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ESAIM: COCV 30 (2024) 86 ESAIM: Control, Optimisation and Calculus of Variations
https://doi.org/10.1051/cocv/2024074 www.esaim-cocv.org
QUANTITATIVE UNIQUENESS ESTIMATES FOR STOCHASTIC
PARABOLIC EQUATIONS ON THE WHOLE EUCLIDEAN SPACE
Yuanhang Liu1, Donghui Yang1, Xingwu Zeng2,*and Can Zhang2
Abstract. In this paper, a quantitative estimate of unique continuation for the stochastic heat equa-
tion with bounded potentials on the whole Euclidean space is established. This paper generalizes the
earlier results in [X. Zhang. Differ. Integral Equ. 21 (2008) 81–93] and [Q. L¨u and Z. Yin ESAIM
Control Optim. Calc. Var. 21 (2015) 378–398] from a bounded domain to an unbounded one. The
proof is based on the locally parabolic-type frequency function method. An observability estimate from
measurable sets in time for the same equation is also derived.
Mathematics Subject Classification. 60H15, 93B05.
Received March 03, 2024. Accepted September 15, 2024.
1. Introduction
The study of unique continuation for solutions to deterministic partial differential equations comes from the
classical Cauchy–Kovalevskaya theorem (see, e.g., [1]). Besides in the theory of partial differential equations, it is
of great significance in both Inverse Problem and Control Theory (see, for instance, [2–4]). The classical unique
continuation property is of a qualitative nature, ensuring that the solution within a given domain can be uniquely
determined by its value within a suitable subdomain. After establishing the unique continuation property, a
natural question arises: Can one develop a method to recover the solution within the domain only based on
the values of the solution within the subdomain? The ill-posedness of the non-characteristic Cauchy problem
is widely known, indicating that a minor error in the data within the subdomain can lead to uncontrollable
ramifications on the solution within the domain (see, for example, [5]). Hence, the stability estimate for the
solution is of importance. For an introduction to this sub ject, we refer the reader to [2].
There are rich references addressing to unique continuation not only for deterministic parabolic equations
(see, e.g., [6–11]), but also for the stochastic counterpart in bounded domains. The result in [12] first showed
that a solution to the stochastic parabolic equation (without boundary condition) evolving in a bounded domain
G⊂RN(N∈N) would vanish identically P-a.s., provided that it vanishes in G0×(0, T ), P-a.s., where G0⊆G.
In [13], the author obtained an interpolation inequality for stochastic parabolic equations by Carleman estimates,
which implied a conditional stability result for stochastic parabolic equations. In [14], the authors proved that
a solution to the stochastic parabolic equation (with a partial homogeneous Dirichlet boundary condition on
arbitrary open subset Γ0of ∂G) evolving in Gvanishes P-a.s., provided that its normal derivative equals zero
Keywords and phrases: Stochastic parabolic equation, unique continuation, unbounded domain.
1School of Mathematics and Statistics, Central South University, Changsha 410083, PR China.
2School of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR China.
*Corresponding author: xingwuzeng@whu.edu.cn
©
The authors. Published by EDP Sciences, SMAI 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
2Y. LIU ET AL.
in Γ0×(0, T ), P-a.s. In [15], the authors established a unique continuation property for stochastic parabolic
equations evolving in a domain G⊂RN. They demonstrated that the solution can be uniquely determined
based on its values on any open subdomain of Gat each single point of time. Moreover, when Gis convex
and bounded, they also provided a quantitative version of unique continuation. In [16], the authors proved
a qualitative unique continuation at two points in time for a stochastic parabolic equation with a randomly
perturbed potential. This result can be considered as a variant of Hardy’s uncertainty principle for stochastic
parabolic evolutions. In [17], the authors proved a local unique continuation property for stochastic hyperbolic
equations without boundary conditions to solve a local state observation problem.
More recently, in [18], the authors established a two-ball and one-cylinder inequality based on a new Carleman
estimate with both time and space boundary observation terms for the stochastic parabolic equations in a
bounded domain, see [18], Section 3 for more details. They utilized these quantitative unique continuation
properties to obtain the stability estimate for the determination of the unknown time-varying boundaries.
The unique continuation estimate for deterministic partial differential equations in an unbounded domain has
been also widely studied over the last decade. In [19], the author proved a unique continuation estimate for the
Kolmogorov equation in the whole space by a spectral inequality and a decay inequality on the Fourier transform
of the solution. In [20], the authors proved that the unique continuation estimate for the pure heat equation
in Rnholds if and only if the unbounded observable set is thick set. In [21,22], the authors proved a global
interpolation inequality for solutions of the heat equation with bounded potential at one point of time variable
using the parabolic-type frequency function method. In [23], the authors proved a H¨older-type interpolation
inequalities of unique continuation for fractional order parabolic equations with space-time dependent potentials
on a thick set. However, to the best of our knowledge, the question of the unique continuation estimate in an
unbounded domain for the stochastic counterpart is still open.
The observability inequality for stochastic parabolic equations on a bounded domain has been extensively
studied over the past decades. In the case that the observation time is the entire time interval and the observation
spatial region is a nonempty open subset, we refer the reader to [24] and the references therein. In those works,
the proofs are almost based on the method of Carleman estimates. Alternatively, when the observation time
region constitutes only a subset of positive Lebesgue measure within the time interval, and the observation
spatial region is a nonempty open subset, we refer the reader to [25]. In a more general context, when the
observation subdomain constitutes a measurable subset of positive measure in both space and time variables,
we refer the reader to [26]. There are few existing results on the observability inequality for stochastic parabolic
equations in an unbounded domain.
The main contribution of this paper is that we establish the quantitative estimate of unique continuation
for the stochastic heat equation with bounded and time-dependent potentials on the whole space, by using
the locally parabolic-type frequency function method. More precisely, we prove a H¨older-type interpolation
inequality for stochastic parabolic equations (see Thm. 2.1 below), which extends a result already given in
[15], Theorem 1.6 from bounded to unbounded domains. This result seems to be discussed for the first time.
As a direct application, we obtain an observability inequality from measurable sets in time for the stochastic
parabolic equation.
We remark that the parabolic-type frequency function method has been well developed in [27], Theorem
6 and [10], Lemma 5 for the deterministic case, while in [15], Theorem 1.6 for the stochastic case. In this
paper, we first employ the parabolic frequency function method to derive a locally quantitative estimate of
unique continuation for the stochastic heat equation with a bounded potential, where we carefully quantify
the dependence of the constant on the L∞-norm of the involved potentials. Next, by the aforementioned local
result and the geometry of the observation subdomains, we obtain a globally quantitative estimate at a single
time point for the solutions of the stochastic heat equation with bounded potentials. Finally, we employ the
telescoping method to establish the observability inequality.
The rest of this paper is organized as follows. Section 2provides the formulation of the primary problem and
states the main result Theorem 2.1. In Section 3, we introduce several auxiliary lemmas, which are instrumental
in proving our main theorem. Section 4is dedicated to the proof of Theorem 2.1, while Section 5focuses on
deriving the observability inequality, i.e., Corollary 2.2.
QUANTITATIVE UNIQUENESS ESTIMATES FOR STOCHASTIC PARABOLIC EQUATIONS 3
2. Problem formulation and main result
Let (Ω,F,F,P) with F≜{Ft}t≥0be a complete filtered probability space on which a one dimensional
standard Brownian motion {W(t)}t≥0is defined.
Let T > 0 and Hand Vbe two separable Hilbert spaces with inner products ⟨·,·⟩H,⟨·,·⟩Vand norms ∥ ·∥H,
∥·∥V, respectively.
By L2
Ft(Ω; H), t≥0, p∈[1,∞), we denote the space consisting of all H-valued, Ft-measurable random
variables ξsuch that E∥ξ∥2
H<+∞.
By Lp
F(Ω; Lq(0, T ;H)), p, q ∈[1,∞), we denote the space consisting of all H-valued, F-adapted processes
X(·) such that E∥X(·)∥p
Lq(0,T ;H)<+∞.
By L∞
F(0, T ;V), we denote the space consisting of all V-valued, F-adapted bounded processes.
