CONVERGENCE OF SOLUTIONS TO THE STOCHASTIC
p-LAPLACE EQUATIONS AS p GOES TO 1
IOANA CIOTIR AND JONAS M. TÖLLE
Abstract. We prove that the solutions to the stochastic p-Laplace equation
on a bounded domain O ⊂ d , d = 1, 2, with Dirichlet boundary conditions
and additive Gaussian noise converge, as p → 1, -a.s. uniformly in time in
L2 (O) to the solution to the multi-valued stochastic 1-Laplace equation with
Dirichlet boundary conditions and additive Gaussian noise. Due to the highly
singular limit case, solutions are defined via stochastic variational inequalities. Convergence of invariant measures of the associated stochastic flows is
investigated.
R
P
1. Introduction
In this paper, we are investigating the following family, p ∈ (1, 2], of stochastic
diffusion equations on L2 (O),
h
i

p
p−1

in (0, T ) × O,
 dXp (t) = div |∇Xp |d sgn(∇Xp ) dt + Q dW (t)

(Ep )
Xp (t) = 0
on (0, T ) × ∂O,



Xp (0) = x
in O.
The purpose of this paper is to prove that, for p → 1, the sequence {Xp }p
of solutions to the equations (Ep ) is convergent to the solution of the following
(multi-valued) stochastic diffusion equation in L2 (O)
p

dX
(t)
∈
div
[sgn
(∇X
(t))]
dt
+
Q dW (t)
in (0, T ) × O,

1
1

(E1 )
X1 (t) = 0
on (0, T ) × ∂O,


X1 (0) = x
in O.
In both cases, O is a bounded open subset of Rd , d = 1, 2, such that its boundary
∂O is sufficiently smooth. Here, |·|d denotes the Euclidean norm and the multid
valued sign-function sgn : Rd → 2R is defined by
 z

,
if z 6= 0,
d
sgn (z) = |z|

y ∈ Rd | |y|d 6 1 ,
if z = 0.
Furthermore, W (t) is a cylindrical Wiener process on L2 (O) of the form
∞
X
W (t) =
γn (t)en , t > 0,
n=1
2000 Mathematics Subject Classification. 60H15; 35K67, 49J45.
Key words and phrases. Stochastic evolution equation, p-Laplace equation, 1-Laplace equation,
variational convergence.
The research was supported by the Collaborative Research Center 701 (SFB 701) of the German
Science Foundation (DFG).
Both authors would like to thank Viorel Barbu and Michael Röckner for helpful comments.
The second author would like to thank Wei Liu for useful discussions on the stochastic p-Laplace
equation.
1
2
I. CIOTIR AND J. M. TÖLLE
where {γn } is a sequence of mutually independent real Brownian motions on a
filtered probability space (Ω, F , {Ft }t>0 , P) and {en } is an orthonormal basis of
L2 (O). We shall make further specifications. Q is assumed to be a linear, continuous, non-negative, symmetric operator on L2 (O) with eigenbasis {en } and corresponding sequence of eigenvalues {λn }. Let (−∆, dom(−∆)) be the Dirichlet
Laplacian in L2 (O), in particular, dom(−∆) = H 2 (O) ∩ H01 (O). Assume for simplicity that {en } is an eigenbasis of −∆ with corresponding sequence of eigenvalues
{µn }. We shall assume that
∞
X
(1.1)
λ1+κ
n µn < ∞
n=1
for some κ > 0. For the situation considered in this paper, it is enough to set
Q := (−∆)−1−δ with δ > 21 + κ for d = 1 and δ > 1 + κ for d = 2.
The singular diffusion operators in equations (Ep ), (E1 ) are called p-Laplacian
and 1-Laplacian respectively. In the deterministic case, i.e. if Q ≡ 0, both equations
of evolution-type are covered by the theory of nonlinear semigroups in Hilbert space
[9]. Both operators are used in image restoration, see [5, Ch. 3] for a comprehensive
treatment.
The deterministic p-Laplace equation arises from geometry, quasi-regular mappings, fluid dynamics and plasma physics, see [12, 13]. In [15], (Ep ) with Q ≡ 0 is
suggested as a model of motion of non-Newtonian fluids.
Due to the singularities in the diffusivity terms |∇Xp |dp−1 sgn(∇Xp ), sgn(∇X1 )
resp., we shall define the operators involved variationally. Equation (Ep ) can informally be rewritten as follows
p

p
in (0, T ) × O,

 dXp (t) + ∇Φ (Xp (t)) dt = Q dW (t)
(1.2)
Xp (t) = 0
on (0, T ) × ∂O,


Xp (0) = x
in O,
where ∇Φp denotes the Gâteaux differential of the convex functional
Z
1
Φp (u) :=
|∇u|pd dξ, u ∈ W01,p (O),
p O
and where W01,p (O) denotes the standard first-order Sobolev space of p-integrable
functions with Dirichlet boundary conditions.
Remark 1.1. By the classical Sobolev-embedding theorem(s), if d = 1, 2 and if
p ∈ (1, 2],
W01,p (O) ⊂ L2 (O)
continuously. See e.g. [1, Theorem 5.4].
For p = 1, the situation is more complicated. We would like to find a convex
functional Φ1 such that equation (E1 ) can be written as
p

