asser-tion

limk→∞

Ttf(xk) = Ttf(∂) = 0,

we can let k →∞ in inequality (2.39) to obtain that

lim supk→∞

Ttkf(xk) = 0.

This contradicts inequality (2.38).The proof of Theorem 2.14 is now complete. �

A family {Tt}t≥0 of bounded linear operators acting on the space C0(K) iscalled a Feller semigroup on K if it satisfies the following three conditions:

(i) Tt+s = Tt · Ts, t, s ≥ 0 (the semigroup property); T0 = I.(ii) The family {Tt} is strongly continuous in t for all t ≥ 0:

lims↓0‖Tt+sf − Ttf‖∞ = 0, f ∈ C0(K).

(iii) The family {Tt} is non-negative and contractive on C0(K):

f ∈ C0(K), 0 ≤ f(x) ≤ 1 on K =⇒ 0 ≤ Ttf(x) ≤ 1 on K.

Rephrased, Theorem 2.14 gives a characterization of Feller semigroups interms of Markov transition functions:

Theorem 2.15. If pt(x, ·) is a uniformly stochastically continuous C0-transition function on K and satisfies condition (L), then its associatedoperators {Tt}t≥0, defined by formula (2.28), form a Feller semigroup on K.

Conversely, if {Tt}t≥0 is a Feller semigroup on K, then there exists a uni-formly stochastically continuous C0-transition pt(x, ·) on K, satisfying condi-tion (L), such that formula (2.28) holds.

The most important applications of Theorem 2.15 are of course in thesecond statement.


2.2.6 Generation Theorems of Feller Semigroups

In this subsection we prove various generation theorems of Feller semigroupsby using the Hille–Yosida theory of semigroups.

If {Tt}t≥0 is a Feller semigroup on K, we define its infinitesimal generatorA by the formula

Au = limt↓0

Ttu− ut

, u ∈ C0(K), (2.40)

provided that the limit (2.40) exists in the space C0(K). More precisely, thegenerator A is a linear operator from C0(K) into itself defined as follows:

(1) The domain D(A) of A is the set

D(A) = {u ∈ C0(K) : the limit (2.40) exists} .

(2) Au = limt↓0 Ttu−ut , u ∈ D(A).

The next theorem is a version of the Hille–Yosida theorem adapted to thepresent context (cf. [Ta2, Theorem 9.3.1 and Corollary 9.3.2]):

Theorem 2.16 (Hille–Yosida). (i) Let {Tt}t≥0 be a Feller semigroup onK and let A be its infinitesimal generator. Then we have the following fourassertions:

(a) The domain D(A) is dense in the space C0(K).(b) For each α > 0, the equation (αI − A)u = f has a unique solution u

in D(A) for any f ∈ C0(K). Hence, for each α > 0 the Green operator(αI −A)−1 : C0(K)→ C0(K) can be defined by the formula

u = (αI −A)−1f, f ∈ C0(K).

(c) For each α > 0, the operator (αI −A)−1 is non-negative on C0(K):

f ∈ C0(K), f(x) ≥ 0 on K =⇒ (αI −A)−1f(x) ≥ 0 on K.

(d) For each α > 0, the operator (αI −A)−1 is bounded on C0(K) with norm

‖(αI −A)−1‖ ≤ 1α.

(ii) Conversely, if A is a linear operator from C0(K) into itself satisfyingcondition (a) and if there is a non-negative constant α0 such that, for allα > α0, conditions (b) through (d) are satisfied, then A is the infinitesimalgenerator of some Feller semigroup {Tt}t≥0 on K.

Proof. In view of the Hille–Yosida theory (see [Yo, Chapter IX, Section 7]), itsuffices to show that the semigroup {Tt}t≥0 is non-negative if and only if itsresolvents (Green operators) {(αI −A)−1}α>α0 are non-negative.


The “only if” part is an immediate consequence of the following expressionof (αI −A)−1 in terms of the semigroup {Tt}:

(αI −A)−1 =∫ ∞

0

exp[−αt]Tt dt, α > 0.