By Lq
F(Ω; C([0, T ]; H)), q∈[1,∞), we denote the space consisting of all H-valued, F-adapted continuous
processes X(·) such that E∥X(·)∥q
C([0;T];H)<+∞.
In the sequel, we simply denote Lp
F(Ω; Lp(0, T ;H)) by Lp
F(0, T ;H) with p∈[1,∞). All the above spaces are
equipped with the canonical quasi-norms.
We consider the following stochastic heat equation with a time and space dependent potential on the whole
Euclidean space
dφ−∆φdt=aφdt+bφdW(t),in RN×(0,+∞),
φ(0) = φ0,in RN,(2.1)
where φ0∈L2
F0(Ω; L2(RN)), a∈L∞
F(0,+∞;L∞(RN)) and b∈L∞
F(0,+∞;W1,∞(RN)). The well-posedness of
stochastic evolution equations is well-known (see e.g., [28], Thm. 3.14), and the equation (2.1) admits a unique
solution φ∈L2
F(Ω; C([0, T ]; L2(RN))) ∩L2
F(0, T ;H1(RN)).
Here and throughout this paper, for r > 0 and x0∈RN, we use Br(x0) to denote the closed ball centered
at x0and of radius r; and Qr(x0) to denote the smallest cube centered at x0so that Br(x0)⊂Qr(x0). Let
int(Qr(x0)) be the interior of Qr(x0). Write ∥a∥∞≜∥a∥L∞
F(0,+∞;L∞(RN)) and ∥b∥∞≜∥b∥L∞
F(0,+∞;W1,∞(RN)).
We always denote by C(·) a generic positive constant depending on what are enclosed in the brackets.
The main result of this paper can be stated as follows.
Theorem 2.1. Let 0< r < R < +∞and T > 0. Assume that there is a sequence {xi}i≥1⊂RNso that
RN=[
i≥1
QR(xi)with int(QR(xi)) \int(QR(xj)) = ∅for each i=j∈N.
Let
ω:= [
i≥1
ωiwith ωibeing an open set and Br(xi)⊂ωi⊂BR(xi)for each i∈N.
Then there are two constants C:= C(R)>0and θ:= θ(r, R)∈(0,1) such that for any φ0∈L2
F0(Ω; L2(RN)),
the corresponding solution φof (2.1) satisfies
EZRN|φ(x, T )|2dx≤eC(T−1+T+T(∥a∥∞+∥b∥2
∞)+∥a∥2/3
∞+∥b∥2
∞+1)EZRN|φ0(x)|2dxθ
×EZω|φ(x, T )|2dx1−θ
.
(2.2)
4Y. LIU ET AL.
As an immediate application of the above theorem, an observability inequality from measurable sets in time
for the solution of (2.1) can be derived.
Corollary 2.2. Let E⊂(0, T )be a Lebesgue measurable subset with a positive measure. Under the assump-
tions in Theorem 2.1, there exist positive constants C=C(r, R), and e
C=e
C(r, R, E)so that for any φ0∈
L2
F0(Ω; L2(RN)), the corresponding solution φof (2.1) satisfies
EZRN|φ(x, T )|2dx≤ee
CeC(T+T(∥a∥∞+∥b∥2
∞)+∥a∥2/3
∞+∥b∥2
∞+1)EZω×E|φ(x, t)|2dxdt.
Remark 2.3. Similar results as in Theorem 2.1 and Corollary 2.2 have been obtained in [15], Theorems 1.6
and 1.10 on a convex and bounded domain. In this paper, we get more sharper estimates and extend them to
the case of unbounded domains.
3. Preliminary lemmas
In this section, we give three auxiliary results that will be used later. The first two lemmas are standard
estimates for solutions of (2.1). For the sake of completeness we provide their detailed proofs in the Appendix.
Lemma 3.1. There is a constant C1>1so that for any φ0∈L2
F0(Ω; L2(RN)), the solution φof (2.1) satisfies
sup
t∈[T−τ1,T ]
EZBr(x0)
φ2(x, t)dx+EZT
T−τ1ZBr(x0)|∇φ(x, s)|2dxds
≤C1(R−r)−2+ (τ2−τ1)−1+∥a∥∞+∥b∥2
∞EZT
T−τ2ZBR(x0)
φ2(x, s)dxds,
(3.1)
for all 0< r < R < +∞,0< τ1< τ2< T and x0∈RN.
Lemma 3.2. There is a constant C2>0so that for any φ0∈L2
F0(Ω; L2(RN)),the solution φof (2.1) satisfies
sup
t∈[T−τ,T ]
EZBR(x0)|∇φ(x, t)|2dx≤C2R−4+τ−2+∥a∥2
∞+∥b∥4
∞EZT
T−2τZB2R(x0)
φ2(x, s)dxds, (3.2)
for all 0< R < +∞,0< τ < T /2and x0∈RN.
The following auxiliary lemma is basically motivated by [10], Lemma 3 and [21], Lemma 2.3.
Lemma 3.3. Let 0<2r≤R < +∞and δ∈(0,1]. Then there are two constants C3:= C3(r, δ)>0and
C4:= C4(r, δ)>0so that for any 0< τ1< τ2< T ,x0∈RN,φ0∈L2
F0(Ω; L2(RN)) \ {0}, the quantity
h0=C3"ln(1 + C4) + 1+2C1(1 + 1
r2)1 + 1
τ2−τ1
+∥a∥2/3
∞+∥b∥2
∞+4C3
T
+ (2∥a∥∞+∥b∥2
∞)T+ ln
ERT
T−τ2RQR(x0)φ2(x, t)dxdt
ERBr(x0)φ2(x, T )dx
#−1(3.3)
QUANTITATIVE UNIQUENESS ESTIMATES FOR STOCHASTIC PARABOLIC EQUATIONS 5
(where φsatisfies the equation (2.1) with φ0∈L2
F0(Ω; L2(RN)) \ {0}, and C1>1is the constant given by
Lemma 3.1), has the following two properties:
(i)
0<1+4C3T−1+ (2∥a∥∞+∥b∥2
∞)T+∥a∥2/3
∞+∥b∥2
∞h0< C3.(3.4)
(ii)There is a constant C5:= C5(r, δ)> C3so that
e(2∥a∥∞+∥b∥2
∞)TEZT
T−τ2ZQR(x0)
φ2(x, s)dxds≤e1+ C5
h0EZB(1+δ)r(x0)
φ2(x, t)dx(3.5)
for each t∈[T−min{τ2, h0}, T ].
Remark 3.4. In fact, by the unique continuation property and the backward uniqueness for the stochastic
parabolic equations, if φ0∈L2
F0(Ω; L2(RN)) \ {0}, then EZBr(x0)
φ2(x, T )dx = 0. The proof of the unique
continuation property for the equation (2.1) is similar with the proof of Theorem 1.2 in [15], and the backward
uniqueness for the equation (2.1) could be shown by borrowing some ideas from the proof of Lemma 3.1 in [16].
Proof. For each r′>0, we write Br′:= Br′(x0) and Qr′:= Qr′(x0). Since B2r⊂QRand
e2C1(1+r−2)[1+(τ2−τ1)−1+∥a∥2/3
∞+∥b∥2
∞]≥C1r−2+ (τ2−τ1)−1+∥a∥∞+∥b∥2
∞,
by (3.1) (where Ris replaced by 2r), we have
e2C1(1+r−2)[1+(τ2−τ1)−1+∥a∥2/3
∞+∥b∥2
∞]ERT
T−τ2RQRφ2dxdt
ERBrφ2(x, T )dx
≥C1r−2+ (τ2−τ1)−1+∥a∥∞+∥b∥2
∞ERT
T−τ2RB2rφ2dxdt
ERBrφ2(x, T )dx≥1.
Hence, (3.4) follows immediately from (3.3).
We now turn to the proof of (3.5). Let h > 0, β(x) = |x−x0|2and η∈C∞
0(B(1+δ)r) be such that
0≤η(·)≤1 in B(1+δ)rand η(·) = 1 in B(1+3δ/4)r.
