Kazuaki Taira. Boundary Value Problems and Markov Processes

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2.2 Markov Processes and Feller Semigroups 35

and

(x, E)=χ

(x).

This represents Brownian motion with a sticking barrier at x =0.When

a Brownian particle reaches the boundary point 0 for the ﬁrst time, instead

of 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 p

(x, ·), under which condi-

tions on p

(x, ·) does there exist a Markov process with transition function

(x, ·) whose paths are almost surely continuous?

A Markov process X =(x

, F, F

)issaidtoberight continuous pro-

vided that we have, for each x ∈ K,

{ω ∈ Ω : the mapping t → x

(ω) is a right continuous function

from [0, ∞)intoK

∂

} =1.

Furthermore, we say that X is continuous provided that we have, for each

x ∈ K,

{ω ∈ Ω : the mapping t → x

(ω) is a continuous function

from [0,ζ(ω)) into K

∂

} =1,

where ζ is the lifetime of the process X.

Nowwegivesomeusefulcriteriaforpathcontinuityintermsof

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 and

let p

(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

lim

x→∂

sup

0≤t≤s

(x, E)=0.

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

lim

t↓0

sup

x∈E

(x, K \ U

(x)) = 0,

where U

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

36 2 Semigroup Theory

Then there exists a Markov process X with transition function p

(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) (replacing

condition (M)) are satisﬁed:

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

lim

t↓0

sup

x∈E

(x, K \U

(x)) = 0,

or equivalently

sup

x∈E

(x, K \U

(x)) = o(t) as t ↓ 0.

Then there exists a Markov process X with transition function p

(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 of

a Markov process are right continuous, then the transition function p

(x, ·)

satisﬁes the condition

lim

t↓0

(x, U

(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 random

times.

We shall formulate precisely this “strong” Markov property. Let X =

, F, F

) be a Markov process. A mapping τ: Ω → [0, ∞] is called a

stopping time or Markov time with respect to {F

} if it satisﬁes the condition

{τ ≤ t} = {ω ∈ Ω : τ(ω) ≤ t}∈F

for all t ∈ [0, ∞).

Intuitively, this means that the events {τ ≤ t} depend on the process only up

to time t, but not on the “future” after time t. It should be noticed that any

non-negative constant mapping is a stopping time.

If τ is a stopping time with respect to {F

},welet

= {A ∈F: A ∩{τ ≤ t}∈F

for all t ∈ [0, ∞)}.

Intuitively, we may think of F

as the “past” up to the random time τ.Itis

easy to verify that F

is a σ-algebra. If τ ≡ t

for some constant t

≥ 0, then

reduces to F

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

:[0,t] × Ω −→ K

∂

2.2 Markov Processes and Feller Semigroups 37

by the formula

(s, ω)=x

(ω).

A Markov process X =(x

, F, F

)issaidtobeprogressively measurable

with respect to {F

} if the mapping Φ

is B

[0,t]

×F

∂

-measurable for each

t ∈ [0, ∞], that is, if we have the condition

−1

(E)={Φ

∈ E}∈B

[0,t]

×F

for all E ∈B

∂

Here B

[0,t]

is the σ-algebra of all Borel sets in the interval [0,t]andB

∂

is the

σ-algebra in K

∂

generated by B. It should be noticed that if X is progressively

measurable and if τ is a stopping time, then the mapping x

: ω → x

τ (ω)

(ω)is

∂

-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 =

, F, F

)hasthestrong Markov property with respect to {F

} if the

following condition is satisﬁed: For all h ≥ 0, x ∈ K

∂

, E ∈B

∂

and all stopping

times τ,wehavetheformula

τ +h

∈ E |F

} = p

,E),

or equivalently,

(A ∩{x

τ +h

∈ E})=



τ (ω)

(ω),E) dP

(ω) for all A ∈F

We shall state a simple criterion for the strong Markov property in terms

of 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 as

an 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 continuous

functions f (x)onK;thespaceC(K) is a Banach space with the supremum

norm

f

∞

=sup

x∈K

|f(x)|.

