Jost J. Partial Differential Equations

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7.2 Inﬁnitesimal Generators of Semigroups 157

Banach-space-valued case. The convergence of the improper integral follows

from the estimate

lim

K,M→∞

)



λe

−λs

vds

)

≤ lim

K,M→∞



λe

−λs

T

vds

≤ lim

K,M→∞

v



λe

−λs

=0,

which holds because of the contraction property and the completeness of B.

Since



∞

λe

−λs

ds =



∞

−



−λs



ds =1, (7.2.8)

v is a weighted mean of the semigroup {T

} applied to v.Since

J

v≤



∞

λe

−λs

T

vds

≤v



∞

λe

−λs

by the contraction property

≤v

(7.2.9)

by (7.2.8), J

: B → B is a bounded linear operator with norm J

≤1.

Lemma 7.2.2: For al l v ∈ B, we have

lim

λ→∞

v = v. (7.2.10)

Proof: By (7.2.8),

v − v =



∞

λe

−λs

v − v)ds.

For δ>0, let

)



λe

−λs

v − v)ds

)



∞

λe

−λs

v − v)ds

)

Now let ε>0 be given. Since T

v is continuous in s, there exists δ>0such

that

T

v − v <

for 0 ≤ s ≤ δ

and thus also

158 7. The Heat Equation, Semigroups, and Brownian Motion

≤



λe

−λs

ds <

by (7.2.8). For each δ>0, there also exists λ

∈ R such that for all λ ≥ λ

≤



∞

λe

−λs

(T

v + v) ds

≤ 2 v



∞

λe

−λs

ds (by the contraction property)

This easily implies (7.2.10). 

Theorem 7.2.1: Let {T

}

t≥0

be a contracting semigroup with inﬁnitesimal

generator A. Then D(A) is dense in B.

Proof: We shall show that for all λ>0 and all v ∈ B,

v ∈ D(A). (7.2.11)

Since by Lemma 7.2.2,

v : λ>0,v ∈ B}

is dense in B, this will imply the assertion. We have

− Id)J

v =



∞

λe

−λs

t+s

vds−



∞

λe

−λs

vds

since T

is continuous and linear



∞

λe

λt

−λσ

vdσ−



∞

λe

−λs

vds

λt

− 1



∞

λe

−λσ

vdσ−



λe

−λs

vds

λt

− 1



v −



λe

−λσ

vdσ



−



λe

−λs

vds.

The last term, the integral being continuous in s,fort → 0 tends to

−λT

v = −λv, while the ﬁrst term in the last line tends to λJ

v.This

implies

v = λ (J

− Id) v for all v ∈ B, (7.2.12)

which in turn implies (7.2.11). 

For a contracting semigroup {T

}

t≥0

, we now deﬁne operators

7.2 Inﬁnitesimal Generators of Semigroups 159

: D(D

)(⊂ B) → B

v := lim

h→0

t+h

− T

) v, (7.2.13)

where D(D

) is the subspace of B where this limit exists.

Lemma 7.2.3: v ∈ D(A) implies v ∈ D(D

), and we have

v = AT

v = T

Av for t ≥ 0. (7.2.14)

Proof: The second equation has already established shown in Lemma 7.2.1.

We thus have for v ∈ D(A),

lim

h0

t+h

− T

) v = AT

v = T

Av. (7.2.15)

Equation (7.2.15) means that the right derivative of T

v with respect to t

exists for all v ∈ D(A) and is continuous in t. By a well-known calculus

lemma, this then implies that the left derivative exists as well and coincides

with the right one, implying diﬀerentiability and (7.2.14). (The proof of the

calculus lemma goes as follows: Let f :[0, ∞) → B be continuous, and

suppose that for all t ≥ 0, the right derivative d

f(t) := lim

h0

(f(t +

h) − f(t)) exists and is continuous. The continuity of d

f implies that on

every interval [0,T] this limit relation even holds uniformly in t.Inorderto

conclude that f is diﬀerentiable with derivative d

f, one argues that

lim

h0

)

(f(t) − f(t − h)) − d

f(t)