Applying first the Itˆo formula to e−β/hη2φ2, and then integrating over B(1+δ)rand taking the expectation, we
get
1
2
d
dtEZB(1+δ)r
e−β/h(ηφ)2dx+EZB(1+δ)r∇φ· ∇(e−β/h η2φ)dx
=EZB(1+δ)r
ae−β/h(ηφ)2dx+1
2EZB(1+δ)r
η2e−β/hb2φ2dx.
(3.6)
Since
∇(e−β/hη2φ) = −1
he−β/hη2φ∇β+ 2e−β/hηφ∇η+ e−β/h η2∇φ,
6Y. LIU ET AL.
by (3.6), we have
1
2
d
dtEZB(1+δ)r
e−β/h(ηφ)2dx+EZB(1+δ)r
e−β/h|η∇φ|2dx
=EZB(1+δ)r
1
he−β/hη2φ∇β· ∇φdx+EZB(1+δ)r−2e−β/hηφ∇η· ∇φdx
+EZB(1+δ)r
ae−β/h(ηφ)2dx+1
2EZB(1+δ)r
η2e−β/hb2φ2dx
≤EZB(1+δ)r
e−β/(2h)|η∇φ|2
h|x−x0|e−β/(2h)η|φ|+ 2|∇η|e−β/(2h)|φ|dx
+∥a∥∞EZB(1+δ)r
e−β/h(ηφ)2dx+1
2∥b∥2
∞EZB(1+δ)r
e−β/h(ηφ)2dx.
This, along with Cauchy–Schwarz inequality, implies that
d
dtEZB(1+δ)r
e−β/h(ηφ)2dx≤4(1 + δ)2r2
h2+ 2∥a∥∞+∥b∥2
∞EZB(1+δ)r
e−β/h(ηφ)2dx
+ 4EZx:(1+3δ/4)r≤√β(x)≤(1+δ)r|∇η|2e−β/h φ2dx,
which indicates that
d
dtEZB(1+δ)r
e−β/h(ηφ)2dx≤4(1 + δ)2r2
h2+ 2∥a∥∞+∥b∥2
∞EZB(1+δ)r
e−β/h(ηφ)2dx
+ 4∥∇η∥2
∞e−(1+3δ/4)2r2
hEZB(1+δ)r
φ2dx.
Here and throughout the proof of Lemma 3.3,∥∇η∥∞:= ∥∇η∥L∞(B(1+δ)r). From the latter it follows that
d
dt"e−4(1+δ)2r2
h2+2∥a∥∞+∥b∥2
∞tEZB(1+δ)r
e−β/h|ηφ|2dx#
≤4∥∇η∥2
∞e−4(1+δ)2r2
h2+2∥a∥∞+∥b∥2
∞te−(1+3δ/4)2r2
hEZB(1+δ)r
φ2dx.
Integrating the above inequality over (t, T ), we get
EZB(1+δ)r
e−β/h|ηφ(x, T )|2dx
≤e4(1+δ)2r2
h2+2∥a∥∞+∥b∥2
∞(T−t)EZB(1+δ)r
e−β/h|ηφ(x, t)|2dx
+ 4e4(1+δ)2r2
h2+2∥a∥∞+∥b∥2
∞(T−t)∥∇η∥2
∞e−(1+3δ/4)2r2
hEZT
tZB(1+δ)r
φ2(x, s)dxds.
(3.7)
QUANTITATIVE UNIQUENESS ESTIMATES FOR STOCHASTIC PARABOLIC EQUATIONS 7
We simply write b1:= 4(1 + δ)2, b2:= (1 + 3δ/4)2and b3:= (1 + δ/2)2.It is clear that 1 < b3< b2< b1. Recall
that t≤T. We now suppose h > 0 to be such that
0< T −(b2−b3)h
b1≤t.
Then b1(T−t)/h2≤(b2−b3)/h and (3.7) yields
EZB(1+δ)r
e−β/h|ηφ(x, T )|2dx≤e(b2−b3)r2
he(2∥a∥∞+∥b∥2
∞)TEZB(1+δ)r
e−β/h|ηφ(x, t)|2dx
+ 4∥∇η∥2
∞e(2∥a∥∞+∥b∥2
∞)Te−b3r2
hEZT
tZB(1+δ)r
φ2(x, s)dxds.
Since η(·) = 1 in Br, the following estimate holds
EZBr|φ(x, T )|2dx≤e(b2−b3+1)r2
he(2∥a∥∞+∥b∥2
∞)TEZB(1+δ)r
e−β/h|ηφ(x, t)|2dx
+ 4∥∇η∥2
∞e(2∥a∥∞+∥b∥2
∞)Te−(b3−1)r2
hEZT
tZB(1+δ)r
φ2(x, s)dxds,
(3.8)
whenever 0 < T −(b2−b3)h/b1≤t≤T. Recall that h0< T from (3.4). We choose has follows:
h=b1
b2−b3
h0
=b1C3/(b2−b3)
ln (1 + C4)e[1+2C1(1+ 1
r2)](1+ 1
τ2−τ1+∥a∥2/3
∞+∥b∥2
∞)+ 4C3
T+(2∥a∥∞+∥b∥2
∞)TERT
T−τ2RQRφ2dxdt
ERBrφ2(x,T )dx
with C3:= (b2−b3)(b3−1)r2/b1and C4:= 4∥∇η∥2
∞. Then for any t∈[T−min{τ2, h0}, T ], we have
4∥∇η∥2
∞e(2∥a∥∞+∥b∥2
∞)Te−(b3−1)r2
hZT
t
EZB(1+δ)r
φ2(x, s)dxds
=
C4e(2∥a∥∞+∥b∥2
∞)TERT
tRB(1+δ)rφ2(x, s)dxds
(1 + C4)e[1+2C1(1+ 1
r2)](1+ 1
τ2−τ1+∥a∥2/3
∞+∥b∥2
∞)+ 4C3
T+(2∥a∥∞+∥b∥2
∞)TERT
T−τ2RQRφ2(x,s)dxds
ERBrφ2(x,T )dx
≤1
eEZBr
φ2(x, T )dx.
(3.9)
The last inequality is implied by the facts that (1 + δ)r≤2r≤Rand B(1+δ)r⊂QR.
On one hand, by (3.8) and (3.9), we get
1−1
eEZBr
φ2(x, T )dx≤e(b2−b3+1)(b2−b3)r2
b1h0e(2∥a∥∞+∥b∥2
∞)TEZB(1+δ)r|φ(x, t)|2dx(3.10)
8Y. LIU ET AL.
for each T−min {τ2, h0} ≤ t≤T. On the other hand, by (3.3), we see
ERT
T−τ2RQRφ2(x, s)dxds
ERBrφ2(x, T )dx≤eC3
h0,
which, combined with (3.10), indicates that
1−1
ee−C3
h0EZT
T−τ2ZQR
φ2(x, s)dxds≤e(b2−b3+1)(b2−b3)r2
b1h0e(2∥a∥∞+∥b∥2
∞)TEZB(1+δ)r|φ(x, t)|2dx
for each T−min {τ2, h0} ≤ t≤T. Since (2∥a∥∞+∥b∥2
∞)T h0< C3(see (3.4)), the desired estimate (3.5) follows
from the latter inequality immediately with C5:= 3C3+ (b2−b3+ 1)(b2−b3)r2/b1.
4. Proof of Theorem 2.1
In this section, we shall study the quantitative version of unique continuation for the solution of (2.1), i.e.,
Theorem 2.1. In what follows, for each λ > 0, and x0∈RN, we define
Gλ(x, t)≜1
(T−t+λ)N/2e−|x−x0|2
4(T−t+λ), t ∈[0, T ], x ∈RN.(4.1)
It is clear that
∂tGλ(x, t)+∆Gλ(x, t)=0,∇Gλ(x, t) = −x−x0
2(T−t+λ)Gλ(x, t),
∆Gλ(x, t) = −N
2(T−t+λ)Gλ(x, t) + |x−x0|2
4(T−t+λ)2Gλ(x, t),
∂xixjGλ(x, t) = (xi−x0i)(xj−x0j)
4(T−t+λ)2Gλ(x, t), i =j.