1
in (0, T ) × O,

 dX1 (t) ∈ −∂Φ (X1 (t)) dt + Q dW (t)
(1.3)
X1 (t) = 0
on (0, T ) × ∂O,


X1 (0) = x
in O,
where ∂Φ1 is the subdifferential of Φ1 .
We shall need the spaces BV (O) and BV0 (O). For f ∈ L1loc (O), define the total
variation
Z
∞
d
kDf k (O) = sup
f div ψ dξ ψ ∈ C0 O; R . |ψ|d 6 1
O
CONVERGENCE OF SOLUTIONS TO THE STOCHASTIC p-LAPLACE EQUATIONS
3
BV (O) is defined to be equal to {f ∈ L1 (O) | kDf k (O) < ∞}. It is a Banach
space with norm kf kBV (O) := kDf k (O) + kf kL1 (O) . Denote the d − 1-dimensional
Hausdorff measure on ∂O by H d−1 . For f ∈ BV (O) there is an element f O ∈
L1 (∂O, dH d−1 ) called the trace such that
Z
Z
Z
f div ψ dξ = −
hψ, d[Df ]id +
hψ, νid f O dH d−1 ∀ψ ∈ C 1 (O; Rd ),
O
O
∂O
where [Df ] denotes the distributional gradient of f on O (which is a Rd -valued
Radon measure here) and ν denotes the outer unit normal on ∂O. With BV0 (O)
we denote the subspace of BV (O) of elements with zero trace.
Remark 1.2. By [2, Corollary 3.49], if d = 1, 2, then
W01,1 (O) ⊂ BV0 (O) ⊂ L2 (O)
continuously.
For spaces of functions of bounded variation, see e.g. [2, Ch. 3].
We shall return to equation (1.3). Recall that the subdifferential ∂Φ1 in L2 (O)
is defined by η ∈ ∂Φ1 (x) iff
Z
(1.4)
Φ1 (x) − Φ1 (y) 6
η (x − y) dξ, ∀y ∈ dom Φ1 .
O
1
One possible choice for Φ is the (homogeneous) energy
(R
|∇u|d dξ,
if u ∈ W01,1 (O),
O
e
Φ(u) :=
+∞,
if u ∈ L2 (O) \ W01,1 (O).
In this case, if u ∈ W01,1 (O), and if U := − div(sgn(∇u)) ⊂ L2 (O), then we have
e and U = ∂ Φ(u).
e
that u ∈ dom ∂ Φ
e fails to be lower semi-continuous in L2 (O) which is a necessary inHowever, Φ
gredient for the theory. Therefore, it is convenient to consider its relaxed functional
in L2 (O), which is equal to
(
kDuk (O),
if u ∈ BV0 (O),
1
Φ (u) :=
+∞,
if u ∈ L2 (O) \ BV0 (O).
e in
Φ1 is proper, convex and lower semi-continuous in L2 (O) and an extension of Φ
1
1
e
e
the sense that dom Φ ⊃ dom Φ and Φ 6 Φ.
Following the approach of Barbu, Da Prato and Röckner [8], we shall give the
definition of a solution for equations (Ep ), p ∈ [1, 2].
Definition 1.3. Set Vp := W01,p (O), p ∈ (1, 2], V1 := BV0 (O). A stochastic process
X = X (t, x) with P-a.s. continuous sample paths in H := L2 (O) is said to be a
solution to equation (Ep ), p ∈ [1, 2] if
X ∈ CW ([0, T ] ; H) ∩ Lp ((0, T ) × Ω, Vp ) ,
X (0) = x ∈ H
and
6
Z t
1
2
kX (t) − Y (t)k2 +
(Φp (X (s)) − Φp (Y (s))) ds
2
0
Z t
1
2
kx − Y (0)k2 +
(G (s) , X (s) − Y (s))2 ds, t ∈ [0, T ] ,
2
0
for all G (t) ∈ L2W (0, T ; H) and Y ∈ CW ([0, T ] ; H) ∩ Lp ((0, T ) × Ω; Vp ) satisfying
the equation
p
(1.5)
dY (t) + G (t) dt = Q dW (t) , t ∈ [0, T ] .
4
I. CIOTIR AND J. M. TÖLLE
Suppose for a while that 1 < p < 2, d = 1, 2. Arguing as in [17, Example 4.1.9,
Theorem 4.2.4], we can easily prove existence and uniqueness of the solution Xp for
equation (Ep ), in the usual (strong) variational sense, as in Prévôt, Röckner [17,
Definition 4.2.1]. Now, by Lemma A.1 in the Appendix A, we see that Xp is also a
solution in the sense of the definition above.
Regarding equation (E1 ), well-posedness of the problem as well as existence and
uniqueness of the solution were proved by Barbu, Da Prato and Röckner in [8].
We are now able to formulate the main result of this paper.
Theorem 1.4. The sequence of solutions {Xp }p to equations (Ep ) is convergent
for p → 1 to the solution X1 of equation (E1 ), strongly in L2 (O), uniformly on
[0, T ], P − a.s., i.e.,
lim sup kXp (t) − X1 (t)k2 = 0,
p→1 t∈[0,T ]
P − a.s.
We remark that our proof comprehends the situation where the limit equation
could possibly be chosen any (Ep0 ), p0 ∈ [1, 2]. This suggests that our convergence
result can as well be considered a continuity result (in the parameter p).
Our proof uses methods from variational convergence of convex functionals, see
