Convergence of invariant measures for singular stochastic diffusion equations

0.25 MB·English

by  Ioana Ciotir

CONVERGENCE OF INVARIANT MEASURES FOR SINGULAR
STOCHASTIC DIFFUSION EQUATIONS
IOANA CIOTIR
Department of Mathematics, Faculty of Economics and Business Administration,
2
“Al. I. Cuza” University, Bd. Carol no. 9–11, Ia¸si, Romania
1
0
2

n JONAS M. TO¨LLE
a
J
Institut fu¨r Mathematik, Technische Universita¨t Berlin (MA 7-5), Straße des 17.
3
1 Juni 136, 10623 Berlin, Germany

]
R Abstract. Itisprovedthatthesolutionstothesingularstochasticp-Laplace
equation, p∈(1,2)andthesolutions tothestochastic fastdiffusionequation
P
withnonlinearityparameterr∈(0,1)onaboundedopendomainΛ⊂ dwith
. R
h Dirichletboundaryconditionsarecontinuousinmean,uniformlyintime,with
t respect to the parameters p and r respectively (in the Hilbert spaces L2(Λ),
a H−1(Λ)respectively). Thehighlysingularlimitcasep=1istreatedwiththe
m
helpofstochasticevolutionvariationalinequalities,where -a.s. convergence,
P
[ uniformlyintime,isestablished.

It is shown that the associated unique invariant measures of the ergodic
1
semigroupsconvergeintheweaksense(ofprobabilitymeasures).
v
9
3
8
2 1. Introduction
.
1
Let Λ ⊂ d be a bounded open domain with Lipschitz boundary ∂Λ. Let
0
R
2 {W(t)}t>0 be a U-valued cylindrical Wiener process on some filtered probability
1 space (Ω,F,{F(t)} , ), where U is a separable Hilbert space.
t>0
: WeareinterestedinthPefollowingtwo(familiesof)stochasticdiffusionequations,
v
i the stochastic p-Laplacianequation, p∈(1,∞), B ∈L2(U,L2(Λ)),
X
r dXp(t)=div |∇Xp(t)|p−2∇Xp(t) dt+BdW (t) in (0,T)×Λ,
a
(PLp) Xp(t)=0 h i on (0,T)×∂Λ,
 X (0)=x∈L2(Λ) in Λ.
p

E-mail addresses: [email protected], [email protected].
2000 Mathematics Subject Classification. 60H15;35K67,37L40, 49J45.
Keywordsandphrases. Stochasticevolutionequation,stochasticdiffusionequation,p-Laplace
equation, 1-Laplace equation, total variation flow, fast diffusion equation, ergodic semigroup,
uniqueinvariantmeasure,variationalconvergence.
Financial support of the DFG Collaborative Research Center 701 (SFB 701) Spectral Struc-
tures and Topological Methods in Mathematics (Bielefeld) and the DFG Research Group 718
(Forschergruppe 718) Analysis and Stochastics in Complex Physical Systems (Berlin–Leipzig) is
gratefullyacknowledged.
Bothauthors wouldliketothank ViorelBarbuandMichaelRo¨ckner forhelpful comments. The
authorsaregratefulfortheremarksoftworefereeswhichhelpedinimprovingthepaper.
1
2 I.CIOTIRANDJ.M.TO¨LLE
The deterministic p-Laplace equation arises from geometry, quasi-regular map-
pings,fluid dynamicsandplasmaphysics,see [19,20]. In [27],(PL )withB ≡0 is
p
suggestedasamodelofmotionofnon-Newtonianfluids. See [28]forthe stochastic
equation.
We are also interested in the stochastic fast diffusion equation r ∈ (0,∞), B ∈
L (U,H−1(Λ)),
2
dY (t)=∆ |Y (t)|r−1Y (t) dt+BdW(t), in (0,T)×Λ,
r r r
(FDr) Yr(t)=0,(cid:0) (cid:1) on (0,T)×∂Λ,
 Y (0)=y ∈H−1(Λ), in Λ,
r
which modelsdiffusion in plasma physics, curvature flows and self-organized criti-
cality in sandpile models, see e.g. [12, 14, 36, 41] and the references therein.
