Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes
Authors: van Neerven Jan , Veraar Mark 
Published: July 2021

We prove a new Burkholder–Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if $$(S(t,s))_{0\leqslant s\le t\leqslant T}$$

(
S

(
t
,
s
)

)

0
⩽
s
≤
t
⩽
T

is a $$C_0$$

C
0

-evolution family of contractions on a 2-smooth Banach space X and $$(W_t)_{t\in [0,T]}$$

(

W
t

)

t
∈
[
0
,
T
]

is a cylindrical Brownian motion on a probability space $$(\Omega ,{\mathbb {P}})$$

(
Ω
,
P
)

adapted to some given filtration, then for every $$0<p<\infty $$

0
<
p
<
∞

there exists a constant $$C_{p,X}$$

C

p
,
X

such that for all progressively measurable processes $$g: [0,T]\times \Omega \rightarrow X$$

g
:
[
0
,
T
]
×
Ω
→
X

the process $$(\int _0^t S(t,s)g_s\,\mathrm{d} W_s)_{t\in [0,T]}$$

(

∫
0
t

S

(
t
,
s
)

g
s

d

W
s

)

t
∈
[
0
,
T
]

has a continuous modification and $$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}\Big \Vert \int _0^t S(t,s)g_s\,\mathrm{d} W_s \Big \Vert ^p\leqslant C_{p,X}^p {\mathbb {E}} \Bigl (\int _0^T \Vert g_t\Vert ^2_{\gamma (H,X)}\,\mathrm{d} t\Bigr )^{p/2}. \end{aligned}$$

E

sup

t
∈
[
0
,
T
]

‖

∫
0
t

S

(
t
,
s
)

g
s

d

W
s

‖

p

⩽

C

p
,
X

p

E

(

∫
0
T

‖

g
t

‖

γ
(
H
,
X
)

2

d
t

)

p
/
2

.

Moreover, for $$2\leqslant p<\infty $$

2
⩽
p
<
∞

one may take $$C_{p,X} = 10 D \sqrt{p},$$

C

p
,
X

=
10
D

p

,

where D is the constant in the definition of 2-smoothness for X. The order $$O(\sqrt{p})$$

O
(

p

)

coincides with that of Burkholder’s inequality and is therefore optimal as $$p\rightarrow \infty $$

p
→
∞

. Our result improves and unifies several existing maximal estimates and is even new in case X is a Hilbert space. Similar results are obtained if the driving martingale $$g_t\,\mathrm{d} W_t$$

g
t

d

W
t

is replaced by more general X-valued martingales $$\,\mathrm{d} M_t$$

d

M
t

. Moreover, our methods allow for random evolution systems, a setting which appears to be completely new as far as maximal inequalities are concerned. As a second application, for a large class of time discretisation schemes (including splitting, implicit Euler, Crank-Nicholson, and other rational schemes) we obtain stability and pathwise uniform convergence of time discretisation schemes for solutions of linear SPDEs $$\begin{aligned} \,\mathrm{d} u_t = A(t)u_t\,\mathrm{d} t + g_t\,\mathrm{d} W_t, \quad u_0 = 0, \end{aligned}$$

d

u
t

=
A

(
t
)

u
t

d
t
+

g
t

d

W
t

,

u
0

=
0
,

where the family $$(A(t))_{t\in [0,T]}$$

(
A

(
t
)

)

t
∈
[
0
,
T
]

is assumed to generate a $$C_0$$

C
0

-evolution family $$(S(t,s))_{0\leqslant s\leqslant t\leqslant T}$$

(
S

(
t
,
s
)

)

0
⩽
s
⩽
t
⩽
T

of contractions on a 2-smooth Banach spaces X. Under spatial smoothness assumptions on the inhomogeneity g, contractivity is not needed and explicit decay rates are obtained. In the parabolic setting this sharpens several know estimates in the literature; beyond the parabolic setting this seems to provide the first systematic approach to pathwise uniform convergence to time discretisation schemes.