(4.2)
For δ∈(0,1], R > 0, we denote R0:= (1 + 2δ)R. Let χ∈C∞
0(BR0) be such that
0≤χ(·)≤1 in BR0and χ(·) = 1 in B(1+3δ/2)R.(4.3)
We set
u:= χφ, F := au −φ∆χ−2∇φ· ∇χ. (4.4)
Then one can verify that
du−∆udt=Fdt+budW(t) in BR0×(0, T ).(4.5)
QUANTITATIVE UNIQUENESS ESTIMATES FOR STOCHASTIC PARABOLIC EQUATIONS 9
Define
Hλ,R0(t) = EZBR0(x0)|u(x, t)|2Gλ(x, t)dx,
Dλ,R0(t) = EZBR0(x0)|∇u(x, t)|2Gλ(x, t)dx,
Nλ,R0(t) = 2Dλ,R0(t)
Hλ,R0(t),whenever Hλ,R0(t)= 0.
(4.6)
Throughout this section, we always work under the assumption Hλ,R0(t)= 0, for any t∈[0, T ], any x0∈Rn
and any R0>0.
Lemma 4.1. For the function Hλ,R0(·)defined in (4.6), involving the solution φto the equation (2.1)over the
ball BR0(x0), it holds that
d
dtHλ,R0(t) = −2Dλ,R0(t)+2EZBR0(x0)
uF Gλ(x, t)dx+EZBR0(x0)
b2u2Gλ(x, t)dx. (4.7)
For simplicity, we denote
∥b∥2
L∞
F(0,+∞;W1,∞(BR0(x0))) := ∥b∥2
BR0.
Next, we introduce the following monotonicity of the parabolic-type frequency function associated with
stochastic parabolic equations.
Lemma 4.2. For the function Nλ,R0(·)defined in (4.6), involving the solution φto the equation (2.1)over the
ball BR0(x0), it follows that
d
dtNλ,R0(t)≤1
T−t+λ+ 2∥b∥2
BR0(x0)Nλ,R0(t)+2∥b∥2
BR0(x0)+
ERBR0(x0)F2Gλ(x, t)dx
Hλ,R0(t).(4.8)
Remark 4.3. Lemmas 4.1 and 4.2 were proved in [15], Lemma 2.1 and [15], Lemma 2.2 for a bounded and
convex domain. By a similar argument, the same results can be obtained. Hence, we omit the detailed proofs
here.
We then have the following two-ball and one-cylinder inequality, which is inspired by [29], Theorem 2 and
[21], Lemma 3.2. Its proof here is adapted from [10], Lemma 4 by using Lemma 3.3 instead.
Lemma 4.4. Let 0< r < R < +∞and δ∈(0,1]. Then there are three positive constants C6:= C6(R, δ), C7:=
C7(R, δ)and γ:= γ(r, R, δ)∈(0,1) so that for any x0∈RNand any φ0∈L2
F0(Ω; L2(RN)), the solution φof
(2.1) satisfies
EZBR(x0)|φ(x, T )|2dx
≤"C6e[1+2C1(1+ 1
R2)](1+ 4
T+∥a∥2/3
∞+∥b∥2
∞)+ C7
T+(2∥a∥∞+∥b∥2
∞)TEZT
T/2ZQ2R0(x0)
φ2(x, t)dxdt#γ
× 2EZBr(x0)|φ(x, T )|2dx!1−γ
,
where C1is the constant given by Lemma 3.1.
10 Y. LIU ET AL.
Remark 4.5. A similar result is obtained in [18], Theorem 3.1 for the stochastic parabolic equation on a
time-varying domain. Their proof is based on the Carleman estimate, while ours is based on the parabolic-type
frequency function and quantify the dependence of the constant on the L∞-norm of the involved potentials.
Proof of Lemma 4.4.For each r′>0, we denote Br′:= Br′(x0) and Qr′:= Qr′(x0). Furthermore, we define
g:= −2∇χ· ∇φ−φ∆χ. (4.9)
Step 1. Note that gis supported on {x: (1 + 3δ/2)R≤ |x−x0| ≤ R0}.Recall that χ(·) = 1 in B(1+δ)R(see
(4.3)). We can easily check that
ERBR0u(x, t)g(x, t)Gλ(x, t)dx
H(t)
=
ERBR0\B(1+3δ/2)Rχφ(−2∇χ· ∇φ−φ∆χ)e−|x−x0|2
4(T−t+λ)dx
ERBR0|χφ(x, t)|2e−|x−x0|2
4(T−t+λ)dx
≤e−K1
T−t+λ
ERBR0\B(1+3δ/2)R2|φ∇χ· ∇φ|+|∆χ|φ2dx
ERB(1+δ)Rφ2(x, t)dx
≤e−K1
T−t+λ
2∥∇χ∥∞(ERBR0φ2(x, t)dx)1
2(ERBR0|∇φ(x, t)|2dx)1
2+∥∆χ∥∞ERBR0φ2(x, t)dx
ERB(1+δ)Rφ2(x, t)dx,
(4.10)
where K1:= [(1 + 3δ/2)R]2/4−[(1 + δ)R]2/4 and ∥∇χ∥∞:= ∥∇χ∥L∞(BR0)and ∥∆χ∥∞:= ∥∆χ∥L∞(BR0).
On one hand, by Lemma 3.1 (where r, R, τ1and τ2are replaced by R0,2R0, T/4 and T /2, respectively), we
have
EZBR0
φ2(x, t)dx≤ K2(1 + T−1+∥a∥∞+∥b∥2
∞)EZT
T/2ZB2R0
φ2(x, t)dxdtfor each t∈[3T/4, T ],(4.11)
where K2:= K2(R)>0.By Lemma 3.2 (where Rand τare replaced by R0and T/4, respectively), we get that
for each t∈[3T/4, T ],
EZBR0|∇φ(x, t)|2dx≤ K3(1 + T−2+∥a∥2
∞+∥b∥4
∞)EZT
T/2ZB2R0
φ2(x, s)dxds, (4.12)
where K3:= K3(R)>0.By (3.5) in Lemma 3.3 (where r, R, τ1and τ2are replaced by R, 2R0, T/4 and T /2,
respectively), it holds that
e(2∥a∥∞+∥b∥2
∞)TEZT
T/2ZB2R0
φ2dxds≤e(2∥a∥∞+∥b∥2
∞)TEZT
T/2ZQ2R0
φ2dxds
≤e1+ C5
h0EZB(1+δ)R
φ2(x, t)dxfor each t∈[T−h0, T ].
(4.13)
QUANTITATIVE UNIQUENESS ESTIMATES FOR STOCHASTIC PARABOLIC EQUATIONS 11
Here, we used the fact that h0< T/4 (see (3.4) in Lemma 3.3). It follows from (4.10)–(4.13) that
ERBR0u(x, t)g(x, t)Gλ(x, t)dx
H(t)
≤e−K1
T−t+λK4(1 + T−2+∥a∥3/2
∞+∥b∥4
∞)ERT
T/2RB2R0φ2(x, s)dxds
ERB(1+δ)Rφ2(x, t)dx
≤K4e−K1
T−t+λe1+ C5
h0(1 + T−2) for each t∈[T−h0, T ].
(4.14)
where K4:= K4(R, δ)>0.
On the other hand, by similar arguments as those for (4.14), we have
ZT
t
ERBR0|g(x, s)|2Gλ(x, s)dx
H(s)ds≤ZT
t
ERBR0| − 2∇χ· ∇φ−φ∆χ|2dx
ERB(1+δ)R|φ(x, s)|2dxe−K1
T−s+λds
≤ZT
t
8∥∇χ∥2
∞ERBR0|∇φ|2dx+ 2∥∆χ∥2
∞ERBR0φ2dx
ERB(1+δ)R|φ(x, s)|2dxe−K1
T−s+λds
≤K5(1 + T−2+∥a∥2
∞+∥b∥4
∞)ZT
t
ERT
T/2RB2R0φ2dxds
ERB(1+δ)R|φ(x, s)|2dxe−K1
T−s+λds
≤K5(1 + T−2+∥a∥2
∞+∥b∥4
∞)e1+ C5
h0e−(2∥a∥∞+∥b∥2
∞)TZT
t
e−K1
T−s+λds
≤K5(1 + T−2)e1+ C5
h0e−K1
T−t+λ(T−t) for each t∈[T−h0, T ],
(4.15)
where K5:= K5(R, δ)>0.