The above equations considered are called singular for p∈(1,2), r ∈(0,1) and
degenerate for p∈(2,∞), r ∈(1,∞) (porous medium equation). In this paper, we
shall investigate the former case.
For p=1, equation (PL ) can be heuristically written as a stochastic evolution
1
inclusion, B ∈L (U,H−1(Λ)),
2
dX (t)∈div[Sgn(∇X (t))]dt+BdW (t) in (0,T)×Λ,
1 1
(PL1) X1(t)=0 on (0,T)×∂Λ,
 X (0)=x∈L2(Λ) in Λ,
1
where Sgn:d →2Rd is defined by
R
u
, if u∈ d\{0},
Sgn(u):= |u| R

v ∈ d ||v|61 , if u=0.

R
A precise characterizationof(cid:8)the 1-Laplace op(cid:9)erator can be found in [2, 3, 37]. A
typical2-dimensionalexample for the so-calledtotalvariationflow canbe found in
image restoration,see [1, 3, 6] and the references therein.
We shall, however, take use of the stochastic evolution variational inequality-
formulation as in [11].
We are particularly interested in continuity of the solutions in the parameters p
andr,especiallyforthecasep→1. StochasticTrotter-typeresultsinthisdirection
havebeen obtainedby the firstnamedauthorin [15,16,17]. However,for the case
p → 1, we shall need the theory of Mosco convergence of convex functionals as in
[4],sincenostrongcharacterizationofthelimitisavailable(whichcouldbetreated
by Yosida-approximation methods). For B = 0 (i.e., the deterministic equation),
the convergenceofsolutionsto the evolutionproblem(PL ) wasprovedin[23, 40].
p
See also [39, Ch. 8.3].
With the help of a uniqueness result for invariant measures of the equations
considered, obtained by Liu and the second named author [29], we prove tightness
andtheweakconvergence(weakcontinuity)ofinvariantmeasuresassociatedtothe
ergodic semigroups of the equations (PL ) and (FD ). See [9, 10, 18, 22] for other
p r
result in this direction.
Organization of the paper. In Section 2, we prove that the solutions to the
basic examples are continuous in the parameters p and r resp.
InSection3,TheresultofSection2iscombinedwiththeuniquenessofinvariant
measuresprovedin[29]inordertoobtaintheweakcontinuityofinvariantmeasures
in the parameters p and r resp.
InSection4,weproveaconvergenceresultforthestochasticp-Laplaceequation
as p → 1, using another notion of a solution. For the limit p = 1, however,
CONVERGENCE OF INVARIANT MEASURES 3
uniqueness of the invariant measure is an open question. The matter is further
investigated in [22].
The Appendix collects some well-known results on Mosco (variational) conver-
gence and Mosco convergence in Lp-spaces, needed for the proof in Section 4.
2. Convergence of solutions
Compare with [16, Theorem 2].
Theorem 2.1. Let {p } ⊂ 1∨ 2d ,2 , n ∈ , p ∈ 1∨ 2d ,2 such that
n 2+d N 0 2+d
pn → p0. Let Xn := Xpn, n ∈(cid:16)N, X0 :=iXp0 be the solutio(cid:16)ns to (PLpin), n ∈ N,
(PL ) resp. Then for x∈L2(Λ).
p0
lim sup kX (t)−X (t)k2 =0.
n E"t∈[0,T] n 0 L2(Λ)#
Proof. For p ∈ (1,∞), define a : d → d by a (x) := |x|p−2x. Furthermore,
p p
R R
let A : W1,p(Λ) → (W1,p)∗(Λ) be defined by A (y) := −div[a (∇y)], where
p 0 0 p p
y ∈W1,p(Λ). To be more specific,
0
hA (y),zi = ha (∇y),∇zi dξ, ∀z ∈W1,p(Λ).
(W1,p)∗ p W1,p p 0
ZΛ
We first consider the following approximating equations for (PL )
p
dXε(t)+Aε Xε(t) dt=BdW (t)
p p p
(2.1)
( (cid:0) Xpε(cid:1)(0)=x
where for any u∈L2(Λ),
Aε(u)=−(1−ε∆)−1div aε ∇(1−ε∆)−1u
p p
h (cid:16) (cid:17)i
and aε is the Yosida approximationof a i.e., for any r∈ d,
p p R
1
aε(r)= 1−(1+εa )−1(r) .
p ε p
(cid:16) (cid:17)
In particular, for u,v ∈L2(Λ),
Aε(u),v = aε(∇R u),∇R (v) dξ,
p L2(Λ) p ε ε
ZΛ
where R :=(1−ε(cid:0)∆)−1 is t(cid:1)he resolvent(cid:10)of the Dirichlet La(cid:11)placian.