Attouch’s book [3] for an introduction. We shall prove the analytic facts used for
the main result in Section 2. Section 3 contains the proof of Theorem 1.4. In
Section 4, we prove the convergence of ergodic invariant measures in the strongly
monotone situation. The proof of tightness relies on a compactness argument in
BV . Ergodicity of the semigroup associated to the singular stochastic p-Laplace
equations has been studied in [16]. For all p ∈ (1, 2) it remains an open question.
In this paper, we prove the following.
Theorem 1.5. Let Xp = Xp (t, x) be the solution to equation (Ep ), p ∈ [1, 2]. Let
p0 ∈ [1, 2], {pn } ⊂ (p0 , 2] such that limn pn = p0 . Let
Ptp ϕ(x) := E [ϕ (Xp (t, x))]
ϕ ∈ Cb (L2 (O))
be the semigroup associated to equation (Ep ). Suppose that Ptpn , n ∈ N, Ptp0 are
ergodic, i.e. admit unique invariant measures νpn , n ∈ N, νp0 on L2 (O). Then
νpn → νp0
in the weak sense.
Notation. Throughout this paper we denote by H the Hilbert space L2 (O) with
R
1/2
the scalar product (f, g)2 := O f g dξ and the norm kf k2 := (f, f )2 . The
1,p
p
spaces L (O) , p > 1, and W0 (O) are the standard spaces of integrable functions
and Sobolev spaces on O with Dirichlet boundary conditions. We set H01 (O) :=
W01,2 (O) and H 2 (O) := W 2,2 (O). Depending on the context, we shall use the notation k·k∞ both to denote the supremum-norm and the essential supremum norm.
|·|d and h·, ·id denote the Euclidean norm and Euclidean scalar product respectively.
For p ∈ [1, ∞], let p0 := p/(p − 1) be the conjugate exponent. The letter C will be
used to denote several positive constants.
2
By L2W (0, T ; H) and CW
([0, T ]; H) we denote the space of all square integrable
(respectively continuous) functions from [0, T ] to L2 (Ω; H) which are adapted to
{Ft }t>0 .
For p ∈ (1, ∞), define ap : Rd → Rd by ap (x) := |x|p−1
sgn(x). Furthermore,
d
1,p
−1,p0
let Ap : W0 (O) → W0
(O) be defined by Ap (y) := − div [ap (∇y)], where
y ∈ W01,p (O). To be more specific,
Z
0 hAp (y), zi
=
hap (∇y), ∇zid dξ, ∀z ∈ W01,p (O) .
−1,p
1,p
W
W
O
CONVERGENCE OF SOLUTIONS TO THE STOCHASTIC p-LAPLACE EQUATIONS
5
Furthermore, for p ∈ [1, ∞), we shall define j p : Rd → R by j p (x) := p1 |x|d .
Obviously, if p > 1, each j p is a convex C 1 -function such that ap = ∇j p and
p
Φp : W01,p (O) → [0, ∞),
Z
p
Φ (y) :=
j p (∇y) dξ, ∀y ∈ W01,p (O)
O
is Fréchet differentiable and satisfies ∂Φp = ∇Φp = Ap .
2. Some results on variational convergence
Before we prove Theorem 1.4, we need some preparations. The results of this section from variational convergence of convex functionals are only partly represented
in the literature. We also collect the necessary facts in Appendix B.
Let p ∈ [1, 2]. Let j p : Rd → R be as in the last
part of the introduction.
1
|x − y|2d be its regularization. For
For ε > 0, let jεp (x) := inf y∈Rd j p (y) + 2ε
u ∈ L2 (O; Rd ), set
Z
Ψp (u) :=
j p (u) dξ.
O
2
Ψp is a continuous convex functional on L (O; Rd ) for each p ∈ [1, 2].
Lemma 2.1. For ε > 0, let Ψpε be the Moreau-Yosida regularization of Ψp in
L2 (O; Rd ). Then
Z
Ψpε (v) =
jεp (v) dξ ∀v ∈ L2 (O; Rd ).
O
Proof. Let v ∈ L (O; R ). Fix a representative v of v. For ξ ∈ O, x ∈ Rd , let
1
fv (ξ, x) := j p (x) + |v(ξ) − x|2d .
2ε
Obviously, fv satisfies Carathéodory’s conditions and is hence a normal integrand.
Therefore, we can apply [18, Theorem 14.60] for the space L2 (O; Rd ) in order to
obtain
Z
Z
Z
p
Ψε (v) =
inf
fv (ξ, u(ξ)) dξ =
inf fv (ξ, x) dξ =
jεp (v(ξ)) dξ,
2
d
u∈L2 (O;
R
d)
O x∈
O
R
d
O
where both integrals are finite.
We would like to prove a convergence result, which shall be useful later. See the
appendix for the terminology. Compare also with [4].
Lemma 2.2. Let {pn } ⊂ [1, 2] such that limn pn = 1. Then
in the Mosco sense in L2 (O; Rd ).