)

≤ lim

h0

)

(f((t − h)+h) − f(t − h)) − d

f(t − h)

)

+ lim

h0

)

f(t − h) − d

f(t)

)

=0. )



Theorem 7.2.2: For λ>0, the operator (λ Id −A):D(A) → B is invertible

(A being the inﬁnitesimal generator of a contracting semigroup), and we have

(λ Id −A)

−1

= R(λ, A):=

, (7.2.16)

i.e.,

(λ Id −A)

−1

v = R(λ, A)v =



∞

−λs

vds. (7.2.17)

160 7. The Heat Equation, Semigroups, and Brownian Motion

Proof: In order that (λ Id −A) be invertible, we need to show ﬁrst that

(λ Id −A) is injective. So, we need to exclude that there exists v

∈ D(A),

=0,with

λv

= Av

. (7.2.18)

For such a v

, we would have by (7.2.14)

= T

= λT

, (7.2.19)

and hence

= e

λt

. (7.2.20)

Since λ>0, for v

= 0 this would violate the contraction property

T

≤v

,

however. Therefore, (λ Id −A)isinvertibleforλ>0. In order to obtain

(7.2.16), we start with (7.2.12), i.e.,

v = λ(J

− Id)v,

and get

(λ Id −A)J

v = λv. (7.2.21)

Therefore, (λ Id −A) maps the image of J

bijectively onto B. Since this

image is dense in D(A) by (7.2.11), and since (λ Id −A) is injective, (λ Id −A)

then also has to map D(A) bijectively onto B.Thus,D(A) has to coincide

with the image of J

, and (7.2.21) then implies (7.2.16). 

Lemma 7.2.4 (resolvent equation): Under the assumptions of Theorem

7.2.2, we have for λ, μ > 0,

R(λ, A) − R(μ, A)=(μ − λ)R(λ, A)R(μ, A). (7.2.22)

Proof:

R(λ, A)=R(λ, A)(μ Id −A)R(μ, A)

= R(λ, A)((μ − λ)Id+(λ Id −A))R(μ, A)

=(μ − λ)R(λ, A)R(μ, A)+R(μ, A).



We now want to compute the inﬁnitesimal generators of the two examples

we have considered with the help of the preceding formalism. We begin with

the translation semigroup: B here is the Banach space of bounded, uniformly

7.2 Inﬁnitesimal Generators of Semigroups 161

continuous functions on [0, ∞], and T

f(x)=f(x + t)forf ∈ B, x, t ≥ 0. We

then have

f)(x)=



∞

λe

−λs

f(x + s)ds =



∞

λe

−λ(s−x)

f(s)ds, (7.2.23)

and hence

f)(x)=−λf(x)+λ(J

f)(x). (7.2.24)

By (7.2.12), the inﬁnitesimal generator satisﬁes

f(x)=λ(J

f − f)(x), (7.2.25)

and consequently

f =

f. (7.2.26)

At the end of the proof of Theorem 7.2.2, we have seen that the image of J

coincides with D(A), and we thus have

Ag =

g for all g ∈ D(A). (7.2.27)

We now intend to show that D(A) contains precisely those g ∈ B for which

g belongs to B as well. For such a g, we deﬁne f ∈ B by

g(x) − λg(x)=−λf (x). (7.2.28)

By (7.2.24), we then also have

f)(x) − λJ

f(x)=−λf(x). (7.2.29)

Thus

ϕ(x):=g(x) − J

f(x)

satisﬁes

ϕ(x)=λϕ(x), (7.2.30)

whence ϕ(x)=ce

λx

, and since ϕ ∈ B, necessarily c =0,andsog = J

We thus have veriﬁed that the inﬁnitesimal generator A is given by

(7.2.27), with the domain of deﬁnition D(A) containing precisely those g ∈ B

for which

g ∈ B as well.

162 7. The Heat Equation, Semigroups, and Brownian Motion

We now want to study the heat semigroup according to the same pattern.

