Kazuaki Taira. Boundary Value Problems and Markov Processes

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9.1 General Existence Theorem for Feller Semigroups 129

Proof. (i) (a) By making use of Friedrichs’ molliﬁers, we ﬁnd that the H¨older

space C

(D)isdenseinC(D) and further that non-negative functions can

be approximated by non-negative smooth functions. Hence, by Lemma 9.2 it

follows that the operator G

: C

(D) → C

2+θ

(D) can be uniquely extended

to a non-negative, bounded linear operator G

: C(D) → C(D)withnorm

G

 = G

1

∞

Furthermore, since the function G

1 satisﬁes the conditions



(A −α)G

1=−1inD,

1=0 on∂D,

by applying Theorem A.2 with A := A − α we obtain that

G

 = G

1

∞

≤

(b) This assertion follows from formula (9.3), since the space C

(D)is

dense in C(

D) and since the operator G

: C(D) → C(D) is bounded.

that equation (9.6) holds true for all f ∈ C

(D). Indeed, it suﬃces to note

that the function

v = G

f − G

f +(α − β)G

f ∈ C

2+θ

(D)

satisﬁes the conditions



(α − A)v =0 inD,

v =0 on∂D,

so that

v =0 inD.

Therefore, we obtain that the resolvent equation (9.6) holds true for all f ∈

D), since the space C

(D)isdenseinC(D) and since the operators G

and G

are bounded.

(d) First, let f(x) be an arbitrary function in C

(D) satisfying the bound-

ary condition f |

∂D

= 0. Then it follows from an application of the uniqueness

theorem for problem (9.2) (Theorem 9.1) that we have, for all α, β,

f − αG

f = G

((β − A)f) − βG

Indeed, the both sides satisfy the same equation (α − A)u = −Af in D and

have the same boundary value 0 on ∂D. Thus we have, by estimate (9.5),

f −αG

f

∞

≤

(β − A)f

∞

f

∞

so that

lim

α→+∞

f − αG

f

∞

=0. (9.10)

130 9 Proof of Theorem 1.3, Part (ii)

Now let f(x) be an arbitrary function in C(D) satisfying the boundary

condition f|

∂D

= 0. By means of molliﬁers, we can ﬁnd a sequence { f

} in

(D) such that



−→ f in C(D)asj →∞,

=0 on∂D.

Then we have, by estimate (9.5) and assertion (9.10) with f := f

f −αG

f

∞

≤f − f



∞

+ f

− αG



∞

+ αG

− f )

∞

≤ 2f − f



∞

+ f

− αG



∞

and hence

lim sup

α→+∞

f −αG

f

∞

≤ 2f −f



∞

This proves the desired assertion (9.8), since f − f



∞

→ 0asj →∞.

To prove assertion (9.7), let f(x) be an arbitrary function in C(

D)andlet

x be an arbitrary point of D.Ifwetakeafunctionψ(y)inC(

D) such that

⎧

⎨

⎩

0 ≤ ψ(y) ≤ 1on

ψ(y) = 0 in a neighborhood of x,

ψ(y)=1 neartheset∂D,

then it follows from the non-negativity of G

and estimate (9.5) that

0 ≤ αG

ψ(x)+αG

(1 − ψ)(x)=αG

1(x) ≤ 1. (9.11)

However, by applying assertion (9.8) to the function 1 − ψ(y)wehavethe

assertion

lim

α→+∞

αG

(1 − ψ)(x)=(1− ψ)(x) = 1 for all x ∈ D.

In view of inequalities (9.11), this implies that

lim

α→+∞

αG

ψ(x) = 0 for all x ∈ D.

Thus, since we have the inequalities

−f

∞

ψ ≤ fψ ≤f

∞

ψ on D,

it follows that, for x ∈ D,

|αG

(fψ)(x)|≤f

∞

· αG

ψ(x) −→ 0asα → +∞.

Therefore, by applying assertion (9.8) to the function (1−ψ(y))f (y)weobtain

that

f(x) = ((1 − ψ)f )(x) = lim

α→+∞

αG

((1 − ψ)f)(x)

= lim

α→+∞

αG

f(x) for all x ∈ D.

9.1 General Existence Theorem for Feller Semigroups 131

This proves the desired assertion (9.7).

(ii) (a



) Since the space C

2+θ

(∂D)isdenseinC(∂D), by Lemma 9.3

it follows that the operator H

: C

2+θ

(∂D) → C

2+θ

(D) can be uniquely

extended to a non-negative, bounded linear operator H

: C(∂D) → C(D).

Furthermore, by applying Theorem A.2 with A := A−α we have the assertion

H

 = H

1

∞

=1.



) This assertion follows from formula (9.4), since the space C

2+θ

(∂D)

is dense in C(∂D) and since the operator H

: C(∂D) → C(D) is bounded.