M
Ψpn −→ Ψ1
Proof. Let us prove (M1) in Definition B.1 first. Let un ∈ L2 (O; Rd ), n ∈ N,
u ∈ L2 (O; Rd ) such that un * u weakly in L2 (O; Rd ). W.l.o.g. lim inf n Ψpn (un ) <
+∞. Extract a subsequence (also denoted by {un }) such that
lim inf Ψpn (un ) = lim Ψpn (un ).
n
n
Let v ∈ L (O; R ). Clearly,
Z
Z
lim
hun , vid dξ =
hu, vid dξ.
∞
d
n
O
O
Also, by Hölder’s inequality,
Z
pn
1 p
hun , vid dξ 6 Ψpn (un )|O|pn −1 kvkLn∞ (O;Rd ) ,
pn
O
6
I. CIOTIR AND J. M. TÖLLE
(here |O| =
R
O
dξ). Upon taking the limit n → ∞, we get that
Z
hu, vi dξ 6 lim inf Ψpn (un ) kvk ∞
L (O;Rd ) .
d
n
O
Taking the supremum over all v ∈ L∞ (O; Rd ) with kvkL∞ (O;Rd ) 6 1 and using the
l.s.c. property of the supremum, we get that
Z
Ψ1 (u) =
|u|d dξ 6 lim inf Ψpn (un ).
O
n
Since the same argument works for any subsequence of {un }, we have proved (M1).
We are left to prove (M2) in Definition B.1. Let u ∈ L2 (O; Rd ). Clearly for a.e.
ξ∈O
1
lim |u(ξ)|pdn = |u(ξ)|d .
n pn
But for all p ∈ [1, 2],
1 p
|u| 6 1O + |u|2d ∈ L1 (O).
p d
Hence an application of Lebesgue’s dominated convergence theorem yields
lim Ψpn (u) = Ψ1 (u).
n
(M2) is proved.
Theorem B.2, Corollary B.3 and Lemmas 2.1, 2.2 together give:
Corollary 2.3. Let {pn } ⊂ [1, 2] such that limn pn = 1. Let ε > 0. Then for
u ∈ L2 (O; Rd ), we have that
Z
Z
jε1 (u) dξ.
(2.1)
lim
jεpn (u) dξ =
n
O
O
Furthermore, if un * u converges weakly in L2 (O; Rd ), we have that
Z
Z
(2.2)
lim inf
jεpn (un ) dξ >
jε1 (u) dξ.
n
O
O
For each ε > 0, let Rε := (1 − ε∆)−1 be the resolvent of the (Dirichlet) Laplace
operator (−∆, dom(−∆)), where dom(−∆) = H01,2 (O) ∩ H 2,2 (O). For p ∈ [1, 2],
ε > 0, let
Z
Φpε (u) :=
O
jεp (∇Rε u) dξ,
u ∈ L2 (O).
Lemma 2.4. Let {pn } ⊂ [1, 2] such that limn pn = 1. Let ε > 0. Then for
u ∈ L2 (O), we have that
lim Φpε n (u) = Φ1ε (u).
(2.3)
n
Furthermore, if un * u converges weakly in L2 (O), we have that
lim inf Φpε n (un ) > Φ1ε (u).
(2.4)
Also, each
n
Φpε ,
p ∈ [1, 2], ε > 0, is continuous w.r.t. the weak topology of L2 (O).
Proof. Since Rε maps to dom(−∆) ⊂ H01 (O), it is clear that ∇Rε u ∈ L2 (O; Rd )
and hence (2.3) follows from (2.1).
Let un ∈ L2 (O), n ∈ N, u ∈ L2 (O), such that un * u weakly in L2 (O). If we
can proof that ∇Rε un * ∇Rε u weakly in L2 (O; Rd ), we can apply (2.2) and (2.4)
follows. Indeed, we even have that ∇Rε un → ∇Rε u strongly in L2 (O; Rd ). To see
this, one possibility is to proceed as follows: Equip dom(−∆) with the graph norm
1/2
2
2
k·kdom(−∆) := k·kL2 (O) + k∆·kL2 (O)
.
CONVERGENCE OF SOLUTIONS TO THE STOCHASTIC p-LAPLACE EQUATIONS
7
Now, by definition of the resolvent, and the fact that it is an L2 (O)-contraction,
we see that Rε : L2 (O) → dom(−∆) is strongly (and hence weakly) continuous.
By [19, §4.2.4, Theorem], k·kdom(−∆) is an equivalent norm to the Sobolev-norm
of H 2 (O). By the Rellich-Kondrachov Theorem (see e.g. [1, Theorem 6.2]), the
embedding H 2 (O) ⊂ H 1 (O) is compact. Hence Rε un → Rε u strongly in H 1 (O)
and so the corresponding gradients converge strongly in L2 (O; Rd ) as claimed.
The last part follows by repeating the compactness argument above and the
strong L2 (O; Rd )-continuity of the Ψpε ’s.
3. Proof of Theorem 1.4
We first consider the following approximating equations for (Ep )
(
p
dXpε (t) + Aεp Xpε dt = Q dW (t)
(3.1)
Xpε (0) = x
where for any u ∈ L2 (O),
Aεp (u) = − (1 − ε∆)
−1
i
h −1
div aεp ∇ (1 − ε∆) u
and aεp is the Yosida approximation of ap i.e., for any r ∈ Rd ,
1
−1
1 − (1 + εap ) (r) .
aεp (r) =
ε
2
In particular, for u, v ∈ L (O),
Z
ε
Aεp (u), v 2 =
ap (∇Rε u), ∇Rε (v) d dξ,
O
where Rε := (1 − ε∆)−1 .
We shall consider a similar approximation for equation (E1 )
(
p
dX1ε (t) + Aε (X1ε ) dt = Q dW (t)
(3.2)
X1ε (0) = x
where for any u ∈ L2 (O),
−1
Aε (u) = − (1 − ε∆)
with
β ε (r) =