Step 2. In this step, the aim is to give an upper bound for the term λNλ,R0(T) (i.e., (4.24) below). By
Lemma 4.2, the second equality in (4.4) and (4.9), we get
d
dtNλ,R0(t)≤1
T−t+λ+ 2∥b∥2
BR0Nλ,R0(t)+2∥b∥2
BR0+
ERBR0|(au +g)(x, t)|2Gλ(x, t)dx
H(t),
which indicates that
d
dt[(T−t+λ)Nλ,R0(t)]
≤2(T−t+λ)∥b∥2
BR0Nλ,R0(t) + 2(T−t+λ)∥b∥2
BR0+ (T−t+λ)
ERBR0|(au +g)(x, t)|2Gλ(x, t)dx
H(t)
≤2(T−t+λ)∥b∥2
BR0Nλ,R0(t) + 2(T−t+λ)∥b∥2
BR0+ 2(T−t+λ) ∥a∥2
∞+
ERBR0|g(x, t)|2Gλ(x, t)dx
H(t)!,
12 Y. LIU ET AL.
this, along with Gronwall’s inequality implies that
λNλ,R0(T)≤(T−t+λ)Nλ,R0(t)e2∥b∥2
BR0(T−t)+ 2e2∥b∥2
BR0(T−t)∥a∥2
∞+∥b∥2
BR0ZT
t
(T−s+λ)ds
+ 2e2∥b∥2
BR0(T−t)ZT
t
(T−s+λ)
ERBR0|g(x, s)|2Gλ(x, s)dx
H(s)ds.
Hence, for any 0 < T −2ε≤t<T (where εwill be determined later), we have
λ
2ε+λNλ,R0(T)≤Nλ,R0(t)e4∥b∥2
BR0ε+ 4e4∥b∥2
BR0ε∥a∥2
∞+∥b∥2
BR0ε
+ 4e4∥b∥2
BR0εZT
t
ERBR0|g(x, s)|2Gλ(x, s)dx
H(s)ds,
(4.16)
this, along with Lemma 4.1, (4.4) and (4.9) implies that
d
dtH(t) + λ
2ε+λNλ,R0(T)H(t)
≤e4∥b∥2
BR0ε2∥a∥∞+∥b∥2
BR0+ 4ε∥a∥2
∞+ 4ε∥b∥2
BR0H(t)
+H(t)e4∥b∥2
BR0ε ERBR0u(x, t)g(x, t)Gλ(x, t)dx
H(t)+ 2 ZT
t
ERBR0|g(x, s)|2Gλ(x, s)dx
H(s)ds!.
(4.17)
Next, on one hand, it follows from (4.14) and (4.15) that
ERBR0u(x, t)g(x, t)Gλ(x, t)dx
H(t)+ 2 ZT
t
ERBR0|g(x, s)|2Gλ(x, s)dx
H(s)ds
≤K6(1 + 2ε)1 + T−2e−K1
2ε+λeC5+K1
h0
:=Qh0,ε,λ for each 0 < T −2ε≤t<T with 2ε∈(0, h0],
(4.18)
where K6:= K6(R, δ)>0.This, along with (4.17), implies that
d
dtH(t)≤ −λ
2ε+λNλ,R0(T)−e4∥b∥2
BR0ε2∥a∥∞+∥b∥2
BR0+ 4ε∥a∥2
∞+ 4ε∥b∥2
BR0+Qh0,ε,λH(t),
which indicates that
d
dt
e λ
2ε+λNλ,R0(T)−e4∥b∥2
BR0
ε2∥a∥∞+∥b∥2
BR0+4ε∥a∥2
∞+4ε∥b∥2
BR0+Qh0,ε,λ!t
H(t)
≤0
for each 0 < T −2ε≤t<T with 2ε∈(0, h0]. Integrating the above inequality over (T−2ε, T −ε), we obtain
eελ
2ε+λNλ,R0(T)H(T−ε)≤ee4∥b∥2
BR0
ε2∥a∥∞+∥b∥2
BR0+4ε∥a∥2
∞+4ε∥b∥2
BR0+Qh0,ε,λεH(T−2ε).
QUANTITATIVE UNIQUENESS ESTIMATES FOR STOCHASTIC PARABOLIC EQUATIONS 13
This yields
eελ
2ε+λNλ,R0(T)≤ee4∥b∥2
BR0
ε2∥a∥∞+∥b∥2
BR0+4ε∥a∥2
∞+4ε∥b∥2
BR0+Qh0,ε,λε
×
ERBR0|u(x, T −2ε)|2e−|x−x0|2
4(2ε+λ)dx
ERBR0|u(x, T −ε)|2e−|x−x0|2
4(ε+λ)dx
.
(4.19)
On the other hand, by (4.3), the first equality in (4.4), (4.11) and noting that e−|x−x0|2
4(2ε+λ)≤1, R0>(1 + δ)R
and B2R0⊂Q2R0, we see
ERBR0|u(x, T −2ε)|2e−|x−x0|2
4(2ε+λ)dx
ERBR0|u(x, T −ε)|2e−|x−x0|2
4(ε+λ)dx≤e((1+δ)R)2
4(ε+λ)ERBR0|φ(x, T −2ε)|2dx
ERB(1+δ)R|φ(x, T −ε)|2dx
≤e((1+δ)R)2
4εK2(1 + T−1+∥a∥∞+∥b∥2
∞)ERT
T/2RQ2R0φ2(x, t)dxdt
ERB(1+δ)R|φ(x, T −ε)|2dx,
which, combined with (ii) of Lemma 3.3 (where r, R, τ1and τ2are replaced by R, 2R0, T /4 and T/2, respectively),
indicates that
ERBR0|u(x, T −2ε)|2e−|x−x0|2
4(2ε+λ)dx
ERBR0|u(x, T −ε)|2e−|x−x0|2
4(ε+λ)dx≤e((1+δ)R)2
4εK2(1 + T−1+∥a∥∞+∥b∥2
∞)e1+ C5
h0
e(2∥a∥∞+∥b∥2
∞)T
≤K2e((1+δ)R)2
4ε1 + T−1e1+ C5
h0.
(4.20)
Then, it follows from (4.19) and (4.20) that for each ε∈(0, h0/2],
λNλ,R0(T)≤2ε+λ
εe4∥b∥2
BR0ε2∥a∥∞+∥b∥2
BR0+ 4ε∥a∥2
∞+ 4ε∥b∥2
BR0+Qh0,ε,λε
+(1 + δ)2R2
4ε+1+C5
h0
+ ln K21 + T−1.
(4.21)
Finally, we choose λ= 2µε with µ∈(0,1) (which will be determined later) and 2ε=K1h0/[2(C5+K1)] so that
Qh0,ε,λ (see (4.18)) satisfies
Qh0,ε,λ =K6(1 + 2ε)1 + T−2e−K1
2ε+λeC5+K1
h0=K6(1 + 2ε)1 + T−2e−2(C5+K1)
h0(µ+1) eC5+K1
h0
=K6(1 + 2ε)1 + T−2eC5+K1
h0(µ−1
µ+1 )≤ K6(1 + 2ε)1 + T−2.
(4.22)
Since 2ε≤h0, by (4.21) and (4.22), we get
λNλ,R0(T)≤4e2h0∥b∥2
∞h0∥a∥∞+1
2h0∥b∥2
∞+h2
0∥a∥2
∞+h2
0∥b∥2
∞+1
2K6(1 + 2ε)1 + T−2h0
+ 4 1 + C5
h0
+K21 + T−1+C5+K1
K1h0
(1 + δ)2R2.
(4.23)
14 Y. LIU ET AL.
According to (i) of Lemma 3.3 (where r, R, τ1and τ2are replaced by R, 2R0, T/4 and T /2, respectively), it
is clear that
h0< C3, h0< T, h0T∥a∥∞< C3, h0∥b∥2
∞< C3, h3
0∥b∥2
∞< C2
3and h3
0∥a∥2
∞< C3
3.