) We ﬁnd from the uniqueness theorem for problem (9.2) that equation

(9.9) holds true for all ϕ ∈ C

2+θ

(∂D). Indeed, it suﬃces to note that the

function

w = H

ϕ − H

ϕ +(α − β)G

ϕ ∈ C

2+θ

(D)

satisﬁes the conditions



(α − A)w =0 inD,

w =0 on∂D,

so that

w =0 inD.

Therefore, we obtain that the desired equation (9.9) holds true for all ϕ ∈

C(∂D), since the space C

2+θ

(∂D)isdenseinC(∂D) and since the operators

and H

are bounded.

The proof of Theorem 9.4 is now complete. 

Summing up, we have the following diagrams for the operators G

and

−−−−→ C(D)

⏐

(D)

−−−−→ C

2+θ

(D)

C(∂D)

−−−−→ C(D)

⏐

2+θ

(∂D)

−−−−→ C

2+θ

(D)

Now we consider the following boundary value problem in the framework

of the spaces of continuous functions.



(α − A)u = f in D,

Lu =0 on∂D.

(9.12)

To do this, we introduce three operators associated with problem (9.12).

132 9 Proof of Theorem 1.3, Part (ii)

(I) First, we introduce a linear operator

A : C(

D) −→ C(D)

as follows.

(a) The domain D(A)ofA is the space C

(D).

(b) Au =

i,j=1

(x)

∂

∂x

i=1

(x)

∂u

∂x

+ c(x)u, u ∈D(A).

Then we have the following:

Lemma 9.5. The operator A has its minimal closed extension

A in the space

D).

Proof. We apply part (i) of Theorem 2.18 to the operator A.

Assume that a function u ∈ C

(D) takes a positive maximum at an interior

point x

of D:

u(x

)=max

x∈D

u(x) > 0.

Then it follows that

∂u

∂x

)=0, 1 ≤ i ≤ N,



i,j=1

)

∂

∂x

) ≤ 0,

since the matrix (a

(x)) is positive deﬁnite. Hence we have the assertion

Au(x



i,j=1

)

∂

∂x

)+c(x

)u(x

) ≤ 0.

This implies that the operator A satisﬁes condition (β) of Theorem 2.18 with

:= D and K := D. Therefore, Lemma 9.5 follows from an application of

the same theorem.

The proof of Lemma 9.5 is complete. 

Remark 9.1. Since the injection: C(

D) →D



(D) is continuous, we have the

formula

Au =



i,j=1

(x)

∂

∂x



i=1

(x)

∂u

∂x

+ c(x)u, u ∈ C(D),

where the right-hand side is taken in the sense of distributions. The operators

A and

A can be visualized as follows:

−−−−→ C(D)

⏐

(D)

−−−−→ C(D)

9.1 General Existence Theorem for Feller Semigroups 133

The extended operators G

: C(D) → C(D)andH

: C(∂D) → C(D),

α>0, still satisfy formulas (9.3) and (9.4) respectively in the following sense:

Lemma 9.6. (i) For any f ∈ C(

D), we have the assertions



f ∈D(A),

(αI −

A)G

f = f in D.

(ii) For any ϕ ∈ C(∂D), we have the assertions



ϕ ∈D(A),

(αI −

A)H

ϕ =0in D.

Here D(

A) is the domain of the closed extension A.

Proof. (i) By making use of Friedrichs’ molliﬁers, we can choose a sequence

} in C

(D) such that f

→ f in C(D)asj →∞. Then it follows from the

boundedness of G

that

−→ G

f in C(D),

and further that

(α − A)G

= f

−→ f in C(D).

Hence we have the assertions



f ∈D(A),

(αI −

A)G

f = f in D.

since the operator

A : C(D) → C(D)isclosed.

(ii) Similarly, part (ii) is proved, since the space C

2+θ

(∂D)isdensein

C(∂D) and since the operator H

: C(∂D) → C(D) is bounded.

The proof of Lemma 9.6 is complete. 

Corollary 9.7. Every function u in D(

A) canbewritteninthefollowing

form:

u = G



(αI −

A)u



+ H

(u|

∂D

) for all α>0. (9.13)

Proof. We let

w = u − G



(αI −

A)u



− H

(u|

∂D

Then it follows from Lemma 9.6 that the function w is in D(

A) and satisﬁes

the conditions



(αI −

A)w =0 inD,

w =0 on∂D.

However, in light of Remark 9.1, by applying Lemma 5.1 with λ

:= α and

Theorem 4.9 with A := A −α to the Dirichlet case (μ(x



) ≡ 0andγ(x



) ≡−1

on ∂D) we obtain that

w ∈ C

∞

(D).

134 9 Proof of Theorem 1.3, Part (ii)

Therefore, it follows from an application of Theorem 9.1 with A := A−α that

w =0.