h i
−1
div β ε ∇ (1 − ε∆) u .
r
, if |r|d 6 ε,
ε
r

, if |r|d > ε.

|r|d
In particular, for u, v ∈ L2 (O),
(Aε (u), v)2 =
Z
O
hβ ε (∇Rε u), ∇Rε (v)id dξ.
Note that β ε is the Yosida approximation of the sign function, i.e., for any
r ∈ Rd ,
1
−1
β ε (r) =
1 − (1 + ε sgn) (r) .
ε
In particular, β ε = ∇j ε , where jε is the convex function defined by



2
|r|d
, if |r|d 6 ε,
ε
2ε
j (r) =

 |r| − ε , if |r| > ε.
d
d
2
8
I. CIOTIR AND J. M. TÖLLE
We shall use the following strategy to prove the main result
kXp (t) − X1 (t)k2
6 Xp (t) − Xpε (t)2 + Xpε (t) − X1ε (t)2 + kX1ε (t) − X1 (t)k2
P-a.s. and uniformly in t ∈ [0, T ] .
At this point we need to prove the following lemma. We introduce the notation
−1
rεp (r) := (1 + εap ) (r).
Lemma 3.1. Under our assumptions, if we let Xpε be the solution to (3.1) and
epε := (1 − ε∆)−1 Xpε , we have that
X
(3.3)
E
Z tZ p
p
epε (s) dξ ds 6 Ct kxk2 + Tr Q ,
rε ∇X
2
0
d
O
for all t ∈ [0, T ].
Proof. We know by the definiton of ap that
p
hap (r) , rid > |r|d .
2
On the other hand we have by Itō’s formula, applied to the function u 7→ kuk2 ,
that
Z tZ D E
2
epε (s) , ∇X
epε (s) dξ ds
aεp ∇X
(3.4)
E Xpε (t)2 + 2E
0
d
O
2
6 Ct kxk2 + Tr Q .
By the definition of the Yosida approximation we have that
aεp (r) = ap (rεp (r))
and
ε
1
2
ap (r) , r d = aεp (rεp (r)) , rεp (r) d + |r − rεp (r)|d .
ε
We rewrite as follows
Z tZ D E
e ε (s) , ∇X
e ε (s) dξ ds
E
aεp ∇X
p
p
d
0
O
Z tZ D E
e ε (s) , rp ∇X
e ε (s)
> E
ap rεp ∇X
dξ ds
p
ε
p
d
0
O
Z tZ p
p
epε (s) dξ ds.
> E
rε ∇X
0
O
d
Plugging into (3.4) proves (3.3).
Step I. We have from [8, equation (4.8)] that
lim sup kX1ε (t) − X1 (t)k2 = 0,
ε→0t∈[0,T ]
P-a.s.
CONVERGENCE OF SOLUTIONS TO THE STOCHASTIC p-LAPLACE EQUATIONS
Step II. We shall prove now that
lim sup Xp (t) − Xpε (t)2 = 0,
9
P-a.s. uniformly in p ∈ (1, 2) .
ε→0t∈[0,T ]
epε = (1 − ε∆)−1 Xpε and X
epλ = (1 − λ∆)−1 Xpλ . Then by (3.1), we have
We set X
that
1
Xpε (t) − Xpλ (t)2
2
2
Z tZ D E
epε (s) − aλp ∇X
epλ (s) , ∇X
epε (s) − ∇X
epλ (s) dξ ds = 0.
+
aεp ∇X
0
d
O
epε (s) = uε and ∇X
epλ (s) = uλ and using
Setting ∇X
−1
aεp (u) ∈ ap (1 + εap ) (u) ,
we get by the monotonicity of ap that
ε ε
ap (u ) − aλp uλ , uε − uλ d
> aεp (uε ) − aλp uλ , εaεp (uε ) − λaλp uλ d .
This leads to
(3.5)
1
Xpε (t) − Xpλ (t)2
2
2
Z tZ 2
2 ε
λ
ε
λ
e
e
6
ε ap ∇Xp (s) + λ ap ∇Xp (s) dξ ds.
0
d
O
d
We can now prove that
Z tZ 2
ε
e ε (s) dξ ds 6 Ct
ap ∇X
p
(3.6)
0
d
O
P-a.s.
for some Ct independent of p and ε.
0
Using Jensen’s inequality (for t 7→ tp /2 ) and taking into account that |ap (r)|d 6
p−1
|r|d , we obtain
Z tZ 2
ε
epε (s) dξ ds
ap ∇X
0
d
O
Z t Z 2/p0
p0
1−2/p0
p
ε
ep (s) dξ ds
6(t |O|)
ap rε ∇X
(3.7)
0
d
O
Z t Z 2/p0
p
p
ε
ep (s) dξ ds
6(1 + t |O|)
rε ∇X
d
0
O
Z t Z p
p
ε
e
6Ct + Ct
∇
X
(s)
dξ
ds
,
rε
p
0
d
O
R
where |O| = O dξ.
Now by Lemma 3.1 we have (3.6) for a constant Ct independent of p and ε, and
passing to the limit for ε, λ → 0 in (3.5) we get that
lim sup Xp (t) − Xpε (t) = 0, P-a.s. uniformly in p ∈ (1, 2) .
ε→0t∈[0,T ]
2
10
I. CIOTIR AND J. M. TÖLLE
Step III. In order to complete the proof we still need to show that for all ε > 0
fixed we have
lim sup Xpε (t) − X1ε (t)2 = 0, P-a.s.
p→1t∈[0,T ]
To this aim, we consider the definiton of the solution for equations
(
p
dXpε (t) + Aεp Xpε dt = Q dW (t)
Xpε (0) = x
as
6