ε
We shall use the following strategy ( -a.s.)
P
kX (t)−X (t)k2
n 0 L2(Λ)
63kX (t)−Xε(t)k2 +3kXε(t)−Xε(t)k2
n n L2(Λ) n 0 L2(Λ)
+3kXε(t)−X (t)k2
0 0 L2(Λ)
=:I (n,ε)+I (n,ε)+I (ε).
1 2 3
uniformly in t∈[0,T].
At this point we need to prove the following lemma. We introduce the notation
rp(r):=(1+εa )−1(r).
ε p
Lemma 2.2. Under our assumptions, if we let Xε be the solution to (2.1) and
p
Xε :=(1−ε∆)−1Xε, we have that
p p
t p
(e2.2) rp ∇Xε(s) dξds6C kxk2 +kBk2 ,
E ε p t L2(Λ) HS
Z0 ZΛ(cid:12) (cid:16) (cid:17)(cid:12) (cid:16) (cid:17)
(cid:12) e (cid:12)
(cid:12) (cid:12)
4 I.CIOTIRANDJ.M.TO¨LLE
for all t∈[0,T].
Proof. We know by the definition of a that
p
ha (r),ri>|r|p.
p
On the other hand we have by Ito¯’s formula, applied to the function u 7→
kuk2 , that
L2(Λ)
t
(2.3) E Xε(t) 2 +2E aε ∇Xε(s) ,∇Xε(s) dξds
p L2(Λ) p p p
(cid:13) (cid:13) Z0 ZΛD (cid:16) (cid:17) E
(cid:13) (cid:13) 6C kxk2 +kBek2 . e
t L2(Λ) HS
By the definition of the Yosida(cid:16)approximation we(cid:17)have that
aε(r)=a (rp(r))
p p ε
and
1
aε(r),r = aε(rp(r)),rp(r) + |r−rp(r)|2.
p p ε ε ε ε
We rewrite as(cid:10)follows (cid:11) (cid:10) (cid:11)
t
E aε ∇Xε(s) ,∇Xε(s) dξds
p p p
Z0 ZΛD (cid:16) (cid:17) E
t
> E a rpe∇Xε(s) e,rp ∇Xε(s) dξds
p ε p ε p
Z0 ZΛD (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17)E
> E t rp ∇Xε(se) p dξds. e
ε p
Z0 ZΛ(cid:12) (cid:16) (cid:17)(cid:12)
Plugging into (2.3) proves ((cid:12)(cid:12)2.2). e (cid:12)(cid:12) (cid:3)
We shall prove now that -a.s.
P
lim sup X (t)−Xε(t) 2 =0, uniformly in p∈ 2d ,2 .
ε→0t∈[0,T] p p L2(Λ) d+2
(cid:13) (cid:13) (cid:16) (cid:17)
We set Xε =(1(cid:13)−ε∆)−1Xε an(cid:13)d Xλ =(1−λ∆)−1Xλ. Then by (2.1), we have
p p p p
that
e e
1 Xε(t)−Xλ(t) 2
2 p p L2(Λ)
t(cid:13) (cid:13)
+ (cid:13) aε ∇Xε(s(cid:13)) −aλ ∇Xλ(s) ,∇Xε(s)−∇Xλ(s) dξds=0 -a.s..
p p p p p p P
Z0 ZΛD (cid:16) (cid:17) (cid:16) (cid:17) E
Setting ∇Xε(es)=uε and ∇Xeλ(s)=uλeand using e
p p
aε(u)∈a (1+εa )−1(u) ,
e p e p p
(cid:16) (cid:17)
we get by the monotonicity of a that
p
aε(uε)−aλ uλ ,uε−uλ
p p
> (cid:10)aεp(uε)−aλp(cid:0)uλ(cid:1),εaεp(uε)(cid:11)−λaλp uλ .