This proves the desired formula (9.13).

The proof of Corollary 9.7 is complete. 

(II) Secondly, we introduce a linear operator

: C(D) −→ C(∂D)

as follows.

(a) The domain D





of LG

is the H¨older space C

(D)with0<θ<1.

(b) LG

f = L





, f ∈D





Then we have the following:

Lemma 9.8. The operator LG

, α>0, can be uniquely extended to a non-

negative, bounded linear operator

: C(D) → C(∂D).

Proof. Let f(x) be an arbitrary function in D(LG

)=C

(D) such that

f(x) ≥ 0on

D. Then we have the assertions

⎧

⎨

⎩

f ∈ C

2+θ

(D),

f ≥ 0onD,

∂D

=0 on∂D,

and hence

f = μ(x



)

∂

∂n

f)+γ(x



= μ(x



)

∂

∂n

f) ≥ 0on∂D.

This proves that the operator LG

is non-negative.

By the non-negativity of LG

,wehave,forallf ∈D(LG

−LG

f

∞

≤ LG

f ≤ LG

f

∞

on ∂D.

This implies the boundedness of LG

with norm

LG

 = LG

1

∞

Recall that the space C

(D)isdenseinC(D) and that non-negative func-

tions can be approximated by non-negative smooth functions. Hence we ﬁnd

that the operator LG

can be uniquely extended to a non-negative, bounded

linear operator

: C(D) → C(∂D).

The proof of Lemma 9.8 is complete. 

9.1 General Existence Theorem for Feller Semigroups 135

The operators LG

and LG

can be visualized as follows:

−−−−→ C(∂D)

⏐

(D)

−−−−→ C

1+θ

(∂D)

The next lemma states a fundamental relationship between the operators

and LG

for α, β>0:

Lemma 9.9. For any f ∈ C(

D), we have the formula

f − LG

f +(α − β)LG

f =0 for all α, β>0. (9.14)

Proof. Choose a sequence {f

} in C

(D) such that f

→ f in C(D)asj →∞,

just as in Lemma 9.6. Then, by using the resolvent equation (9.6) with f := f

we have the formula

− LG

+(α − β)LG

=0.

Hence, the desired formula (9.14) follows by letting j →∞, since the operators

, LG

and G

are all bounded.

The proof of Lemma 9.9 is complete. 

(III) Finally, we introduce a linear operator

: C(∂D) −→ C(∂D)

as follows.

(a) The domain D(LH

)ofLH

is the space C

2+θ

(∂D).

(b) LH

ψ = L (H

ψ), ψ ∈D(LH

Then we have the following:

Lemma 9.10. The operator LH

, α>0, has its minimal closed extension

in the space C(∂D).

Proof. We apply part (i) of Theorem 2.18 to the operator LH

.Todothis,it

suﬃces to show that the operator LH

satisﬁes condition (β



)withK := ∂D

(or condition (β)withK := K

= ∂D) of the same theorem.

Assume that a function ψ in D(LH

)=C

2+θ

(∂D) takes its positive max-

imum at some point x



of ∂D. Since the function H

ψ is in C

2+θ

(D)and

satisﬁes



(A −α)H

ψ =0 inD,

ψ = ψ on ∂D,

by applying the weak maximum principle (Theorem A.1) with A := A − α to

the function H

ψ we ﬁnd that the function H

ψ takes its positive maximum

136 9 Proof of Theorem 1.3, Part (ii)

at the boundary point x



∈ ∂D. Thus we can apply the boundary point lemma

(Lemma A.3) with A := A − α to obtain that

∂

∂n

ψ)(x



) < 0.

Hence we have the inequality

ψ(x



N−1



i,j=1



)

∂

∂x



)+μ(x



)

∂

∂n

ψ)(x



)

+γ(x



)ψ(x



)

≤ 0.

This veriﬁes condition (β



) of Theorem 2.18. Therefore, Lemma 9.10 follows

from an application of the same theorem.

The proof of Lemma 9.10 is complete. 

Remark 9.2. The operator

enjoys the following property:

If a function ψ in the domain D





takes its positive maximum

at some point x



of ∂D, then we have the inequality

ψ(x



) ≤ 0. (9.15)

The operators LH

and LH

can be visualized as follows:

C(∂D)

−−−−→ C(∂D)

⏐

2+θ

(∂D)

−−−−→ C

1+θ

(∂D)

The next lemma states a fundamental relationship between the operators

and LH

for α, β>0:

Lemma 9.11. The domain D





does not depend on α>0;so

we denote by D the common domain. Then we have the formula

ψ −LH

ψ +(α −β)LG

ψ =0 for all α, β>0 and ψ ∈D. (9.16)

Proof. Let ψ(x



) be an arbitrary function in D(LH

), and choose a sequence

{ψ

} in D(LH

)=C

2+θ

(∂D) such that, as j →∞,



−→ ψ in C(∂D),

−→ LH

ψ in C(∂D).