Z t
1
Xpε (t) − Y (t)2 +
Φpε Xpε (s) − Φpε (Y (s)) ds
2
2
0
Z t
1
2
kx − Y (0)k2 +
G (s) , Xpε (s) − Y (s) 2 ds,
2
0
for all t ∈ [0, T ] , P-a.s.
We take Y = X1ε , the solution of equation
(
p
dX1ε (t) + Aε (X1ε ) dt = Q dW (t)
X1ε (0) = x.
and using the definiton of the subdifferential we get that
(3.8)
1
Xpε (t) − X1ε (t)2
2
2
Z t
+
Φpε Xpε (s) − Φpε (X1ε (s)) + Φ1ε (X1ε (s)) − Φ1ε Xpε (s) ds
0
1
2
6 kx − X1ε (0)k2 = 0,
2
for t ∈ [0, T ] and P-a.s. By estimate (3.4), we can extract a subsequence {pn }
with limn pn = 1 such that for Xnε := Xpεn we have that for dt-a.a. t ∈ [0, T ],
Xnε (t) * Z ε (t) weakly in L2 (O) P-a.s. for some dt⊗ P-measurable Z ε that satisfies
sup kZ ε (t)k2 6 lim inf sup kXn (t)k2
n
t∈[0,T ]
t∈[0,T ]
P-a.s.
We shall need following lemma. Set Φnε := Φpε n .
Lemma 3.2.
Φnε (X1ε (·)) − Φnε (Xnε (·)) + Φ1ε (Xnε (·)) − Φ1ε (X1ε (·))
is
P-a.s. bounded above by a function in L∞ (0, T ).
Proof. Set u := Xnε (·), v := X1ε (·). Recall that in our notation, Rε := (1 − ε∆)−1 .
Let us treat the term Φ1ε (u) − Φ1ε (v) first. By the definition of the subgradient
it is bounded by (∇Φ1ε (u), u − v)2 . But this term is equal to
Z
hβ ε (∇Rε (u)), ∇Rε (u − v)id dξ.
O
Since |β |d 6 1, we get that the latter is bounded by k∇Rε (u − v)kL2 (O;Rd ) . By
the proof of Lemma 2.4, ∇Rε is a bounded operator from L2 (O) to L2 (O; Rd ).
We get that
ε
Φ1ε (Xnε (·)) − Φ1ε (X1ε (·)) 6 C sup kXnε (·)k2 + C kX1ε (·)k2
which is
P-a.s. in L
∞
n
(0, T ) again by estimate (3.4).
CONVERGENCE OF SOLUTIONS TO THE STOCHASTIC p-LAPLACE EQUATIONS
11
We continue with the term Φnε (v) − Φnε (u). By the definition of the subgradient
it is bounded by (∇Φnε (v), v − u)2 , which is equal to
Z
ε
ap (∇Rε (v)), ∇Rε (v − u) d dξ.
O
Noticing that rεp is a contraction on
to get that the latter is bounded by
Rd , we can use a similar estimate as in (3.7)
C + C k∇Rε (v)kL2 (O;Rd ) k∇Rε (v − u)kL2 (O;Rd ) .
Arguing as above, we see that this term is bounded by
2
C + C sup kXnε (·)k2 kX1ε (·)k2 + C kX1ε (·)k2 ,
n
which is
P-a.s. in L∞ (0, T ) by estimate (3.4).
We take the limit superior in (3.8) and continue investigating
Z t
n ε
Φε (X1 (s)) − Φnε (Xnε (s)) + Φ1ε (Xnε (s)) − Φ1ε (X1ε (s)) ds.
lim sup
n
0
By Lemma 3.2, we can apply (reverse) Fatou’s lemma such that it is sufficient to
prove that
lim sup Φnε (X1ε (s)) − Φnε (Xnε (s)) + Φ1ε (Xnε (s)) − Φ1ε (X1ε (s)) 6 0
n
P-a.s. and for ds-a.e. s ∈ [0, T ]. At this point, we apply Lemma 2.4 and get that
lim sup Φnε (X1ε (s)) − Φnε (Xnε (s)) + Φ1ε (Xnε (s)) − Φ1ε (X1ε (s))
n
6 lim sup Φnε (X1ε (s)) − lim inf Φnε (Xnε (s)) + lim sup Φ1ε (Xnε (s)) − Φ1ε (X1ε (s))
n
1
6Φε (X1ε (s))
n
−
Φ1ε (Z ε (s))
+
Φ1ε (Z ε (s))
−
n
1
Φε (X1ε (s))
=0,
P-a.s. and for ds-a.e. s ∈ [0, T ].
Final step. Going back to
kXp (t) − X1 (t)k2
6 Xp (t) − Xpε (t)2 + Xpε (t) − X1ε (t)2 + kX1ε (t) − X1 (t)k2
P-a.s. and uniformly in t ∈ [0, T ], we can complete the proof using Steps I–III as
follows. Let δ > 0. Pick ε0 > 0, independent of p, such that the first and the third
term are less than δ/3. Having fixed ε0 in such a way, we can pick p such that the
second term is less than δ/3.
The proof of Theorem 1.4 is complete.
4. Convergence of invariant measures
For each p ∈ [1, 2], let γp > 0. Under similar assumptions, replace (Ep ) (p ∈
[1, 2]) by the equation