This leads to (cid:10) (cid:0) (cid:1) (cid:0) (cid:1)(cid:11)
(2.4) 1 Xε(t)−Xλ(t) 2
2 p p L2(Λ)
(cid:13) t (cid:13) 2 2
(cid:13) 6 ε(cid:13) aε ∇Xε(s) +λ aλ ∇Xλ(s) dξds -a.s..
p p p p P
Z0 ZΛ(cid:18) (cid:12) (cid:16) (cid:17)(cid:12) (cid:12) (cid:16) (cid:17)(cid:12) (cid:19)
We can now prove that P(cid:12)(cid:12) -a.s. e (cid:12)(cid:12) (cid:12)(cid:12) e (cid:12)(cid:12)
CONVERGENCE OF INVARIANT MEASURES 5
t 2
(2.5) aε ∇Xε(s) dξds6C
p p t
Z0 ZΛ(cid:12) (cid:16) (cid:17)(cid:12)
for some Ct independent of p a(cid:12)(cid:12)nd ε. e (cid:12)(cid:12)
UsingJensen’sinequality(fort7→tp/(2p−2))andtakingintoaccountthat|a (r)|6
p
|r|p−1, we obtain
t 2
aε ∇Xε(s) dξds
p p
Z0 ZΛ(cid:12) (cid:16) (cid:17)(cid:12)
6(t|Λ|)1−(cid:12)(cid:12)((2p−2)e/p) (cid:12)(cid:12)t a rp ∇Xε(s) p/(p−1) dξds (2p−2)/p
p ε p
(2.6) (cid:18)Z0 ZΛ(cid:12) (cid:16) (cid:16) (cid:17)(cid:17)(cid:12) (cid:19)
6(1+t|Λ|) t rp ∇X(cid:12)(cid:12)ε(s) p dξeds (2p−(cid:12)(cid:12)2)/p
ε p
(cid:18)Z0 ZΛ(cid:12) (cid:16) (cid:17)(cid:12) (cid:19)
6C +C t rp(cid:12)(cid:12) ∇Xεe(s) p(cid:12)(cid:12)dξds ,
t t ε p
(cid:18)Z0 ZΛ(cid:12) (cid:16) (cid:17)(cid:12) (cid:19)
where |Λ|= Λ dξ. (cid:12)(cid:12) e (cid:12)(cid:12)
Now by Lemma 2.2 we have (2.5) for a constantC independent of p and ε, and
t
R
passing to the limit for ε,λ→0 in (2.4) we get that -a.s.
P
lim sup X (t)−Xε(t) 2 =0, uniformly in p∈ 1∨ 2d ,2 .
ε→0t∈[0,T] p p L2(Λ) (cid:18) d+2 (cid:19)
(cid:13) (cid:13)
As a consequ(cid:13)ence, I (n,ε) a(cid:13)nd I (ε) tend to zero as ε↓0, uniformly in n.
1 3
For I (n,ε), using the monotonicity of aε we have
2 pn
1 Xε (t)−Xε (t) 2
2 pn p0 L2(Λ)
(cid:13)t (cid:13)
+ (cid:13) aε ∇Xε(cid:13)(s) −aǫ ∇Xε (s) ,∇Xε (s)−∇Xε (s) dξds60.
Z0 ZΛD pn(cid:16) p0 (cid:17) p0(cid:16) p0 (cid:17) pn p0 Ed
Since e e e e
1 Xε (t)−Xε (t) 2
2 pn p0 L2(Λ)
(cid:13)t (cid:13)
6 (cid:13) (1−ε∆)−1d(cid:13)iv aε ∇Xε (s) −aǫ ∇Xε (s) Xε (s)−Xε (s) dξds
pn p0 p0 p0 pn p0
Z0 ZtΛh (cid:16) (cid:16) e (cid:17) (cid:16) e (cid:17)(cid:17)i(cid:2) 2 (cid:3) 1/2
6 (1−ε∆)−1divaε ∇Xε (s) −(1−ε∆)−1divaǫ ∇Xε (s) dξds
pn p0 p0 p0
(cid:18)Z0 ZΛ(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17) (cid:19)
t e 1/2 e
× Xε (s)−Xε (s) 2dξds
pn p0
(cid:18)Z0 ZΛ (cid:19)
(cid:0) (cid:1)
We only need to prove that
t 2 1/2
(1−ε∆)−1divaε ∇Xε (s) −(1−ε∆)−1divaǫ ∇Xε (s) dξds →0
pn p0 p0 p0
(cid:18)Z0 ZΛ(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17) (cid:19)
and that follows from e e
(2.7) Aε (u)→Aε (u), strongly in L2((0,T)×Λ),
pn p0
where Aε (u)=(1−ε∆)−1divaε (u) (as in (2.1)).
