On the 𝐿_{𝑝}-boundedness of the stochastic singular integral operators and its application to 𝐿_{𝑝}-regularity theory of stochastic partial differential equations
Authors: Kim Ildoo , Kim Kyeong-Hun 
Published: January 2020

In this article we introduce a stochastic counterpart of the Hörmander condition and Calderón-Zygmund theorem. Let

W
t

W_t

be a Wiener process in a probability space

Ω
\Omega

and let

K
(
ω
,
r
,
t
,
x
,
y
)

K(\omega ,r,t,x,y)

be a random kernel which is allowed to be stochastically singular in a domain

O

⊂

R

d

\mathcal {O} \subset \mathbf {R}^d

in the sense that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper E StartAbsoluteValue integral Subscript 0 Superscript t Baseline integral Underscript StartAbsoluteValue x minus y EndAbsoluteValue greater-than epsilon Endscripts EndAbsoluteValue upper K left-parenthesis omega comma s comma t comma y comma x right-parenthesis StartAbsoluteValue d y d upper W Subscript s Baseline EndAbsoluteValue Superscript p Baseline equals normal infinity for-all t comma p comma epsilon greater-than 0 comma x element-of script upper O period">

E

|

∫
0

t

∫

|

x
−
y

|

>
ε

|

K
(
ω
,
s
,
t
,
y
,
x
)

|

d
y
d

W
s

|

p

=
∞

∀

t
,
p
,
ε
>
0
,

x
∈

O

.

\begin{equation*} \mathbb {E} \left |\int _0^{t} \int _{|x-y|>\varepsilon }|K(\omega , s, t,y,x)|dy dW_s\right |^p = \infty \quad \forall \, t, p,\varepsilon >0,\, x\in \mathcal {O}. \end{equation*}

We prove that the stochastic integral operator of the type
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row double-struck upper T g left-parenthesis t comma x right-parenthesis colon-equal integral Subscript 0 Superscript t Baseline integral Underscript script upper O Endscripts upper K left-parenthesis omega comma s comma t comma y comma x right-parenthesis g left-parenthesis s comma y right-parenthesis d y d upper W Subscript s Baseline EndLayout">

T

g
(
t
,
x
)

≔

∫
0

t

∫

O

K
(
ω
,
s
,
t
,
y
,
x
)
g
(
s
,
y
)
d
y
d

W
s

\begin{align} \mathbb {T} g(t,x) \coloneq \int _0^{t} \int _{\mathcal {O}} K(\omega ,s,t,y,x) g(s,y)dy dW_s \end{align}

is bounded on
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper L Subscript p Baseline equals upper L Subscript p Baseline left-parenthesis normal upper Omega times left-parenthesis 0 comma normal infinity right-parenthesis semicolon upper L Subscript p Baseline left-parenthesis script upper O right-parenthesis right-parenthesis">

L

p

=

L
p

(
Ω
×
(
0
,
∞
)
;

L

p

(

O

)
)

\mathbb {L}_p=L_p \left (\Omega \times (0,\infty ); L_{p}(\mathcal {O}) \right )

for all

p
∈
[
2
,
∞
)

p \in [2,\infty )

if it is bounded on

L

2

\mathbb {L}_2

and the following (which we call stochastic Hörmander condition) holds: there exists a quasi-metric

ρ
\rho

on

(
0
,
∞
)
×

O

(0,\infty )\times \mathcal {O}

and a positive constant

C
0

C_0

such that for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X equals left-parenthesis t comma x right-parenthesis comma upper Y equals left-parenthesis s comma y right-parenthesis comma upper Z equals left-parenthesis r comma z right-parenthesis element-of left-parenthesis 0 comma normal infinity right-parenthesis times script upper O">

X
=
(
t
,
x
)
,
Y
=
(
s
,
y
)
,
Z
=
(
r
,
z
)
∈
(
0
,
∞
)
×

O

X=(t,x), Y=(s,y), Z=(r,z) \in (0,\infty ) \times \mathcal {O}

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript omega element-of normal upper Omega comma upper X comma upper Y Endscripts integral Subscript 0 Superscript normal infinity Baseline left-bracket integral Underscript rho left-parenthesis upper X comma upper Z right-parenthesis greater-than-or-equal-to upper C 0 rho left-parenthesis upper X comma upper Y right-parenthesis Endscripts StartAbsoluteValue upper K left-parenthesis r comma t comma z comma x right-parenthesis minus upper K left-parenthesis r comma s comma z comma y right-parenthesis EndAbsoluteValue d z right-bracket squared d r greater-than normal infinity period">

sup

ω
∈
Ω
,
X
,
Y

∫

0

∞

[

∫

ρ
(
X
,
Z
)
≥

C
0

ρ
(
X
,
Y
)

|

K
(
r
,
t
,
z
,
x
)
−
K
(
r
,
s
,
z
,
y
)

|

 
d
z
]

2

d
r
>
∞
.

\begin{equation*} \sup _{\omega \in \Omega ,X,Y}\int _{0}^\infty \left [ \int _{\rho (X,Z) \geq C_0 \rho (X,Y)} | K(r,t, z,x) - K(r,s, z,y)| ~dz\right ]^2 dr >\infty . \end{equation*}

Such a stochastic singular integral naturally appears when one proves the maximal regularity of solutions to stochastic partial differential equations (SPDEs). As applications, we obtain the sharp

L
p

L_p

-regularity result for a wide class of SPDEs, which includes SPDEs with time measurable pseudo-differential operators and SPDEs defined on non-smooth angular domains.