We say that a function f ∈ C(K) converges to zero as x → ∂ if, for each

ε>0, there exists a compact subset E of K such that

|f(x)| <ε for all x ∈ K \E,

and we then write lim

x→∂

f(x)=0.Welet

(K)=



f ∈ C(K) : lim

x→∂

f(x)=0



The space C

(K) is a closed subspace of C(K); hence it is a Banach space.

Note that C

(K) may be identiﬁed with C(K)ifK is compact.

38 2 Semigroup Theory

Now we introduce a useful convention as follows:

Any real-valued function f (x) on K is extended to

the space K

∂

= K ∪{∂}by setting f (∂)=0.

From this point of view, the space C

(K) is identiﬁed with the subspace of

C(K

∂

) which consists of all functions f(x) satisfying the condition f(∂)=0:

(K)={f ∈ C(K

∂

):f(∂)=0}.

Furthermore, we can extend a Markov transition function p

(x, ·)onK to a

Markov transition function p



(x, ·)onK

∂

by the formulas:

⎧

⎨

⎩



(x, E)=p

(x, E) for all x ∈ K and E ∈B,



(x, {∂})=1−p

(x, K) for all x ∈ K,



(∂,K)=0,p



(∂,{∂})=1.

Intuitively this means that a Markovian particle moves in the space K until

it “dies” at the time when it reaches the point ∂; hence the point ∂ is called

the terminal point.

Now we introduce some conditions on the measures p

(x, ·) related to con-

tinuity in x ∈ K,forﬁxedt ≥ 0:

Deﬁnition 2.3. (i) A Markov transition function p

(x, ·) is called a Fel ler

function if the function

f(x)=



(x, dy)f(y)

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

the condition

f ∈ C(K)=⇒ T

f ∈ C(K).

(ii) We say that p

(x, ·)isaC

-function if the space C

(K) is an invariant

subspace of C(K) for the operators T

f ∈ C

(K)=⇒ T

f ∈ C

(K).

Remark 2.3. The Feller property is equivalent to saying that the measures

(x, ·) depend continuously on x ∈ K in the usual weak topology, for every

ﬁxed 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 p

(x, ·) of a right continuous Mar-

kov process X has the C

-property, then X is a strong Markov process.

2.2 Markov Processes and Feller Semigroups 39

Furthermore, we state a simple criterion for the strong Markov property in

terms of Markov transition functions. To do this, we introduce the following

deﬁnition:

Deﬁnition 2.4. A Markov transition function p

(x, ·)onK 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

lim

t↓0

sup

x∈E

[1 − p

(x, U

(x))] = 0, (2.27)

where U

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

It should be emphasized that every uniformly stochastically continuous

transition function is normal and satisﬁes condition (M) in Theorem 2.10. By

combining part (i) of Theorem 2.10 and Theorem 2.11, we obtain the following

result (see [Dy1, Theorem 6.3]):

Theorem 2.12. Assume that a uniformly stochastically continuous, C

transition function p

(x, ·) satisﬁes condition (L). Then it is the transition

function of some strong Markov process X whose paths are right continuous

and 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, C

transition function p

(x, ·) satisﬁes conditions (L) and (N). Then it is the

transition function of some diﬀusion process X.

This is an immediate consequence of part (ii) of Theorem 2.10 and

Theorem 2.12.

2.2.5 Markov Transition Functions and Feller Semigroups

The Feller or C

-property deals with continuity of a Markov transition func-

tion p

(x, E)inx, and does not, by itself, have no concern with continuity

in t. We give a necessary and suﬃcient condition on p

(x, E)inorderthatits

associated operators {T

}

t≥0

, deﬁned by the formula

f(x)=



(x, dy)f(y),f∈ C

(K), (2.28)

is strongly continuous in t on the space C

(K):

lim

s↓0

T

t+s

f − T

f

∞

=0,f∈ C

(K). (2.29)

40 2 Semigroup Theory

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

Theorem 2.14. Let p

(x, ·) be a C

-transition function on K. Then the as-

sociated operators {T

}

t≥0

, deﬁned by formula (2.28) is strongly continuous

in t on C

(K) if and only if p

(x, ·) is uniformly stochastically continuous on

K and satisﬁes condition (L).

