An 𝐿_{𝑝}-theory for diffusion equations related to stochastic processes with non-stationary independent increment
Authors: Kim Ildoo , Kim Kyeong-Hun , Kim Panki 
Published: September 2017

Let

X
=
(

X
t

)

t
≥
0

X=(X_t)_{t \ge 0}

be a stochastic process which has a (not necessarily stationary) independent increment on a probability space

(
Ω
,

P

)

(\Omega , \mathbb {P})

. In this paper, we study the following Cauchy problem related to the stochastic process

X
X

:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel StartFraction partial-differential u Over partial-differential t EndFraction left-parenthesis t comma x right-parenthesis equals script upper A left-parenthesis t right-parenthesis u left-parenthesis t comma x right-parenthesis plus f left-parenthesis t comma x right-parenthesis comma u left-parenthesis 0 comma dot right-parenthesis equals 0 comma left-parenthesis t comma x right-parenthesis element-of left-parenthesis 0 comma upper T right-parenthesis times bold upper R Superscript d Baseline comma EndLayout">

(*)

∂
u

∂
t

(
t
,
x
)
=

A

(
t
)
u
(
t
,
x
)
+
f
(
t
,
x
)
,

u
(
0
,
⋅
)
=
0
,

(
t
,
x
)
∈
(
0
,
T
)
×

R

d

,

\begin{align} \tag {*}\frac {\partial u}{\partial t}(t,x) = \mathcal {A}(t)u(t,x) +f(t,x), \quad u(0,\cdot )=0, \quad (t,x) \in (0,T) \times \mathbf {R}^d, \end{align}

where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper L Subscript p Baseline left-parenthesis left-parenthesis 0 comma upper T right-parenthesis semicolon upper L Subscript p Baseline left-parenthesis bold upper R Superscript d Baseline right-parenthesis right-parenthesis equals upper L Subscript p Baseline left-parenthesis left-parenthesis 0 comma upper T right-parenthesis semicolon upper L Subscript p Baseline right-parenthesis">

f
∈

L
p

(
(
0
,
T
)
;

L
p

(

R

d

)
)
=

L
p

(
(
0
,
T
)
;

L
p

)

f \in L_p( (0,T) ; L_p(\mathbf {R}^d))=L_p( (0,T) ; L_p)

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row script upper A left-parenthesis t right-parenthesis u left-parenthesis t comma x right-parenthesis equals limit Underscript h down-arrow 0 Endscripts StartFraction double-struck upper E left-bracket u left-parenthesis t comma x plus upper X Subscript t plus h Baseline minus upper X Subscript t Baseline right-parenthesis minus u left-parenthesis t comma x right-parenthesis right-bracket Over h EndFraction period EndLayout">

A

(
t
)
u
(
t
,
x
)
=

lim

h
↓
0

E

[
u
(
t
,
x
+

X

t
+
h

−

X
t

)
−
u
(
t
,
x
)
]

h

.

\begin{align*} \mathcal {A}(t)u(t,x) = \lim _{h \downarrow 0}\frac {\mathbb {E}\left [u(t,x+X_{t+h}-X_t)-u(t,x)\right ]}{h}. \end{align*}

We provide a sufficient condition on

X
X

(see Assumptions 2.1 and 2.2) to guarantee the unique solvability of equation (*) in

L
p

(
[
0
,
T
]
;

H

p

ϕ

)

L_p( [0,T] ; H^\phi _{p})

, where

H

p

ϕ

H^\phi _{p}

is a

ϕ
\phi

-potential space on

R

d

\mathbf {R}^d

(see Definition 2.9). Furthermore we show that for this solution,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row double-vertical-bar u double-vertical-bar Subscript upper L Sub Subscript p Subscript left-parenthesis left-bracket 0 comma upper T right-bracket semicolon upper H Sub Subscript p Sub Superscript phi Subscript right-parenthesis Baseline less-than-or-equal-to upper N double-vertical-bar f double-vertical-bar Subscript upper L Sub Subscript p Subscript left-parenthesis left-bracket 0 comma upper T right-bracket semicolon upper L Sub Subscript p Subscript right-parenthesis Baseline comma EndLayout">

‖
u

‖

L
p

(
[
0
,
T
]
;

H

p

ϕ

)

≤
N
‖
f

‖

L
p

(
[
0
,
T
]
;

L
p

)

,

\begin{align*} \| u\|_{L_p( [0,T] ; H^\phi _{p})} \leq N \|f\|_{L_p\left ( [0,T] ; L_p\right )}, \end{align*}

where

N
N

is independent of

u
u

and

f
f

.