These, together with (4.23), derive that
ελNλ,R0(T)≤4εe2h0∥b∥2
∞h0∥a∥∞+1
2h0∥b∥2
∞+h2
0∥a∥2
∞+h2
0∥b∥2
∞+1
2K6(1 + 2ε)1 + T−2h0
+ 4ε1 + C5
h0
+K21 + T−1+C5+K1
K1h0
(1 + δ)2R2
≤4e2h0∥b∥2
∞h0T∥a∥∞+h3
0∥a∥2
∞+h3
0∥b∥2
∞+1
2K6(1 + 2h0)1 + T−2h2
0
+ 4 h0+C5+K2h01 + T−1+C5+K1
K1
(1 + δ)2R2
≤2e2C32C3+ 2C3
3+ 2C2
3+K6(1 + 2C3)1 + C2
3
+ 4 C3+C5+K2(1 + C3) + C5+K1
K1
(1 + δ)2R2.
Hence, recalling that λ= 2µε, we have
16λ
r2N
4+1
2λNλ,R0(T)≤16µ
r2N
2C3+ελNλ,R0(T)≤2µ(1 + K7),(4.24)
where K7:= K7(r, R, δ, N)>0.
Step 3. We claim that
EZBR0|u(x, T )|2e−|x−x0|2
4λdx≤EZBr|φ(x, T )|2e−|x−x0|2
4λdx+ 2µ(1 + K7)EZBR0|u(x, T )|2e−|x−x0|2
4λdx. (4.25)
Indeed, noting that uis H1(BR0)-value, by [21], Page 1951, also see [9,10,29]. we have
1
16λZBR0|x−x0|2|u(x, T )|2e−|x−x0|2
4λdx
≤N
4ZBR0|u(x, T )|2e−|x−x0|2
4λdx+λZBR0|∇u(x, T )|2e−|x−x0|2
4λdxP−a.s.in BR0.
This implies
EZBR0|u(x, T )|2e−|x−x0|2
4λdx
≤EZBR0\Br
|x−x0|2
r2|u(x, T )|2e−|x−x0|2
4λdx+EZBr|u(x, T )|2e−|x−x0|2
4λdx
≤16λ
r2N
4+1
2λNλ,R0(T)EZBR0|u(x, T )|2e−|x−x0|2
4λdx+EZBr|φ(x, T )|2e−|x−x0|2
4λdx,
(4.26)
QUANTITATIVE UNIQUENESS ESTIMATES FOR STOCHASTIC PARABOLIC EQUATIONS 15
where in the last line, we used the definition of Nλ,R0(T) (see (4.6)) and the fact that u=φin Br(see (4.3)
and (4.4)). Then (4.25) follows from (4.26) and (4.24) immediately.
Step 4. End of the proof. We choose µ= 1/[2(1 + K7)]. Then, λ= 2µε =K1h0/[4(1 + K7)(C5+K1)].By
(4.25), e−|x−x0|2
4λ≤1 and the fact that u=φin BR(see (4.3) and (4.4)), we have
e−R2
4λEZBR|φ(x, T )|2dx≤EZBR|φ(x, T )|2e−|x−x0|2
4λdx≤EZBR0|u(x, T )|2e−|x−x0|2
4λdx
≤EZBr|φ(x, T )|2e−|x−x0|2
4λdx+EZBR0|u(x, T )|2e−|x−x0|2
4λdx
≤2EZBr|φ(x, T )|2dx.
This, along with the definition of h0(see (3.3), where r, R, τ1and τ2are replaced by R, 2R0, T/4 and T /2,
respectively), implies that
EZBR|φ(x, T )|2dx ≤2e (1+K7)(C5+K1)R2
K1h0EZBr|φ(x, T )|2dx
≤h(1 + C4)e[1+2C1(1+R−2)](1+4T−1+∥a∥2/3
∞+∥b∥2
∞)+ 4C3
T+(2∥a∥∞+∥b∥2
∞)T
ERT
T/2RQ2R0φ2dxdt
ERBR|φ(x, T )|2dxi(1+K7)(C5+K1)R2
K1C3×2EZBr|φ(x, T )|2dx.
Hence, we can conclude that the desired estimate of Lemma 4.4 holds with
γ=(1 + K7)(C5+K1)R2
C3K1+ (1 + K7)(C5+K1)R2∈(0,1).
In summary, we finish the proof of this lemma.
Finally, based on Lemma 4.4, we are ready to prove Theorem 2.1.
Proof of Theorem 2.1.By Lemma 4.4 (where r, R and δare replaced by r, √NR and 1/2, respectively), we
obtain
EZQR(xi)|φ(x, T )|2dx≤EZB√N R(xi)|φ(x, T )|2dx
≤b
K1e[1+2C1(1+R−2)](1+4T−1+∥a∥2/3
∞+∥b∥2
∞)+ b
K2T−1+(2∥a∥∞+∥b∥2
∞)TEZT
T/2ZQ4√NR (xi)
φ2dxdtθ
×"2EZBr(xi)|φ(x, T )|2dx#1−θ
,
16 Y. LIU ET AL.
where b
K1:= b
K1(R)>0,b
K2:= b
K2(R)>0 and θ:= θ(r, R)∈(0,1).This, along with Young’s inequality, implies
that for each ε > 0,
EZQR(xi)|φ(x, T )|2dx≤εθ b
K1e[1+2C1(1+R−2)](1+4T−1+∥a∥2/3
∞+∥b∥2
∞)+ b
K2T−1+(2∥a∥∞+∥b∥2
∞)T
×EZT
T/2ZQ4√NR (xi)
φ2dxdt+ 2ε−θ
1−θ(1 −θ)EZBr(xi)|φ(x, T )|2dx.
Then
EZRN|φ(x, T )|2dx=X
i≥1
EZQR(xi)|φ(x, T )|2dx
≤εθ b
K1e[1+2C1(1+R−2)](1+4T−1+∥a∥2/3
∞+∥b∥2
∞)+ b
K2T−1+(2∥a∥∞+∥b∥2
∞)T
X
i≥1
EZT
T/2ZQ4√NR (xi)
φ2dxdt+ 2ε−θ
1−θ(1 −θ)EZω|φ(x, T )|2dx.
(4.27)
Since
X
i≥1
EZT
T/2ZQ4√NR (xi)
φ2dxdt≤b
K3EZT
T/2ZRN
φ2dxdt,
where b
K3>0, it follows from (4.27) that
EZRN|φ(x, T )|2dx≤εθ b
K1b
K3e[1+2C1(1+R−2)](1+4T−1+∥a∥2/3
∞+∥b∥2
∞)+ b
K2T−1+(2∥a∥∞+∥b∥2
∞)T
×EZT
T/2ZRN
φ2dxdt+ 2ε−θ
1−θ(1 −θ)EZω|φ(x, T )|2dxfor each ε > 0.
This implies
EZRN|φ(x, T )|2dx
≤"b
K1b
K3e[1+2C1(1+R−2)](1+4T−1+∥a∥2/3
∞+∥b∥2
∞)+ b
K2T−1+(2∥a∥∞+∥b∥2
∞)TEZT
T/2ZRN
φ2dxdt#θ
×2EZω|φ(x, T )|2dx1−θ
.
(4.28)
Applying the Itˆo formula to φ2and then taking expectation and by Gronwall’s inequality, we have the energy
estimate of the equation (2.1):
EZRN|φ(x, t)|2dx≤e(2∥a∥∞+∥b∥2
∞)tEZRN|φ0(x)|2dxfor each t∈[0, T ].(4.29)
QUANTITATIVE UNIQUENESS ESTIMATES FOR STOCHASTIC PARABOLIC EQUATIONS 17
Finally, by (4.28), we deduce
EZRN|φ(x, T )|2dx≤b
K1b
K3Te[1+2C1(1+R−2)](1+4T−1+∥a∥2/3
∞+∥b∥2
∞)+ b
K2T−1+2(2∥a∥∞+∥b∥2
∞)T
×EZRN|φ0(x)|2dxθ
×2EZω|φ(x, T )|2dx1−θ
.
Hence, (2.2) follows from the latter inequality immediately.
5. Proof of Corollary 2.2
Now, we are able to present the proof of Corollary 2.2 by Theorem 2.1 and the telescoping series method (see
[27,30]). For the convenience of the reader, we provide here the detailed computation.