Proof. (i) The “if” part: Since continuous functions with compact support are

dense in C

(K), it suﬃces to prove the strong continuity of {T

} at t =0:

lim

t↓0

T

f − 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

T

f − f

∞

≤ sup

x∈E

f(x) − f(x)| +sup

x∈K\E

f(x)|

≤ sup

x∈E

f(x) − f(x)| + f 

∞

· sup

x∈K\E

(x, supp f). (2.31)

However, condition (L) implies that, for each ε>0, we can ﬁnd a compact

subset E of K such that, for all suﬃciently small t>0,

sup

x∈K\E

(x, supp f) <ε. (2.32)

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

f(x) − f(x)=



(x)

(x, dy)(f (y) − f(x))



K\U

(x)

(x, dy)(f (y) − f(x)) − f(x)(1 − p

(x, K)),

and hence

sup

x∈E

f(x) − f(x)|

≤ sup

ρ(x,y)<δ

|f(y) − f(x)|+3f 

∞

· sup

x∈E

[1 − p

(x, U

(x))] .

Since the function f(x) is uniformly continuous, we can choose a positive

constant δ such that

sup

ρ(x,y)<δ

|f(y) − f(x)| <ε.

Furthermore, it follows from condition (2.27) with ε := δ (the uniform stochas-

tic continuity of p

(x, ·)) that, for all suﬃciently small t>0,

2.2 Markov Processes and Feller Semigroups 41

sup

x∈E

[1 − p

(x, U

(x))] <ε.

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

sup

x∈E

f(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,

T

f − f 

∞

<ε(1 + 4f 

∞

This proves the desired formula (2.30), that is, the strong continuity (2.29) of

(ii) The “only if” part: For any x ∈ K and ε>0, we deﬁne a continuous

function f

(y) by the formula (see Figure 2.13)

(y)=



1 −

ρ(x, y)ifρ(x, y) ≤ ε,

0ifρ(x, y) >ε.

(2.34)

εε

Fig. 2.13.

Let E be an arbitrary compact subset of K. Then, for all suﬃciently small

ε>0, the functions f

, x ∈ E,areinC

(K) and satisfy the condition

f

− f



∞

≤

ρ(x, z) for all x, z ∈ E. (2.35)

However, for any δ>0, by the compactness of E we can ﬁnd a ﬁnite number

of points x

, x

, ..., x

of E such that

E =



k=1

δε/4

and hence

min

1≤k≤n

ρ(x, x

) ≤

δε

for all x ∈ E.

Thus, by combining this inequality with inequality (2.35) with z := x

obtain that

42 2 Semigroup Theory

min

1≤k≤n

f

− f



∞

≤

for all x ∈ E. (2.36)

Now we have, by formula (2.34),

0 ≤ 1 − p

(x, U

(x)) ≤ 1 −



∂

(x, dy)f

(y)

= f

(x) − T

(x)

≤f

− T



∞

≤f

− f



∞

+ f

− T



∞

+T

− T



∞

≤ 2f

− f



∞

+ f

− T



∞

for all x ∈ E.

In view of inequality (2.36), the ﬁrst term on the last inequality is bounded by

δ/2 for the right choice of k. Furthermore, it follows from the strong continuity

(2.30) of {T

} that the second term tends to zero as t ↓ 0foreachk =1,2,

..., n.

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

sup

x∈E

[1 − p

(x, U

(x))] ≤ δ.

This proves the desired condition (2.27), that is, the uniform stochastic con-

tinuity of p

(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

, a sequence { t

}, t

↓ t (0 ≤ t ≤ s) and a sequence {x

}, x

→ ∂,such

that

,E) ≥ ε

. (2.37)

NowwetakearelativelycompactsubsetU of K containing E,andlet

(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)isinC

(K) and satisﬁes the condi-

tion

f(x)=



(x, dy)f(y) ≥ p

(x, E) ≥ 0.