On the other hand, the “if” part follows from the expression of the semi-group Tt(α) in terms of the Yosida approximation Jα = α(αI −A)−1:

Tt(α) = exp[−αt] exp [αtJα] = exp[−αt]∞∑

n=0

(αt)n

n!Jn

α ,

and the definition of the semigroup Tt:

Tt = limα→∞Tt(α).

The proof of Theorem 2.16 is complete. �

Corollary 2.17. Let K be a compact metric space and let A be the infinitesi-mal generator of a Feller semigroup on K. Assume that the constant function1 belongs to the domain D(A) of A and that we have, for some constant c,

A1(x) ≤ −c on K. (2.41)

Then the operator A′ = A + cI is the infinitesimal generator of some Fellersemigroup on K.

Proof. It follows from an application of part (i) of Theorem 2.16 that, for allα > c the operators

(αI −A′)−1 = ((α− c)I −A)−1

are defined and non-negative on the whole space C(K). However, in view ofinequality (2.41) we obtain that

α ≤ α− (A1 + c) = (αI −A′)1 on K,

so thatα(αI −A′)−11 ≤ (αI −A′)−1(αI −A′)1 = 1 on K.

Hence we have, for all α > c,

‖(αI −A′)−1‖ = ‖(αI −A′)−11‖∞ ≤1α.

Therefore, by applying part (ii) of Theorem 2.16 to the operator A′ wefind that A′ is the infinitesimal generator of some Feller semigroup on K.

The proof of Corollary 2.17 is complete. �


Now we write down explicitly the infinitesimal generators of Feller semi-groups associated with the transition functions in Examples 2.1 through 2.7(cf. [DY]).

Example 2.8 (Uniform motion). K = R and{D(A) = {f ∈ C0(K) : f ′ ∈ C0(K)},Af = vf ′, f ∈ D(A).

Example 2.9 (Poisson process). K = R and{D(A) = C0(K),Af(x) = λ(f(x + 1)− f(x)), f ∈ D(A).

The operator A is not “local”; the value Af(x) depends on the values f(x)and f(x+ 1). This reflects the fact that the Poisson process changes state byjumps.

Example 2.10 (Brownian motion). K = R and{D(A) = {f ∈ C0(K) : f ′ ∈ C0(K), f ′′ ∈ C0(K)},Af = 1

2f′′, f ∈ D(A).

The operator A is “local”, that is, the value Af(x) is determined by thevalues of f in an arbitrary small neighborhood of x. This reflects the fact thatBrownian motion changes state by continuous motion.

Example 2.11 (Brownian motion with constant drift). K = R and{D(A) = {f ∈ C0(K) : f ′ ∈ C0(K), f ′′ ∈ C0(K)},Af = 1

2f′′ +mf ′, f ∈ D(A).

Example 2.12 (Cauchy process). K = R and, the domain D(A) contains C2

functions on K with compact support, and the infinitesimal generator A is ofthe form

Af(x) =1π

∫ +∞

−∞(f(x+ y)− f(x))

dy

y2.

The operator A is not “local”, which reflects the fact that the Cauchyprocess changes state by jumps.

Example 2.13 (Reflecting barrier Brownian motion). K = [0,∞) and{D(A) = {f ∈ C0(K) : f ′ ∈ C0(K), f ′′ ∈ C0(K), f ′(0) = 0},Af = 1

2f′′, f ∈ D(A).


Example 2.14 (Sticking barrier Brownian motion). K = [0,∞) and{D(A) = {f ∈ C0(K) : f ′ ∈ C0(K), f ′′ ∈ C0(K), f ′′(0) = 0},Af = 1

2f′′, f ∈ D(A).

Finally, here are two more examples where it is difficult to begin with atransition function and the infinitesimal generator is the basic tool of describ-ing the process.

Example 2.15 (Sticky barrier Brownian motion). K = [0,∞) and{D(A) = {f ∈ C0(K) : f ′ ∈ C0(K), f ′′ ∈ C0(K), f ′(0)− αf ′′(0) = 0},Af = 1

2f′′, f ∈ D(A).

Here α is a positive constant.This process may be thought of as a “combination” of the reflecting and

sticking Brownian motions. The reflecting and sticking cases are obtained byletting α→ 0 and α→∞, respectively.

Example 2.16 (Absorbing barrier Brownian motion). K = [0,∞) where theboundary point 0 is identified with the point at infinity ∂.