h
h
i
i
p
 dX (t) = div |∇X |p−1 sgn(∇X ) − γ X dt + Q dW (t)
p
p d
p
p p
(EEp )
 X (0) = x,
p
with Dirichlet boundary conditions in O.
By [17, Theorem 4.3.9], if p > 1 and γp > 0, then equation (EEp ) has exactly
one ergodic invariant measure νp with second moments.
The following statement can be easily verified by replacing Φp in the previous
γ
2
section by Φp + 2p k·k2 .
12
I. CIOTIR AND J. M. TÖLLE
Corollary 4.1. Let Xp = Xp (t, x) be the variational solution to equation (EEp ).
If limp→1 γp = γ1 , then
lim sup kXp (t, x) − X1 (t, x)k2 = 0
p→1 t∈[0,T ]
P-a.s.
In the strongly monotone situation, we have the following:
Theorem 4.2. Assume that for each p ∈ [1, 2], γp > k for some k > 0, and that
γp → γ1 as p → 1. Then (EE1 ) has a unique ergodic invariant measure ν1 , the
family of measures {νp }p∈(1,2) is tight and νp → ν1 in the weak sense as p → 1.
Proof. Let us prove tightness of {νp }p∈(1,2) first. Denote the norm of Vp := W01,p (O)
by k·k1,p .
Let Xp = Xp (t, x) be the variational solution to equation (EEp ). By Itō’s
formula, (see [17]),
Z t
2
2
(4.1) E kXp (t, x)k2 + 2E
Ap (X p (s, x)), X p (s, x) Vp ds = kxk2 + t Tr Q,
V∗
0
p
and hence,
(4.2)
1
E
t
Z
0
t
X p (s, x)p ds 6 1 kxk2 + 1 Tr Q.
2
1,p
2t
2
Here X p is some suitable progressively measurable Vp -valued version of Xp . By
the Krylov–Bogoliubov theorem (and a truncation argument), passing to the limit
t → +∞ yields the estimate
Z
Tr Q
p
(4.3)
kxk1,p νp (dx) 6
∀p ∈ (1, 2).
2
H
By [14, Ch. 5.1], Vp ⊂ BV0 (O) for every p. We have that
Z
Z
p
kxk1,p =
|∇x|p dx >
|∇x| dx − |O| = kDxk (O) − |O|.
O
O
Let θ > 0. Set
o
n
Bθ := x ∈ H kDxk (O) 6 θ−1 + |O| .
By [2, Corollary 3.49], Bθ is compact in H. Now, for any p ∈ (1, 2), using (4.3),
Z
Tr Q
p
νp (Bθc ) = νp kD·k (O) − |O| > θ−1 6 θ
kxk1,p νp (dx) 6 θ
.
2
H
Let pn → 1, pn ∈ (1, 2). Let µ be a weak accumulation point of {νpn }n∈N , such
that νpnk → µ weakly in H for some subsequence {pnk }. Denote by
Ptp ϕ(x) := E[ϕ(Xp (t, x))] ϕ ∈ Cb (H)
the semigroup associated to equation (EEp ).
Let {Tl } be exactly the sequence of positive numbers such that Tl ↑ +∞ which
is used in the proof of [8, Theorem 5.1]. By the Krylov–Bogoliubov theorem, we
have for all ϕ ∈ Cb (H) that
Z
Z
1 Tl p n k
Pt ϕ(x) dt
ϕ(x) νpnk (dx) = lim
l Tl 0
H
Z
1 Tl p n k
= lim
Pt ϕ(x) − Pt1 ϕ(x) dt
l Tl 0
Z
1 Tl 1
+ lim
Pt ϕ(x) dt.
l Tl 0
CONVERGENCE OF SOLUTIONS TO THE STOCHASTIC p-LAPLACE EQUATIONS
13
Since ϕ is bounded, we can apply Lebesgue’s dominated convergence theorem (in
L2 (Ω)) to Corollary 4.1 in order to obtain the convergence
pnk
Pt
ϕ(x) → Pt1 ϕ(x).
By [8, Theorem 5.1], we get the existence of an invariant measure for equation
(EE1 ), which we denote by ν. By the above,
Z
Z
ϕ(x) µ(dx) =
ϕ(x) ν(dx) ∀ϕ ∈ Cb (H).
H
H
We are left to prove ergodicity of ν. Let x, y ∈ H. Let Xp (t, x), Xp (t, y) be solutions to (EEp ) starting in x and y resp. By Itō’s formula and strong monotonicity
we have that P-a.s.
Z t
1
1
2
2
2
kXp (s, x) − Xp (s, y)k2 ds.
kXp (t, x) − Xp (t, y)k2 6 kx − yk2 − γp
2
2
0
By Corollary 4.1, we can pass to the limit p → 1 and get that P-a.s.
Z t
1
1
2
2
2
kX1 (t, x) − X1 (t, y)k2 6 kx − yk2 − γ1
kX1 (s, x) − X1 (s, y)k2 ds.
2
2
0
Hence by Grönwall’s lemma, P-a.s.
2
2
kX1 (t, x) − X1 (t, y)k2 6 kx − yk2 exp(−2γ1 t).
We know that by [8, Theorem 5.1], ν is an invariant measure for the semigroup
Pt1 ϕ := E [ϕ(X1 (t, x))] ,
If ϕ ∈
ϕ ∈ Cb (H),
Cb1 (H),
we have that
Z
Z
Pt ϕ(x) −
|Pt ϕ(x) − Pt ϕ(y)| ν(dy).
ϕ(y) ν(dy) 6
H
H
Furthermore,
Z
Z
6 kϕk
Pt ϕ(x) −
ϕ(y)
ν(dy)
1,∞
H
H
E kX1 (t, x) − X1 (t, y)k22 ν(dy)
Z
6 kϕk1,∞
H
2
kx − yk2 ν(dy) exp(−2γ1 t).
By weak convergence, ν has second moments. Now, since Cb1 (H) is dense in
L2 (H, ν), it follows for any f ∈ L2 (H, ν) that
Z
lim Pt f (x) =
f dν, ∀x ∈ H.
t→+∞
H
Therefore, ν is ergodic and strongly mixing by [11, Theorem 3.4.2].