pn pn
Indeed, we obtain (2.7) by the following arguments:
6 I.CIOTIRANDJ.M.TO¨LLE
Sinceaε (u)→aǫ (u)pointwise,whichfollowsfromLemmaA.5and[4,Propo-
pn p0
sition3.29],andsince aε (u) isboundeda.e.on(0,T)×ΛwegetbyLebesgue’s
pn n
dominated convergence theorem
(cid:8) (cid:9)
divaε (u)−divaǫ (u),v = aε (u)−aǫ (u),∇v →n 0,
pn p0 L2((0,T)×Λ) pn p0 L2((0,T)×Λ)
(cid:10) (cid:11) (cid:10) (cid:11)
for all v ∈L2((0,T)×Λ).
That means
divaε (u)→divaǫ (u), weakly in L2((0,T)×Λ)
pn p0
and this leads to
(1−ε∆)−1divaε (u)→(1−ε∆)−1divaε (u), strongly in L2((0,T)×Λ),
pn p0
which is (2.7).
We have proved that
lim sup kX (t)−X (t)k =0 -a.s.
n t∈[0,T] n 0 L2(Λ) P
The convergence
lim sup kX (t)−X (t)k2 =0
n E"t∈[0,T] n 0 L2(Λ)#
is established by Lebesgue’s dominated convergence theorem and [28, Eq. (1.3)],
where the constant can be controlled uniformly in p by Ito¯’s formula, Poincar´e
inequalityandGr¨onwall’slemma. Wereferto[38]forthep-dependenceofPoincar´e
constants. (cid:3)
Theorem 2.3. Let {r } ⊂ 0∨ d−2,1 , n ∈ , r ∈ 0∨ d−2,1 such that
n d+2 N 0 d+2
r(FnD→)r0re.spL.etTYhenn:=forYyrn∈, nH−∈(cid:16)1(NΛ,),Y0 :=iYr0 be the solutio(cid:16)ns to (FDrin), n ∈ N,
r0
lim sup kY (t)−Y (t)k2 =0.
n E"t∈[0,T] n 0 H−1(Λ)#
Proof. We need to show that
limE sup kY (t)−Y (t)k2 =0.
n "t∈[0,T] n 0 H−1(Λ)#
Using the same approximation as in [10] consider
kY (t)−Y (t)k
n 0 H−1(Λ)
6 kY (t)−Yε(t)k +kYε(t)−Yε(t)k +kYε(t)−Y (t)k
n n H−1(Λ) n 0 H−1(Λ) 0 0 H−1(Λ)
= I +I +I .
1 2 3
ForI andI wehavetheconvergenceuniformlyinr forr >1/2,arguingasin
1 3 n n
[10],Proposition2.6andusingattheendJensen’sinequalityforL2(Λ)⊂L2rn(Λ).
For I note that the pointwise convergence of Ψ (x) = |x|rn−1x to Ψ (x) =
2 rn r0
|x|r0−1x imply the convergence of the resolvent in R and then we get the result
arguing as in [15]. (cid:3)
CONVERGENCE OF INVARIANT MEASURES 7
3. Convergence of invariant measures
In this section, we shall present a result on convergence of invariant measures
associated to equations (PL ), (FD ) respectively.
p r
Let{Xx(t)} bethevariationalsolutionassociatedtoequation(PL )starting
p t>0 p
at x ∈ L2(Λ). Similarly, let {Yy(t)} be the variational solution associated to
r t>0
equation (FD ) starting at y ∈H−1(Λ).
r
Let
PpF(x):= F(Xx(t)) , F ∈C (L2(Λ)), t>0,
t E p b
be the semigroup associated to equation (PL ).