Proof of Corollary 2.2. For any 0 ≤t1< t2≤T, by using Theorem 2.1, we obtain from Young’s inequality
that
E∥φ(t2)∥2
L2(RN)≤εE∥φ(t1)∥2
L2(RN)+e
K1
εαee
K2
t2−t1E∥φ(t2)∥2
L2(ω)for each ε > 0,(5.1)
where e
K1:= e C8
1−θ(T+T(∥a∥∞+∥b∥2
∞)+∥a∥2/3
∞+∥b∥2
∞),e
K2:= C8/(1 −θ) and α:= θ/(1 −θ).
Let lbe a density point of E. According to Proposition 2.1 in [30], for each κ > 1, there exists l1∈(l, T ),
depending on κand E, so that the sequence {lm}m≥1, given by
lm+1 =l+1
κm(l1−l),
satisfies
lm−lm+1 ≤3|E∩(lm+1, lm)|.(5.2)
Next, let 0 < lm+2 < lm+1 ≤t<lm< l1< T . It follows from (5.1) (where t1,t2are replaced by lm+2 and t,
respectively) that
E∥φ(t)∥2
L2(RN)≤εE∥φ(lm+2)∥2
L2(RN)+e
K1
εαee
K2
t−lm+2 E∥φ(t)∥2
L2(ω)for each ε > 0.(5.3)
Similar to (4.29), we have
E∥φ(lm)∥L2(RN)≤e(2∥a∥∞+∥b∥2
∞)TE∥φ(t)∥L2(RN).
This, along with (5.3), implies for each ε > 0,
E∥φ(lm)∥2
L2(RN)≤e(2∥a∥∞+∥b∥2
∞)T εE∥φ(lm+2)∥2
L2(RN)+e
K1
εαee
K2
t−lm+2 E∥φ(t)∥2
L2(ω)!,
which indicates that
E∥φ(lm)∥2
L2(RN)≤εE∥φ(lm+2)∥2
L2(RN)+e
K3
εαee
K2
t−lm+2 E∥φ(t)∥2
L2(ω)for each ε > 0,
18 Y. LIU ET AL.
where e
K3= (e(2∥a∥∞+∥b∥2
∞)T)1+αe
K1. Integrating the latter inequality over E∩(lm+1, lm) gives
|E∩(lm+1, lm)|E∥φ(lm)∥2
L2(RN)≤ε|E∩(lm+1, lm)|E∥φ(lm+2 )∥2
L2(RN)
+e
K3
εαee
K2
lm+1−lm+2 EZlm
lm+1
χE∥φ(t)∥2
L2(ω)dtfor each ε > 0.(5.4)
Here and in the sequel, χEdenotes the characteristic function of E.
Since lm−lm+1 = (κ−1)(l1−l)/κm,by (5.4) and (5.2), we obtain
E∥φ(lm)∥2
L2(RN)≤1
|E∩(lm+1, lm)|e
K3
εαee
K2
lm+1−lm+2 EZlm
lm+1
χE∥φ(t)∥2
L2(ω)dt+εE∥φ(lm+2)∥2
L2(RN)
≤3κm
(l1−l)(κ−1) e
K3
εαee
K21
l1−l
κm+1
κ−1EZlm
lm+1
χE∥φ(t)∥2
L2(ω)dt+εE∥φ(lm+2)∥2
L2(RN)
for each ε > 0. This yields
E∥φ(lm)∥2
L2(RN)≤1
εα
3
κe
K3
e
K2
e2e
K21
l1−l
κm+1
κ−1EZlm
lm+1
χE∥φ(t)∥2
L2(ω)dt+εE∥φ(lm+2)∥2
L2(RN)(5.5)
for each ε > 0. Denote by d:= 2 e
K2/[κ(l1−l)(κ−1)]. It follows from (5.5) that
εαe−dκm+2 E∥φ(lm)∥2
L2(RN)−ε1+αe−dκm+2 E∥φ(lm+2)∥2
L2(RN)≤3
κe
K3
e
K2
EZlm
lm+1
χE∥φ(t)∥2
L2(ω)dt
for each ε > 0.
Choosing ε= e−dκm+2 in the above inequality gives
e−(1+α)dκm+2 E∥φ(lm)∥2
L2(RN)−e−(2+α)dκm+2 E∥φ(lm+2)∥2
L2(RN)≤3
κe
K3
e
K2
EZlm
lm+1
χE∥φ(t)∥2
L2(ω)dt. (5.6)
Taking κ=p(α+ 2)/(α+ 1) in (5.6), we then have
e−(2+α)dκmE∥φ(lm)∥2
L2(RN)−e−(2+α)dκm+2 E∥φ(lm+2)∥2
L2(RN)≤3
κe
K3
e
K2
EZlm
lm+1
χE∥φ(t)∥2
L2(ω)dt.
Changing mto 2m′and summing the above inequality from m′= 1 to infinity give the desired result. Indeed,
e−(2∥a∥∞+∥b∥2
∞)Te−(2+α)dκ2E∥φ(T)∥2
L2(RN)≤e−(2+α)dκ2E∥φ(l2)∥2
L2(RN)
≤
+∞
X
m′=1 e−(2+α)dκ2m′E∥φ(l2m′)∥L2(RN)−e−(2+α)dκ2m′+2 E∥φ(l2m′+2)∥2
L2(RN)
≤3
κe
K3
e
K2
+∞
X
m′=1
EZl2m′
l2m′+1
χE∥φ(t)∥2
L2(ω)dt≤3
κe
K3
e
K2
EZT
0
χE∥φ(t)∥2
L2(ω)dt.
In summary, we finish the proof of Corollary 2.2.
QUANTITATIVE UNIQUENESS ESTIMATES FOR STOCHASTIC PARABOLIC EQUATIONS 19
6. Further comments
6.1. Controllability for the backward stochastic parabolic equation
One could obtain the null controllability result for the backward stochastic parabolic equations by the classical
duality argument as in [24], Theorem 2.2 or [15], Theorem 1.12.
Given T > 0, consider the following controlled backward stochastic heat equation
(dy+ ∆ydt=a1ydt+b1Ydt+χEχωudt+YdW(t),in RN×(0, T ),
y(T) = yT,in RN.(6.1)
Here yT∈L2
FT(Ω; L2(RN)), a1∈L∞
F(0,+∞;L∞(RN)), b1∈L∞
F(0,+∞;W1,∞(RN)) and u∈L2
F(0,+∞;
L2(RN)) is the control. According to [28], Theorem 4.10, the system (6.1) has a unique solution (y(·), Y (·)) ∈
L2
F(Ω; C([0, T ]; L2(RN))) ∩L2
F(0, T ;H1
0(RN)) ×L2
F(0, T ;L2(RN)).
We say system (6.1) is null controllable if for any yT∈L2
FT(Ω; L2(RN)), there exists a control u∈
L2
F(0,+∞;L2(RN)) such that the solution of the system (6.1) with terminal state yTand control usatisfying
that y(0) = 0. We have the following result.
Corollary 6.1. Under the assumption of Theorem 2.1, the system (6.1)is null controllable.
Proof. Consider the following equation:
dˆy−∆ˆydt=−a1ˆydt−b1ˆydW(t),in RN×(0, T ),
ˆy(0) = ˆy0∈L2
F0(Ω; L2(RN)),in RN.(6.2)
We introduce a linear subspace of L2
F(0, T ;L2(ω)):
X≜{ˆy|ω×E: ˆysolves the equation (6.2)},
and define a linear functional Lon Xas follows:
L(ˆy|ω×E) = −EZRN
ˆy(T)yTdx.
By Corollary 2.2, we have that
|L(ˆy|ω×E)|⩽∥ˆy(T)∥L2
FT(Ω;L2(RN))∥yT∥L2
FT(Ω;L2(RN))
⩽ee
C1eC1(T+T(∥a1∥∞+∥b1∥2
∞)+∥a1∥2/3
∞+∥b1∥2
∞+1)∥yT∥L2
FT(Ω;L2(RN)) EZω×E|ˆy(x, t)|2dxdt1
2
.