With uniqueness of invariant measures assumed a priori, Theorem 1.5 can be
proved exactly by the steps above, where γp = 0, p ∈ [1, 2].
Appendix A. Two notions of solutions
Lemma A.1. A solution to equation (Ep ), p ∈ (1, 2], in the sense of Prévôt,
Röckner [17, Definition 4.2.1] is also a solution in the sense of Definition 1.3.
Proof. Let Xp (t, x) be a solution to (Ep ) in the sense of Prévôt and Röckner. Let
bp be its dt⊗ P-equivalence class. Then X
bp ∈ Lp ([0, T ]×Ω, dt⊗ P; Vp )∩L2 ([0, T ]×
X
Ω, dt ⊗ P; H) and P-a.s.
Z tp
Z t
Xp (t, x) = x −
∂Φp (X p (s, x)) ds +
Q dW (s)
0
0
bp .
for any Vp -valued progressively measurable dt ⊗ P-version X p of X
14
I. CIOTIR AND J. M. TÖLLE
Now, let Y ∈ CW ([0, T ]; H) ∩ Lp ((0, T ) × Ω; Vp ) and G ∈ L2W (0, T ; H) such that
P-a.s.
Z tp
Z t
G(s) ds +
Y (t) = Y (0) −
Q dW (s).
0
0
Take the difference of the two equations to obtain the Vp∗ -valued process
Z t
Z t
p
G(s) ds.
Xp (t, x) − Y (t) = x − Y (0) −
∂Φ (X p (s, x)) ds +
0
0
By the Itō-formula [17, Theorem 4.2.5], we get that
Z t
p
1
X p (t, x) − Y (t)2 +
ds
∗ ∂Φ (X p (s, x)), X p (s, x) − Y (s) V
H
V
p
p
2
0
Z t
1
2
= kx − Y (0)kH +
ds
∗ G(s), X p (s, x) − Y (s) V
V
p
p
2
0
bp . Using the
for any Vp -valued progressively measurable dt ⊗ P-version X p of X
definition of the subgradient and the fact that G is H-valued yields
Z t
1
X p (t, x) − Y (t)2 +
(Φp (X p (s, x)) − Φp (Y (s))) ds
H
2
0
Z t
1
2
6 kx − Y (0)kH +
(G(s), X p (s, x) − Y (s))H ds,
2
0
which completes the proof.
Appendix B. Mosco convergence
Let H be a separable Hilbert space. For a proper, convex functional Φ : H →
(−∞, +∞], the Legendre transform Φ∗ is defined by
Φ∗ (y) := sup [(x, y)H − Φ(x)] ,
y ∈ H.
x∈H
For two functionals F, G : H → (−∞, +∞] the infimal convolution F #G is defined
by
(F #G)(y) := inf [F (x) + G(y − x)] , y ∈ H.
x∈H
For a proper, convex, l.s.c. functional Φ : H → (−∞, +∞], for each ε > 0, define
the Moreau-Yosida regularization
1
2
Φε := Φ# k·kH .
2ε
Φε is a continuous convex function. Also, limε&0 Φε = Φ pointwise.
It holds that
ε
2
(B.1)
(Φε )∗ = Φ∗ + k·kH .
2
see e.g. [6, §2.2] and [3, Ch. 3].
Recall following definition.
Definition B.1 (Mosco convergence). Let Φn : H → (−∞, +∞], n ∈ N, Φ :
M
H → (−∞, +∞] be proper, convex, l.s.c. functionals. We say that Φn −→ Φ in the
Mosco sense if
(M1) ∀x ∈ H ∀xn ∈ H, n ∈ N, xn * x weakly in H :
lim inf Φn (xn ) > Φ(x).
(M2) ∀y ∈ H ∃yn ∈ H, n ∈ N, yn → y strongly in H :
lim sup Φn (yn ) 6 Φ(y).
We shall need following theorem.
n
n
CONVERGENCE OF SOLUTIONS TO THE STOCHASTIC p-LAPLACE EQUATIONS
15
Theorem B.2. Let Φn : H → (−∞, +∞], n ∈ N, Φ : H → (−∞, +∞] be proper,
convex, l.s.c. functionals. Then the following conditions are equivalent.
M
(i) Φn −→ Φ.
M
(ii) (Φn )∗ −→ Φ∗ .
(iii) ∀ε > 0, ∀x ∈ H: limn Φnε (x) = Φε (x).
Proof. See [3, Theorems 3.18 and 3.26].
M
M
Corollary B.3. Suppose that Φn −→ Φ. Then for each ε > 0, Φnε −→ Φε , too.
M
M
Proof. Suppose that Φn −→ Φ. By Theorem B.2, (Φn )∗ −→ Φ∗ , too.
M
If we can prove for each ε > 0 that (Φnε )∗ −→ (Φε )∗ , we are done by Theorem B.2.
(M2) in Definition B.1 follows easily, using equation (B.1) and (M2) for {(Φn )∗ }
and Φ∗ .
Let xn ∈ H, n ∈ N, x ∈ H such that xn * x weakly in H. By (B.1), weak lower
semi-continuity of the norm and (M1) in Definition B.1 for {(Φn )∗ } and Φ∗ we get
that
i
h
ε
2
lim inf (Φnε )∗ (xn ) = lim inf (Φn )∗ (xn ) + kxn kH
n
n
2
ε
ε
2
2
n ∗
> lim inf (Φ ) (xn ) + lim inf kxn kH > Φ∗ (x) + kxkH = (Φε )∗ (x).
n
n
2
2
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Department of Mathematics, Faculty of Economics and Business Administration, “Al.
I. Cuza” University, Bd. Carol no. 9–11, Iaşi, Romania
E-mail address: ioana.ciotir@feaa.uaic.ro
Faculty of Mathematics, Universität Bielefeld, Postfach 100 131, D-33501 Bielefeld,
Germany
E-mail address: jtoelle1@math.uni-bielefeld.de