(cid:2) (cid:3) p
Let
QrG(y):= [G(Yy(t))], G∈C (H−1(Λ)), t>0,
t E r b
be the semigroup associated to equation (FD ).
r
Recently, Liu and the second named author obtained the following result:
Proposition 3.1. Suppose that p ∈ 1∨ 2d ,2 , r ∈ 0∨ d−2,1 . Then {Pp}
2+d d+2 t
and {Qrt} are ergodic and admit uniqu(cid:16)e invariantimeasu(cid:16)res µp, νr riespectively. It
holds that µ is supported by W1,p(Λ) and ν is supported by Lr+1(Λ). Also
p 0 r
(3.1) kxkp µ (dx)<+∞,
1,p p
ZL2(Λ)
and
(3.2) kykr+1ν (dy)<+∞.
r+1 r
ZH−1(Λ)
Proof. See [29, Propositions 3.2 and 3.4]. (cid:3)
Theorem 3.2. (i) Let {p } ⊂ 1∨ 2d ,2 , n ∈ , p ∈ 1∨ 2d ,2 such that
n 2+d N 0 2+d
p →p . Set Pn :=Ppn, P0(cid:16):=Pp0. i (cid:16) i
n 0 t t t t
Then the unique invariant measures µ , n ∈ , µ resp. associated to
n 0
N
{Pn}, n∈ , {P0} converge in the weak sense, i.e.
t N t
lim F(x)µ (dx)= F(x)µ (dx) ∀F ∈C (L2(Λ)).
n 0 b
n ZL2(Λ) ZL2(Λ)
(ii) Let {r } ⊂ 0∨ d−2,1 , n ∈ , r ∈ 0∨ d−2,1 such that r → r . Set
n d+2 N 0 d+2 n 0
Qn :=Qrn, Q(cid:16)0 :=Qr0.i (cid:16) i
t t t t
Then the unique invariant measures ν , n ∈ , ν resp. associated to
n 0
N
{Qn}, n∈ , {Q0} converge in the weak sense, i.e.
t N t
lim F(x)ν (dx)= F(x)ν (dx) ∀F ∈C (L2(Λ)).
n 0 b
n ZL2(Λ) ZL2(Λ)
Proof. Let us prove (i) first. By Proposition 3.1, we see that {Pn}, n ∈ , {P0}
t N t
admit unique invariant measures µ , n ∈ , µ resp. Let p := inf p . By the
n 0 1 n n
N
convergence p → p , p ∈ 1∨ 2d ,2 and the embedding W1,p1(Λ) ⊂ L2(Λ) is
n 0 1 2+d 0
compact. (cid:16) i
Let θ >0. Set
K := x∈L2(Λ) kxkp1 6θ−1+|Λ| .
θ 1,p1
(cid:26) (cid:12) (cid:27)
Clearly, Kθ is compact in L2(Λ). Now(cid:12)(cid:12)by (3.1),
(cid:12)
µ {Kc} = µ k·kp1 −|Λ|>θ−1 6 θ kxkpn µ (dx) 6 θkBk2 .
n θ nn 1,p1 o ZL2(Λ) 1,pn n HS
8 I.CIOTIRANDJ.M.TO¨LLE
Hence the family of measures {µ } is tight and has a weak accumula-
n n∈
tion point µ, i.e. µ → µ weakly. ByN the Krylov–Bogoliubov theorem, for
nk
F ∈C (L2(Λ)),
b
e e 1 T
F(x)µ (dx)= lim PnkF(x)dt
ZL2(Λ) nk T→+∞T Z0 t
1 T
= lim (PnkF(x)−P0F(x))dt
T→+∞T Z0 t t
1 T
+ lim P0F(x)dt
T→+∞T Z0 t
=:ε + F(x)µ (dx)
k 0
ZL2(Λ)
By Theorem 2.1 and dominated convergence, ε →0 as k →+∞ and hence
k
F(x)µ(dx)= F(x)µ (dx).