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

f(x

) ≥ p

,E) ≥ ε

. (2.38)

However, we have the inequality

f(x

) ≤T

f − T

f

∞

+ T

f(x

). (2.39)

2.2 Markov Processes and Feller Semigroups 43

Fig. 2.14.

Since the semigroup {T

} is strongly continuous and since we have the asser-

tion

lim

k→∞

f(x

)=T

f(∂)=0,

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

lim sup

k→∞

f(x

)=0.

This contradicts inequality (2.38).

The proof of Theorem 2.14 is now complete. 

A family {T

}

t≥0

of bounded linear operators acting on the space C

(K)is

called a Feller semigroup on K if it satisﬁes the following three conditions:

(i) T

t+s

= T

· T

, t, s ≥ 0 (the semigroup property); T

= I.

(ii) The family {T

} is strongly continuous in t for all t ≥ 0:

lim

s↓0

T

t+s

f − T

f

∞

=0,f∈ C

(K).

(iii) The family {T

} is non-negative and contractive on C

(K):

f ∈ C

(K), 0 ≤ f(x) ≤ 1onK =⇒ 0 ≤ T

f(x) ≤ 1onK.

Rephrased, Theorem 2.14 gives a characterization of Feller semigroups in

terms of Markov transition functions:

Theorem 2.15. If p

(x, ·) is a uniformly stochastically continuous C

transition function on K and satisﬁes condition (L), then its associated

operators {T

}

t≥0

, deﬁned by formula (2.28), form a Feller semigroup on K.

Conversely, if {T

}

t≥0

is a Feller semigroup on K, then there exists a uni-

formly stochastically continuous C

-transition p

(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 the

second statement.

44 2 Semigroup Theory

2.2.6 Generation Theorems of Feller Semigroups

In this subsection we prove various generation theorems of Feller semigroups

by using the Hille–Yosida theory of semigroups.

If {T

}

t≥0

is a Feller semigroup on K, we deﬁne its inﬁnitesimal generator

A by the formula

Au = lim

t↓0

u − u

,u∈ C

(K), (2.40)

provided that the limit (2.40) exists in the space C

(K). More precisely, the

generator A is a linear operator from C

(K) into itself deﬁned as follows:

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

D(A)={u ∈ C

(K) : the limit (2.40) exists}.

(2) Au = lim

t↓0

u−u

, u ∈D(A).

The next theorem is a version of the Hille–Yosida theorem adapted to the

present context (cf. [Ta2, Theorem 9.3.1 and Corollary 9.3.2]):

Theorem 2.16 (Hille–Yosida). (i) Let {T

}

t≥0

be a Feller semigroup on

K and let A be its inﬁnitesimal generator. Then we have the following four

assertions:

(a) The domain D(A) is dense in the space C

(K).

(b) For each α>0, the equation (αI − A)u = f has a unique solution u

in D(A) for any f ∈ C

(K). Hence, for each α>0 the Green operator

(αI −A)

−1

: C

(K) → C

(K) can be deﬁned by the formula

u =(αI −A)

−1

f, f ∈ C

(K).

−1

is non-negative on C

(K):

f ∈ C

(K),f(x) ≥ 0 on K =⇒ (αI − A)

−1

f(x) ≥ 0 on K.

(d) For each α>0, the operator (αI −A)

−1

is bounded on C

(K) with norm

(αI −A)

−1

≤

(ii) Conversely, if A is a linear operator from C

(K) into itself satisfying

condition (a) and if there is a non-negative constant α

such that, for all

α>α

, conditions (b) through (d) are satisﬁed, then A is the inﬁnitesimal

generator of some Feller semigroup {T

}

t≥0

on K.

Proof. In view of the Hille–Yosida theory (see [Yo, Chapter IX, Section 7]), it

suﬃces to show that the semigroup {T

}

t≥0

is non-negative if and only if its

resolvents (Green operators) {(αI −A)

−1

}

α>α

are non-negative.