Let B be the Banach space of bounded, uniformly continuous functions on

,and

f(x)=

(4πt)



−

|x−y|

f(y)dy for t>0. (7.2.31)

We now have

f(x)=



∞

(4πt)

−λt−

|x−y|

dtf(y)dy. (7.2.32)

We compute

ΔJ

f(x)=



∞

(4πt)

−λt−

|x−y|

dtf(y)dy



∞

λe

−λt

∂

∂t



(4πt)

−

|x−y|



dtf(y)dy

= −λf(x) −



∞

∂

∂t



λe

−λt



(4πt)

−

|x−y|

dtf(y)dy

= −λf(x)+λJ

f(x).

It follows as before that

f = ΔJ

f, (7.2.33)

and thus

Ag = Δg for all g ∈ D(A). (7.2.34)

We now want to show that this time, D(A) contains all those g ∈ B for which

Δg is contained in B as well. For such a g, we deﬁne f ∈ B by

Δg(x) − λg(x)=−λf (x) (7.2.35)

and compare this with

ΔJ

f(x) − λJ

f(x)=−λf(x). (7.2.36)

Thus ϕ := g − J

f is bounded and satisﬁes

Δϕ − λϕ =0 forλ>0. (7.2.37)

The next lemma will imply ϕ ≡ 0, whence g = J

f as desired:

Lemma 7.2.5: Let λ>0. There does not exist ϕ ≡ 0 with

Δϕ(x)=λϕ(x) for all x ∈ R

. (7.2.38)

7.2 Inﬁnitesimal Generators of Semigroups 163

Proof: For a solution of (7.2.38), we compute

Δϕ

=2|∇ϕ|

+2ϕΔϕ



with ∇ϕ =



∂

∂x

ϕ,...,

∂

∂x



=2|∇ϕ|

+2λϕ

by (7.2.38). (7.2.39)

Let x

∈ R

. We choose C

-functions η

for R ≥ 1with

0 ≤ η

(x) ≤ 1 for all x ∈ R

, (7.2.40)

(x)=0 for |x − x

|≥R +1, (7.2.41)

(x)=1 for |x − x

|≤R, (7.2.42)

|∇η

(x)| + |Δη

(x)|≤c

with a constant c

that does (7.2.43)

not depend on x and R.

We compute





= η

Δϕ

+ ϕ

Δη

+8η

ϕ∇η

·∇ϕ

≥ 2η

|∇ϕ|

+2λη



Δη



− 2η

|∇ϕ|

− 8 |∇η

by (7.2.39) and the Schwarz inequality

=2λη



Δη

− 8 |∇η



. (7.2.44)

Together with (7.2.40)–(7.2.43), this implies



B(x

,R+1)





≥ 2λ



B(x

,R)

− c



B(x

,R+1)\B(x

,R)

(7.2.45)

where the constant c

does not depend on R.

By assumption, ϕ is bounded, so

≤ K. (7.2.46)

Thus (7.2.45) implies



B(x

,R)

≤

d−1

, (7.2.47)

where the constant c

again is independent of R. Equation (7.2.39) implies

that ϕ is subharmonic. The mean value inequality (cf. Theorem 7.2.2) thus

implies

) ≤



B(x

,R)

≤

λR

(by (7.2.47)) → 0forR →∞.

(7.2.48)

Thus, ϕ(x

) = 0. Since this holds for all x

∈ R

, ϕ has to vanish identically.



164 7. The Heat Equation, Semigroups, and Brownian Motion

Lemma 7.2.6: Let B be a Banach space, L : B → B a continuous linear

operator with L≤1. Then for every t ≥ 0 and each x ∈ B, the series

exp(tL)x :=

∞



ν=0

ν!

(tL)

converges and deﬁnes a continuous semigroup with inﬁnitesimal generator L.

Proof: Because of L≤1, we also have

L

≤1 for all n ∈ N. (7.2.49)

Thus

)



ν=m

ν!

(tL)

)

≤



ν=m

ν!

L

x≤x



ν=m

ν!