0
ZL2(Λ) ZL2(Λ)
As a consequence, for the whole sequence, µ →µ weakly.
e n 0
The proof for (ii) can be carried out by similar arguments. (cid:3)
4. The case p=1
For p = 1, the situation is more complicated. We would like to find a convex
functional Φ1 such that the stochastic 1-Laplace equation
∇X (t)
1
dX (t)=div dt+BdW (t) in (0,T)×Λ,
1
|∇X (t)|
(PL ) (cid:20) 1 (cid:21)
1  X1(t)=0 on (0,T)×∂Λ,
X (0)=x in Λ,
1
can be written as
dX (t)∈−∂Φ1(X (t))dt+BdW (t) in (0,T),
1 1
(4.1)
( X1(0)=x,
where ∂Φ1 is the subdifferential of Φ1.
We shall need the spaces BV(Λ) and BV( d). For f ∈L1 (Λ), define the total
R loc
variation
kDfk(Λ)=sup fdivψdξ ψ ∈C∞ Λ;Rd , |ψ|61
0
(cid:26)ZΛ (cid:12) (cid:27)
BV(Λ) is defined to be equal to {f ∈ L1(Λ(cid:12))| kDfk((cid:0)Λ) <(cid:1)∞}. Denote the d−1-
(cid:12)
dimensional Hausdorff measure on ∂Λ by Hd−1. For f ∈ BV(Λ) there is an
element fΛ ∈L1(∂Λ,dHd−1) called the trace such that
fdivψdξ =− hψ,d[Df]i+ hψ,νifΛdHd−1 ∀ψ ∈C1(Λ; d),
R
ZΛ ZΛ Z∂Λ
where [Df] denotes the distributional gradient of f on Λ (which is a d-valued
R
Radon measure here) and ν denotes the outer unit normal on ∂Λ. BV( d) is
R
defined similarly by setting Λ = d. Define also kDfk( d) in the above manner.
R R
Note that for f ∈ BV(Λ) (extended by zero outside Λ) it holds that f ∈ BV( d)
R
and that
(4.2) kDfk( d)=kDfk(Λ)+ fΛ dHd−1,
R
Z∂Λ
(cid:12) (cid:12)
cf. [1, Theorem 3.87]. (cid:12) (cid:12)
CONVERGENCE OF INVARIANT MEASURES 9
Remark 4.1. By Ambrosio et al. [1, Corollary 3.49], if d∈{1,2}, then
W1,1(Λ)⊂BV(Λ)⊂L2(Λ)
0
continuously. If d=1, then
BV(Λ)⊂⊂L2(Λ)
compactly.
For further results in spaces of functions of bounded variation, we refer to [1,
Ch. 3].
We shall return to equation (4.1). Recall that the subdifferential ∂Φ1 in L2(Λ)
is defined by η ∈∂Φ1(x) iff
(4.3) Φ1(x)−Φ1(y)6 η(x−y) dξ, ∀y ∈domΦ1.
ZΛ
One possible choice for Φ1 is the (homogeneous) energy
|∇u|dξ, if u∈W1,1(Λ),
Φ(u):= Λ 0
(+R ∞, if u∈L2(Λ)\W01,1(Λ).
In this case, if ue∈ W1,1(Λ), and if U := −div(sgn(∇u)) ⊂ L2(Λ), then we have
0
that u∈dom∂Φ and U =∂Φ(u).
However, Φ fails to be lower semi-continuous in L2(Λ) which is a necessary in-
gredientfortheetheory. Thereefore,itisconvenienttoconsideritsrelaxedfunctional
in L2(Λ), wheich is equal to
kDuk( d), if u∈BV(Λ),
Φ1(u):= R
(+∞, if u∈L2(Λ)\BV(Λ),
see equation(4.2) above. Φ1 is proper,convex and lower semi-continuousin L2(Λ)
and an extension of Φ in the sense that domΦ1 ⊃ domΦ and Φ1 6 Φ. Compare
with [3, 24, 37, 40].
Following the appreoach of Barbu, Da Prato and Ro¨ckneer [11], we sehall give the
definition of a solution for equations (PL ), p∈[1,2].
p
Definition 4.2. Set V :=W1,p(Λ), p∈(1,2], V :=BV(Λ). Let Φ1 be defined as
p 0 1
above. For p∈(1,2], let
1
|∇x|pdξ, if u∈W1,p(Λ),
Φp(x):= p 0
+∞ZΛ, if u∈L2(Λ)\W1,p(Λ).
