Perturbed semigroup limit theorems with applications to discontinuous random evolutions
Authors: Kertz Robert P. 
Published: July 2010

For

ε
>
0

\varepsilon > 0

small, let

U
ε

(
t
)

{U^\varepsilon }(t)

and

S
(
t
)

S(t)

be strongly continuous semigroups of linear contractions on a Banach space

L
L

with infinitesimal operators

A
(
ε
)

A(\varepsilon )

and

B
B

respectively, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis epsilon right-parenthesis equals upper A Superscript left-parenthesis 1 right-parenthesis Baseline plus epsilon upper A Superscript left-parenthesis 2 right-parenthesis Baseline plus o left-parenthesis right-parenthesis">

A
(
ε
)
=

A

(
1
)

+

ε

A

(
2
)

+
o
(
)

A(\varepsilon ) = {A^{(1)}} + {\varepsilon A^{(2)}} + o()

as

ε
→
0

\varepsilon \to 0

. Let

{
B
(
u
)
;
u
⩾
0
}

\{ B(u);u \geqslant 0\}

be a family of linear operators on

L
L

satisfying
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B left-parenthesis epsilon right-parenthesis equals upper B plus epsilon normal upper Pi Superscript left-parenthesis 1 right-parenthesis Baseline plus epsilon squared normal upper Pi Superscript epsilon left-parenthesis 2 right-parenthesis Baseline plus o left-parenthesis epsilon squared right-parenthesis">

B
(
ε
)
=
B
+

ε

Π

(
1
)

+

ε
2

Π

ε
(
2
)

+
o
(

ε
2

)

B(\varepsilon ) = B + {\varepsilon \Pi ^{(1)}} + {\varepsilon ^2}{\Pi ^{\varepsilon (2)}} + o({\varepsilon ^2})

as

ε
→
0

\varepsilon \to 0

. Assume that

A
(
ε
)
+

ε

−
1

B
(
)

A(\varepsilon ) + {\varepsilon ^{ - 1}}B()

is the infinitesimal operator of a strongly continuous contraction semigroup

T
ε

(
t
)

{T_\varepsilon }(t)

on

L
L

and that for each
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper L comma limit Underscript lamda right-arrow 0 Endscripts lamda integral Subscript 0 Superscript normal infinity Baseline e Superscript minus lamda t Baseline upper S left-parenthesis t right-parenthesis f d t identical-to upper P f">

f
∈
L
,

lim

λ
→
0

λ

∫
0
∞

e

−
λ
t

S
(
t
)
f
d
t
≡
P
f

f \in L,{\lim _{\lambda \to 0}}\lambda \int _0^\infty {{e^{ - \lambda t}}} S(t)fdt \equiv Pf

exists. We give conditions under which

T
ε

(
t
)

{T_\varepsilon }(t)

converges as

→
0

\to 0

to the semigroup generated by the closure of

P
(

A

(
1
)

+

Π

(
1
)

)

P({A^{(1)}} + {\Pi ^{(1)}})

on
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R left-parenthesis upper P right-parenthesis intersection script upper D left-parenthesis upper A Superscript left-parenthesis 1 right-parenthesis Baseline right-parenthesis intersection script upper D left-parenthesis normal upper Pi Superscript left-parenthesis 1 right-parenthesis Baseline right-parenthesis">

R

(
P
)
∩

D

(

A

(
1
)

)
∩

D

(

Π

(
1
)

)

\mathcal {R}(P) \cap \mathcal {D}({A^{(1)}}) \cap \mathcal {D}({\Pi ^{(1)}})

. If
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P left-parenthesis upper A Superscript left-parenthesis 1 right-parenthesis Baseline plus normal upper Pi Superscript left-parenthesis 1 right-parenthesis Baseline right-parenthesis f equals 0 comma upper B h equals minus left-parenthesis upper A Superscript left-parenthesis 1 right-parenthesis Baseline plus normal upper Pi Superscript left-parenthesis 1 right-parenthesis Baseline right-parenthesis f">

P
(

A

(
1
)

+

Π

(
1
)

)
f
=
0
,
B
h
=
−
(

A

(
1
)

+

Π

(
1
)

)
f

P({A^{(1)}} + {\Pi ^{(1)}})f = 0,Bh = - ({A^{(1)}} + {\Pi ^{(1)}})f

, and we let

V
^

f
=
P
(

A

(
1
)

+

Π

(
1
)

)
h

\hat Vf = P({A^{(1)}} + {\Pi ^{(1)}})h

, then we show that

T
ε

(
t

/

ε
)
f

{T_\varepsilon }(t/\varepsilon )f

converges as

ε
→
0

\varepsilon \to 0

to the strongly continuous contraction semigroup generated by the closure of

V

(
2
)

+

V
^

{V^{(2)}} + \hat V

. From these results we obtain new limit theorems for discontinuous random evolutions and new characterizations of the limiting infinitesimal operators of the discontinuous random evolutions. We apply these results in a model for the approximation of physical Brownian motion and in a model of the content of an infinite capacity dam.