. (7.2.50)

By the Cauchy property of the real-valued exponential series, the last ex-

pression becomes arbitrarily small for suﬃciently large m, n, and thus our

Banach-space-valued exponential series satisﬁes the Cauchy property as well,

and therefore it converges, since B is complete. The limit exp(tL) is bounded,

because by (7.2.50)

)



ν=0

ν!

(tL)

)

≤ e

x

and thus also

exp(tL)x≤e

x. (7.2.51)

As for the real exponential series, we have

∞



ν=0

(t + s)

ν!

x =



∞



μ=0

μ!



∞



σ=0

σ!



x, (7.2.52)

i.e.,

exp((t + s)L)=exptL ◦ exp sL, (7.2.53)

whence the semigroup property. Furthermore ,

)

(exp(hL) − Id) x − Lx

)

≤

∞



ν=2

ν−1

ν!

L

x≤x

∞



ν=2

ν−1

ν!

Since the last expression tends to 0 as h → 0, h is the inﬁnitesimal generator

of the semigroup {exp(tL)}

t≥0

. 

7.2 Inﬁnitesimal Generators of Semigroups 165

In the same manner as (7.2.53), one proves (cf. (7.2.52)) the following lemma:

Lemma 7.2.7: Let L, M : B → B be continuous linear operators satisfying

the assumptions of Lemma 7.2.6, and suppose

LM = ML. (7.2.54)

Then

exp(t(M + L)) = exp(tM) ◦ exp(tL). (7.2.55)



Theorem 7.2.3 (Hille–Yosida): Let A : D(A) → B be a linear operator

whose domain of deﬁnition D(A) is dense in the Banach space B. Suppose

that the resolvent R(n, A)=(n Id −A)

−1

exists for all n ∈ N, and that

)



Id −



−1

)

≤ 1 for all n ∈ N. (7.2.56)

Then A generates a unique contracting semigroup.

Proof: As before, we put



Id −



−1

for n ∈ N (cf. Theorem 7.2.2).

The proof will consist of several steps:

(1) We claim

lim

n→∞

x = x for all x ∈ B, (7.2.57)

and

x ∈ D(A) for all x ∈ B. (7.2.58)

Namely, for x ∈ D(A), we ﬁrst have

x = J

Ax = J

(A − n Id)x + nJ

x = n(J

− Id)x, (7.2.59)

and since by assumption J

Ax≤Ax, it follows that

x − x =

Ax → 0forn →∞.

As D(A) is dense in B and the operators J

are equicontinuous by our

assumptions, (7.2.57) follows. (7.2.59) then also implies (7.2.58).

166 7. The Heat Equation, Semigroups, and Brownian Motion

(2) By Lemma 7.2.6, the semigroup

{exp(sJ

)}

s≥0

exists, because of (7.2.56). Putting s = tn, we obtain the semigroup

{exp(tnJ

)}

t≥0

and likewise the semigroup

(n)

:= exp(tAJ

)=exp(tn(J

− Id)) (t ≥ 0)

(cf. (7.2.59)). By Lemma 7.2.7, we then have

(n)

=exp(−tn)exp(tnJ

). (7.2.60)

Since by (7.2.56)

exp(tnJ

)x≤

∞



ν=0

(nt)

ν!

J

x≤exp(nt) x,

it follows that

)

(n)

)

≤ 1, (7.2.61)

and thus in particular, the operators are equicontinuous in t ≥ 0and

n ∈ N.

(3) For all m, n ∈ N,wehave

= J

. (7.2.62)

Since by (7.2.60), J

commutes with T

(n)

,thenalsoJ

commutes with

(n)

for all n, m ∈ N, t ≥ 0. By Lemmas 7.2.3, 7.2.6, we have for x ∈ B,

(n)

x = AJ

(n)

x = T

(n)

x; (7.2.63)

hence

)

(n)

x − T

(m)

)





(m)

t−s

(n)



)



(m)

t−s

(n)

(AJ

− AJ

) xds

)

≤ t (AJ

− AJ

)x

(7.2.64)

with (7.2.61). For x ∈ D(A), we have by (7.2.59)

(AJ

− AJ

) x =(J

− J

) Ax. (7.2.65)