{D(A) = {f ∈ C0(K) : f ′ ∈ C0(K), f ′′ ∈ C0(K), f(0) = 0},Af = 1

2f′′, f ∈ D(A).

This represents Brownian motion with an absorbing barrier at x = 0; aBrownian particle “dies” at the first moment when it hits the boundary x = 0.Namely, the point 0 is the terminal point.

It is worth pointing out here that a strong Markov process cannot stay ata single position for a positive length of time and then leave that position bycontinuous motion; it must either jump away or leave instantaneously.

We give a simple example of a strong Markov process which changesstate not by continuous motion but by jumps when the motion reaches theboundary.

Example 2.17. K = [0,∞).⎧⎨

⎩

D(A) = {f ∈ C0(K) ∩ C2(K) : f ′ ∈ C0(K), f ′′ ∈ C0(K),f ′′(0) = 2c

∫∞0

(f(y)− f(0))dF (y),Af = 1

2f′′, f ∈ D(A).

Here c is a positive constant and F is a distribution function on the interval(0,∞).

This process may be interpreted as follows. When a Brownian particlereaches the boundary x = 0, it stays there for a positive length of time andthen jumps back to a random point, chosen with the function F , in the interior


(0,∞). The constant c is the parameter in the “waiting time” distribution atthe boundary x = 0. We remark that the boundary condition

f ′′(0) = 2c∫ ∞

0

(f(y)− f(0)) dF (y)

depends on the values of f far away from the boundary x = 0, unlike theboundary conditions in Examples 2.13 through 2.16.

Although Theorem 2.16 asserts precisely when a linear operator A is theinfinitesimal generator of some Feller semigroup, it is usually difficult to verifyconditions (b) through (d). So we give useful criteria in terms of the maximumprinciple (see [BCP], [SU], [Ra], [Ta2, Theorem 9.3.3 and Corollary 9.3.4]):

Theorem 2.18 (Hille–Yosida–Ray). Let K be a compact metric space.Then we have the following two assertions:

(i) Let B be a linear operator from C(K) = C0(K) into itself, and assumethat:(α) The domain D(B) of B is dense in the space C(K).(β) There exists an open and dense subset K0 of K such that if a function

u ∈ D(B) takes a positive maximum at a point x0 of K0, then wehave the inequality

Bu(x0) ≤ 0.

Then the operator B is closable in the space C(K).(ii) Let B be as in part (i), and further assume that:

(β′) If a function u ∈ D(B) takes a positive maximum at a point x′ of K,then we have the inequality

Bu(x′) ≤ 0.

(γ) For some α0 ≥ 0, the range R(α0I − B) of α0I − B is dense in thespace C(K).

Then the minimal closed extension B of B is the infinitesimal generatorof some Feller semigroup on K.

Proof. (i) It suffices to show that:

{un} ⊂ D(B), un → 0 and Bun → v in C(K) =⇒ v = 0.

By replacing v by −v if necessary, we assume, to the contrary, that:The function v(x) takes a positive value at some point of K.Then, since K0 is open and dense in K, we can find a point x0 of K0, a

neighborhood U of x0 contained in K0 and a positive constant ε such that wehave, for all sufficiently large n,

Bun(x) > ε for all x ∈ U. (2.42)


On the other hand, by condition (α) there exists a function h ∈ D(B) suchthat {

h(x0) > 1,h(x) < 0 for all x ∈ K \ U.

Therefore, since un → 0 in C(K), it follows that the function

u′n(x) = un(x) +εh(x)

1 + ‖Bh‖∞

satisfies the conditions

u′n(x0) = un(x0) +εh(x0)

1 + ‖Bh‖∞> 0,


1 + ‖Bh‖∞< 0 for all x ∈ K \ U,

if n is sufficiently large. This implies that the function u′n ∈ D(B) takes itspositive maximum at a point x′n of U ⊂ K0. Hence we have, by condition (β),

Bu′n(x′n) ≤ 0.

However, it follows from inequality (2.42) that

Bu′n(x′n) = Bun(x′n) + εBh(x′n)

1 + ‖Bh‖∞> Bun(x′n)− ε > 0.