Then it follows from the boundedness of H

and LG

that

9.1 General Existence Theorem for Feller Semigroups 137

)=LG

) −→ LG

ψ)inC(∂D).

Therefore, by using formula (9.9) with ϕ := ψ

we obtain that, as j →∞,

= LH

− (α − β)LG

)

−→

ψ − (α − β)LG

ψ)inC(∂D).

This implies that



ψ ∈D(

ψ = LH

ψ −(α − β)LG

ψ),

since the operator

: C(∂D) → C(∂D)isclosed.

Conversely, by interchanging α and β we have the assertion

) ⊂D(LH

and so

)=D(LH

The proof of Lemma 9.11 is complete. 

Now we can prove a general existence theorem for Feller semigroups on

∂D in terms of boundary value problem (9.12). The next theorem asserts that

the operator

is the inﬁnitesimal generator of some Feller semigroup on

∂D if and only if problem (9.12) is solvable for suﬃciently many functions ϕ

in the space C(∂D):

Theorem 9.12. (i) If the operator

, α>0, is the inﬁnitesimal generator

of a Feller semigroup on ∂D, then, for each positive constant λ, the boundary

value problem



(α − A)u =0 in D,

(λ − L)u = ϕ on ∂D

(9.17)

has a solution u ∈ C

2+θ

(D) for any ϕ in some dense subset of C(∂D).

(ii) Conversely, if, for some non-negative constant λ, problem (9.17) has

asolutionu ∈ C

2+θ

(D) for any ϕ in some dense subset of C(∂D), then the

operator

is the inﬁnitesimal generator of some Feller semigroup on ∂D.

Proof. (i) If the operator

generates a Feller semigroup on ∂D, by apply-

ing part (i) of Theorem 2.18 with K := ∂D to the operator

we obtain

that



λI −



= C(∂D)foreachλ>0.

This implies that the range R(λI −LH

) is a dense subset of C(∂D)for

each λ>0. However, if ϕ ∈ C(∂D) is in the range R(λI −LH

)andifϕ =

(λI −LH

) ψ with ψ ∈ C

2+θ

(∂D), then the function u = H

ψ ∈ C

2+θ

(D)is

a solution of problem (9.17). This proves part (i) of Theorem 9.12.

138 9 Proof of Theorem 1.3, Part (ii)

(ii) We apply part (ii) of Theorem 2.18 with K := ∂D to the operator

. To do this, it suﬃces to show that the operator LH

satisﬁes condition

(γ) of the same theorem, since it satisﬁes condition (β



), as is shown in the

proof of Lemma 9.10.

By the uniqueness theorem for problem (9.2), it follows that any function

u ∈ C

2+θ

(D) which satisﬁes the homogeneous equation

(α − A)u =0 inD

canbewrittenintheform:

u = H

(u|

∂D

) ,u|

∂D

∈ C

2+θ

(∂D)=D(LH

) .

Thus we ﬁnd that if there exists a solution u ∈ C

2+θ

(D) of problem (9.17) for

a function ϕ ∈ C(∂D), then we have the assertion

(λI −LH

)(u|

∂D

)=ϕ,

and so

ϕ ∈R(λI −LH

) .

Hence, if, for some non-negative constant λ, problem (9.12) has a solution

u ∈ C

2+θ

(D) for any ϕ in some dense subset of C(∂D), then the range

R(λI −LH

)isdenseinC(∂D). This veriﬁes condition (γ)(withα

:= λ)of

Theorem 2.18. Therefore, part (ii) of Theorem 9.12 follows from an application

of the same theorem.

Now the proof of Theorem 9.12 is complete. 

Remark 9.3. Intuitively, Theorem 9.12 asserts that we can “piece together” a

Markov process (Feller semigroup) on the boundary ∂D with A-diﬀusion in

the interior D to construct a Markov process (Feller semigroup) on the closure

D = D ∪∂D. The situation may be represented schematically by Figure 9.1.

We conclude this section by giving a precise meaning to the boundary

conditions Lu for functions u in D(

A).

We let

D(L)=



u ∈D(

A):u|

∂D

∈D



where D is the common domain of the operators

for all α>0. We remark

that the space D(L)containsC

2+θ

(D), since C

2+θ

(∂D)=D(LH

) ⊂D.

Corollary 9.7 asserts that every function u in D(L) ⊂D(

A) can be written in

the form

u = G



(αI −

A)u



+ H

(u|

∂D

) for all α>0. (9.13)

Then we deﬁne the boundary condition Lu by the formula

Lu =



(αI −

A)u



+ LH

(u|

∂D

) ,u∈D(L). (9.18)

The next lemma justiﬁes deﬁnition (9.18) of Lu for each u ∈D(L):