Therefore, Lis a bounded linear functional on X. By the Hahn–Banach theorem, Lcan be extended to a
bounded linear functional with the same norm on L2
F(0, T ;L2(ω)). For simplicity, we use the same notation for
this extension. By the Riesz representation theorem, there exists a stochastic process ˆu∈L2
F(0, T ;L2(ω)) such
that
EZω×E
ˆyˆudxdt=EZRN
ˆy(T)yTdx. (6.3)
20 Y. LIU ET AL.
Let
u(x, t) = (ˆu(x, t),(x, t)∈ω×E,
0,else.
Then it is obvious that u∈L2
F(0,+∞;L2(RN)), and we claim that this uis the control we need. In fact, for
any yT∈L2
FT(Ω; L2(RN)), for the solution ˆyof equation (6.2) and the solution (y, Y ) of equation (6.1), by the
Itˆo formula, we have that
EZRN
ˆy(T)y(T)dx−EZRN
ˆy0y(0)dx
=EZT
0ZRN
[ˆy(−∆y+a1y+b1Y+χEχωu) + y(∆ˆy−a1ˆy)−b1ˆyY ] dxdt
=EZT
0ZRN
ˆyχEχωudxdt
=EZω×E
ˆyˆudxdt.
(6.4)
Combining (6.3) and (6.4), we get that
EZRN
ˆy0y(0)dx= 0.
Since ˆy0can be chosen arbitrarily, we know that y(0) = 0,P−a.s.in RN.
6.2. Controllability for the forward stochastic parabolic equation
The observability inequality for the solution of forward stochastic parabolic equation we obtained here cannot
imply the controllability result for the same forward stochastic parabolic equation, because the solutions of the
forward and backward stochastic parabolic equations are not equivalent. In fact, the concept of controllability
for the forward stochastic parabolic equation is much more complicated than the deterministic couterpart, which
usually involves a control in the diffusion term of the equation. For this topic, we refer [24,25,31,32] to the
interesting reader.
Acknowledgements
The first two authors are supported by the National Natural Science Foundation of China under grant 11871478, the
Science Technology Foundation of Hunan Province. The last two authors is supported by the National Natural Science
Foundation of China under grant 12422118, and by the Fundamental Research Funds for the Central Universities under
grant 2042023kf0193.
Appendix: A
Proof of Lemma 3.1. For simplicity, we may write Br:= Br(x0) and BR:= BR(x0).Let η∈C∞
0(BR) verifies
0≤η(·)≤1 in BR, η(·) = 1 in Brand |∇η(·)| ≤ C(R−r)−1.(A.1)
Here and throughout the proof of Lemma 3.1,Cdenotes a generic positive constant. Let ξ∈C∞(R) satisfy
0≤ξ(·)≤1,|ξ′(·)| ≤ C(τ2−τ1)−1in R,(A.2)
QUANTITATIVE UNIQUENESS ESTIMATES FOR STOCHASTIC PARABOLIC EQUATIONS 21
ξ(·) = 0 in (−∞, T −τ2] and ξ(·) = 1 in [T−τ1,+∞).(A.3)
Applying the Itˆo formula to η2ξ2φ2, we have
d(η2ξ2φ2)=2ξξ′η2φ2dt + 2η2ξ2φ·[∆φdt +aφdt +bφdW (t)] + η2ξ2b2φ2dt.
Integrating the above equality over BR×(T−τ2, t) for t∈[T−τ1, T ] and taking the expectation, noting that
ξ(T−τ2) = 0, we obtain that
EZBR
η2ξ2(t)φ2(x, t)dx=EZt
T−τ2ZBR2ξξ′η2φ2+ 2η2ξ2φ·(∆φ+aφ) + η2ξ2b2φ2dxds
= 2EZt
T−τ2ZBR
ξξ′η2φ2dxds+ 2EZt
T−τ2ZBR
η2ξ2φ·∆φdxds
+EZt
T−τ2ZBR2aη2ξ2φ2+η2ξ2b2φ2dxds.
(A.4)
Notice that
2EZt
T−τ2ZBR
η2ξ2φ·∆φdxds=−4EZt
T−τ2ZBR
ξ2ηφ∇η· ∇φdxds−2EZt
T−τ2ZBR
η2ξ2|∇φ|2dxds,
and by (A.4) and Young’s inequality, we have
EZBR
η2ξ2(t)φ2(x, t)dx+EZt
T−τ2ZBR
η2ξ2|∇φ|2dxds
≤4EZt
T−τ2ZBR|∇η|2ξ2φ2dxds+ 2EZt
T−τ2ZBR
η2ξξ′φ2dxds
+EZt
T−τ2ZBR2aη2ξ2φ2+η2ξ2b2φ2dxds,
(A.5)
This, along with (A.1)–(A.3), implies that
EZBr
φ2(x, t)dx+EZt
T−τ1ZBr|∇φ|2dxds
≤C(R−r)−2+ (τ2−τ1)−1+∥a∥∞+∥b∥2
∞EZT
T−τ2ZBR
φ2dxds, for each t∈[T−τ1, T ].
Hence, (3.1) follows from the last inequality immediately.
Proof of Lemma 3.2. For each r′>0,we write Br′:= Br′(x0). Let η∈C∞
0(B4R/3) satisfies
0≤η(·)≤1,|∇η(·)| ≤ CR−1,|∆η(·)| ≤ CR−2in B4R/3(A.6)
and
η(·) = 1 in BR.(A.7)
22 Y. LIU ET AL.
Here and throughout the proof of Lemma 3.2,Cdenotes a generic positive constant. Let ξ∈C∞(R) verifies
0≤ξ(·)≤1,|ξ′(·)| ≤ Cτ −1in R,(A.8)
ξ(·) = 0 in (−∞, T −4τ/3] and ξ(·) = 1 in [T−τ, +∞).(A.9)
Applying the Itˆo formula to 1
2η2ξ2φ2
i, where φi=∂xiφ, integrating over B4R/3×(T−4τ /3, t) for t∈[T−
τ, T ], taking the expectation, and noting that ξ(T−4τ /3) = 0, η(·)≡0 on ∂B4R/3. Similar to the calculation
of (A.5), we obtain that
EZB4R/3
η2(x)ξ2(t)φ2
i(x, t)dx+EZt
T−4τ/3ZB4R/3
(ξη∇φi)2dxds
≤2EZt
T−4τ/3ZB4R/3
ξξ′η2φ2
idxds+ 4EZt
T−4τ/3ZB4R/3
(ξφi∇η)2dxds
+ 4EZt
T−4τ/3ZB4R/3
(ξηiφi)2dxds+ 2EZt
T−4τ/3ZB4R/3
(aηξφ)2dxds
+EZt
T−4τ/3ZB4R/3
(ηξφii)2dxds+ 2EZt
T−4τ/3ZB4R/3
(ξηbiφ)2+ (ξηbφi)2dxds.
(A.10)
This, along with (A.6)–(A.9), implies that
sup
t∈[T−τ,T ]
EZBR|φi(x, t)|2dx≤C∥a∥2
∞+∥b∥2
∞EZT
T−4τ/3ZB4R/3
φ2dxds
+Cτ−1+R−2+∥b∥2
∞EZT
T−4τ/3ZB4R/3
φ2
idxds
≤C∥a∥2
∞+∥b∥2
∞EZT
T−4τ/3ZB4R/3
φ2dxds
+Cτ−1+R−2+∥b∥2
∞EZT
T−4τ/3ZB4R/3|∇φ|2dxds.
(A.11)
According to (3.1) of Lemma 3.1 (where r, R, τ1and τ2are replaced by 4R/3,2R, 4τ/3 and 2τ, respectively),
it is clear that
EZT
T−4τ/3ZB4R/3|∇φ|2dxdt≤Cτ−1+R−2+∥a∥∞+∥b∥2
∞EZT
T−2τZB2R
φ2dxdt.
This, along with (A.11), implies that
sup
t∈[T−τ,T ]
EZBR|φi(x, t)|2dx≤Cτ−2+R−4+∥a∥2
∞+∥b∥4
∞EZT
T−4τ/3ZB2R
φ2dxds.
Hence, (3.2) follows from the last inequality by summing in i= 1, ..., n.
QUANTITATIVE UNIQUENESS ESTIMATES FOR STOCHASTIC PARABOLIC EQUATIONS 23
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