This is a contradiction.(ii) We apply part (ii) of Theorem 2.16 to the minimal closed extension B

of B. The proof is divided into six steps.Step 1: First, we show that

u ∈ D(B), (α0I − B)u ≥ 0 on K =⇒ u ≥ 0 on K. (2.43)

By condition (γ), we can find a function v ∈ D(B) such that

(α0I −B)v ≥ 1 on K. (2.44)

Then we have, for any ε > 0,{u+ εv ∈ D(B),(α0I −B)(u + εv) ≥ ε on K.

In view of condition (β′), this implies that the function −(u(x) + εv(x)) doesnot take any positive maximum on K, so that

u(x) + εv(x) ≥ 0 on K.


Thus, by letting ε ↓ 0 in this inequality we obtain that

u(x) ≥ 0 on K.

This proves the desired assertion (2.43).Step 2: It follows from assertion (2.43) that the inverse (α0I − B)−1 of

α0I − B is defined and non-negative on the range R(α0I − B). Moreover, itis bounded with norm

‖(α0I −B)−1‖ ≤ ‖v‖∞. (2.45)

Here v(x) is the function which satisfies condition (2.44).Indeed, since g = (α0I −B)v ≥ 1 on K, it follows that, for all f ∈ C(K),

−‖f‖∞g ≤ f ≤ ‖f‖∞g on K.

Hence, by the non-negativity of (α0I−B)−1 we have, for all f ∈ R(α0I−B),

−‖f‖∞v ≤ (α0I −B)−1f ≤ ‖f‖∞v on K.

This proves the desired inequality (2.45).Step 3: Next we show that

R(α0I −B) = C(K). (2.46)

Let f(x) be an arbitrary element of C(K). By condition (γ), we can find asequence {un} in D(B) such that fn = (α0I −B)un → f in C(K). Since theinverse (α0I−B)−1 is bounded, it follows that un = (α0I−B)−1fn convergesto some function u ∈ C(K), and hence Bun = α0un−fn converges to α0u−fin C(K). Thus we have, by the closedness of B,

{u ∈ D(B),Bu = α0u− f,

so that(α0I −B)u = f.

This proves the desired assertion (2.46).Step 4: Furthermore, we show that

u ∈ D(B), (α0I − B)u ≥ 0 on K =⇒ u ≥ 0 on K. (2.47)

Since R(α0I − B) = C(K), in view of the proof of assertion (2.47) itsuffices to show the following:

If a function u ∈ D(B) takes a positive maximum at a point x′ of K, thenwe have the inequality

Bu(x′) ≤ 0. (2.48)

Assume, to the contrary, that

Bu(x′) > 0.


Since there exists a sequence {un} in D(B) such that un → u and Bun → Buin C(K), we can find a neighborhood U of x′ and a positive constant ε suchthat, for all sufficiently large n,

Bun(x) > ε for all x ∈ U. (2.49)

Furthermore, by condition (α) we can find a function h ∈ D(B) such that{h(x′) > 1,h(x) < 0 for all x ∈ K \ U.

Then it follows that the function


1 + ‖Bh‖∞

satisfies the condition{u′n(x′) > u(x′) > 0,u′n(x) < u(x′) for all x ∈ K \ U,

if n is sufficiently large. This implies that the function u′n ∈ D(B) takes itspositive maximum at a point x′n of U . Hence we have, by condition (β′),

Bu′n(x′n) ≤ 0, x′n ∈ U.

However, it follows from inequality (2.49) that

Bu′n(x′n) = Bun(x′n) + εBh(x′n)

1 + ‖Bh‖∞> Bun(x′n)− ε > 0.

This is a contradiction.Step 5: In view of Steps 3 and 4, we obtain that the inverse (α0I −B)−1

of α0I −B is defined on the whole space C(K), and is bounded with norm

‖(α0I −B)−1‖ = ‖(α0I −B)−11‖∞.

Step 6: Finally, we show that:For all α > α0, the inverse (αI −B)−1 of αI −B is defined on the whole

space C(K), and is non-negative and bounded with norm

‖(αI −B)−1‖ ≤ 1α. (2.50)

We letGα0 = (α0I −B)−1.

First, we choose a constant α1 > α0 such that

(α1 − α0)‖Gα0‖ < 1,


and letα0 < α ≤ α1.