 0
A stochastic process X = Xx with -a.s. continuous sample paths in H :=

P
L2(Λ) is said to be a solution to equation (PL ), p∈[1,2] if
p
X ∈C ([0,T];H)∩Lp((0,T)×Ω,V ), X(0)=x∈H
W p
and
1 t
kX(t)−Y (t)k2 + (Φp(X(s))−Φp(Y (s))) ds
2 L2(Λ)
Z0
1 t
6 kx−Y (0)k2 + (G(s),X(s)−Y (s)) ds, t∈[0,T],
2 L2(Λ) L2(Λ)
Z0
for all G∈L2 (0,T;H)and Y ∈C ([0,T];H)∩Lp((0,T)×Ω;V ) satisfying the
W W p
equation
(4.4) dY (t)+G(t) dt=BdW (t), t∈[0,T].
10 I.CIOTIRANDJ.M.TO¨LLE
Suppose for a while that 1< p< 2, d= 1,2. Arguing as in [31, Example 4.1.9,
Theorem 4.2.4], we can easily prove existence and uniqueness of the solution X
p
for equation (PL ), in the usual (strong) variational sense, as in Pardoux, Krylov,
p
Rozovski˘ı[26,30]. WeshallrefertoPr´evoˆt,Ro¨ckner[31,Definition4.2.1]. ByIto¯’s
formula, we see that X is also a solution in the sense of the definition above.
p
Here, W(t) is a cylindrical Wiener process on L2(Λ) of the form
∞
W(t)= γ (t)e , t>0,
n n
n=1
X
where {γ } is a sequence of mutually independent real Brownian motions on a
n
filtered probability space (Ω,F,{F } , ) and {e } is an orthonormal basis of
t t>0 n
L2(Λ). We shall make further specificatPions. BB∗ is assumed to be a linear,
continuous, non-negative, symmetric operator on L2(Λ) with eigenbasis {e } and
n
corresponding sequence of eigenvalues {λ }. Let (−∆,dom(−∆)) be the Dirichlet
n
Laplacian in L2(Λ), in particular, dom(−∆) = H2(Λ)∩H1(Λ). Assume for sim-
0
plicitythat{e }isaneigenbasisof−∆withcorrespondingsequenceofeigenvalues
n
{µ }. We shall assume that
n
∞
(4.5) λ1+κµ <∞
n n
n=1
X
for some κ > 0. For the situation considered in this paper, it is enough to set
Q:=(−∆)−1−δ with δ > 1 +κ for d=1 and δ >1+κ for d=2.
2
Regarding equation (PL ), well-posedness of the problem as well as existence
1
and uniqueness of the solution were proved by Barbu, Da Prato and Ro¨ckner in
[11].
Remark 4.3. Note that in [11], the space BV (Λ) is introduced, consisting of
0
BV(Λ)-functions with zero trace. They claim, however, that the energy
kDuk(Λ), if u∈BV (Λ),
0
Ψ(u):=
(+∞, if u∈L2(Λ)\BV0(Λ).
is lower semi-continuous which is not the case. Consider, for example, a sequence
u of trace zero Lipschitz functions on Λ with kDu k(Λ)=1 converging in L2(Λ)
n n
to . Then
Λ
1
limΨ(u )=1<+∞=Ψ( ).
n Λ
n 1
Fortunately, all results of [11] remain true, if one replaces Ψ (denoted by Φ in their
paper) by Φ1. We do not repeat the steps taken in the proof of [11] here, but note
that for their existence and uniqueness result relies on an approximation {Ψε} of Ψ
which “does not see” the trace-term in (4.2), i.e. maps L2(Λ) functions on a joint
subspace of BV (Λ) and dom(Φ1). In fact, {Ψε} is defined similarly to (4.6).
0
Other results of stochastic evolution variational inequalities can be found in
[8, 13, 32, 33, 34].
We are now able to formulate the main result of this section.
Theorem 4.4. Let d∈{1,2}. The sequenceof solutions {X } toequations (PL )
p p p
is convergent for p → 1 to the solution X of equation (PL ), strongly in L2(Λ),
1 1
uniformly on [0,T], −a.s., i.e.,
P
lim sup kX (t)−X (t)k =0, P−a.s.
p→1t∈[0,T] p 1 L2(Λ)
There is some evidence that the following conjecture is true, see [21, 22, 25].