Then, for any f ∈ C(K), the Neumann series

u =

(I +

∞∑

n=1

(α0 − α)nGnα0

)Gα0f

converges in C(K), and is a solution of the equation

u− (α0 − α)Gα0u = Gα0f.

Hence we have the assertions{u ∈ D(B),(αI −B)u = f.

This proves that

R(αI −B) = C(K), α0 < α ≤ α1. (2.51)

Thus, by arguing just as in the proof of Step 1 we obtain that, for any α0 <α ≤ α1,

u ∈ D(B), (αI −B)u ≥ 0 on K =⇒ u ≥ 0 on K. (2.52)

By combining assertions (2.51) and (2.52), we find that, for any α0 < α ≤ α1,the inverse (αI −B)−1 is defined and non-negative on the whole space C(K).

We letGα = (αI −B)−1, α0 < α ≤ α1.

Then it follows that the operator Gα is bounded with norm

‖Gα‖ ≤1α. (2.53)

Indeed, in view of assertion (2.48) it follows that if a function u ∈ D(B) takesa positive maximum at a point x′ of K, then we have the inequality

Bu(x′) ≤ 0,

so that

maxx∈K

u(x) = u(x′) ≤ 1α

(αI −B)u(x′) ≤ 1α‖(αI −B)u‖∞. (2.54)

Similarly, if the function u ∈ D(B) takes a negative minimum at a point ofK, then (replacing u(x) by −u(x)), we have the inequality

−minx∈K

u(x) = maxx∈K

(−u(x)) ≤ 1α‖(αI − B)u‖∞. (2.55)


The desired inequality (2.53) follows from inequalities (2.54) and (2.55).Summing up, we have proved assertion (2.50) for all α0 < α ≤ α1.Now we assume that assertion (2.50) is proved for all α0 < α ≤ αn−1,

n = 2, 3, . . .. Then, by taking

αn = 2αn−1 −α1

2, n ≥ 2,

or equivalently

αn =(

2n−2 +12

)α1, n ≥ 2,

we have, for all αn−1 < α ≤ αn,

(α− αn−1)‖Gαn−1‖ ≤α− αn−1

αn−1

≤ αn − αn−1

αn−1

=1

1 + 22−n

< 1.

Hence assertion (2.50) for αn−1 < α ≤ αn is proved just as in the proof ofassertion (2.50) for α0 < α ≤ α1. This proves the desired assertion (2.50).

Consequently, by applying part (ii) of Theorem 2.16 to the operator B weobtain that B is the infinitesimal generator of some Feller semigroup on K.

The proof of Theorem 2.18 is now complete. �

Corollary 2.19. Let A be the infinitesimal generator of a Feller semigroupon a compact metric space K and let M be a bounded linear operator on C(K)into itself. Assume that either M or A′ = A+M satisfies condition (β′). Thenthe operator A′ is the infinitesimal generator of some Feller semigroup on K.

Proof. We apply part (ii) of Theorem 2.18 to the operator A′.First, we remark that A′ = A + M is a densely defined, closed linear

operator from C(K) into itself. Since the semigroup {Tt}t≥0 is non-negativeand contractive on C(K), it follows that if a function u ∈ D(A) takes a positivemaximum at a point x′ of K, then we have the inequality

Au(x′) = limt↓0

Ttu(x′)− u(x′)t

≤ 0.

This implies that if M satisfies condition (β′), so does A′ = A+M .We let

Gα0 = (α0I −A)−1, α0 > 0.

If α0 is so large that

‖Gα0M‖ ≤ ‖Gα0‖ · ‖M‖ ≤‖M‖α0

< 1,


then the Neumann series

u =

(I +

∞∑

n=1

(Gα0M)n

)Gα0f

converges in C(K) for any f ∈ C(K), and is a solution of the equation

u−Gα0Mu = Gα0f.

Hence we have the assertions{u ∈ D(A) = D(A′),(α0I −A′)u = f.

This proves thatR(α0I −A′) = C(K).

Therefore, by applying part (ii) of Theorem 2.18 to the operator A′ weobtain that A′ is the infinitesimal generator of some Feller semigroup on K.

The proof of Corollary 2.19 is complete. �

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