a Xi :1411.5237 1 [ma h.AP] 19 No 2014
LOCAL PATHWISE SOLUTIONS TO STOCHASTIC EVOLUTION
EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS
WITH HURST PARAMETERS H∈(1/3,1/2]
MAR´
IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨
ORN SCHMALFUSS
Abs ac . In his a icle we a e conce ned wi h he s udy o he exis ence
and uniqueness o pa hwise mild solu ions o e olu ions equa ions d i en by a
H¨olde con inuous unc ion wi h H¨olde exponen in (1/3,1/2). Ou s ochas ic
in eg al is a gene aliza ion o he well-known Young in eg al. To be mo e
p ecise, he in eg al is de ined by using a ac ional in eg a ion by pa s o mula
and i in ol es a enso o which we need o o mula e a new equa ion. F om
his i u ns ou ha we ha e o sol e a sys em consis ing in a pa h and an
a ea equa ions. In his pape we p o e he exis ence o a unique local solu ion
o he sys em o equa ions. The esul s can be applied o s ochas ic e olu ion
equa ions wi h a non-linea di usion coe icien d i en by a ac ional B ownian
mo ion wi h Hu s pa ame e in (1/3,1/2], which is pa icula includes whi e
noise.
Sep embe 2, 2016
1. In oduc ion
In his a icle, we shall ocus on he s udy o a local solu ion o he ollowing kind
o s ochas ic e olu ion equa ions
du( ) = Au( )d +G(u( ))dω( ),
u(0) = u0,
(1)
in a Hilbe –space V, whe e he noise inpu ωis a H¨olde con inuous unc ion
wi h H¨olde exponen in he in e al (1/3,1/2), Ais he in ini esimal gene a o
o an analy ic semig oup S(·) on Vand Gis a nonlinea e m sa is ying ce ain
assump ions which will be desc ibed in he nex sec ions. As a pa icula case
o d i ing noises we can conside a ac ional B ownian mo ion BHwi h Hu s
pa ame e H∈(1/3,1/2]. To be mo e p ecise, we will s udy (1) in he sense o
mild solu ions gi en by
(2) u( ) = S( )u0+Z
0
S( − )G(u( ))dω.
Ou in e p e a ion o pa hwise is ha we ob ain a solu ion o hese s ochas ic equa-
ions which does no p oduce excep ional se s depending on he ini ial condi ions.
In he classical heo y o s ochas ic e olu ion equa ions, i.e., s ochas ic e olu ion
2000 Ma hema ics Subjec Classi ica ion. P ima y: 60H15; Seconda y: 60H05, 60G22, 26A33,
26A42.
Key wo ds and ph ases. S ochas ic PDEs, Hilbe - alued ac ional B ownian mo ion, pa hwise
solu ions.
This wo k was pa ially suppo ed by MTM2011-22411, FEDER ounding (M.J. Ga ido-A ienza
and B. Schmal uß), and by NSF0909400 (K. Lu).
1
2 MAR´
IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨
ORN SCHMALFUSS
equa ions (SEEs) d i en by B ownian mo ion B1/2, s ochas ic I o in eg als a e
cons uc ed o be a limi in p obabili y o pa icula andom a iables de ined only
almos su ely, whe e he excep ional se s may depend on he ini ial condi ions,
which is in con adic ion wi h he cocycle p ope y needed o de ine a andom dy-
namical sys em. Pa hwise esul s o ha classical heo y a e only a ailable o he
whi e noise case (G= id) and a ew special cases when u7→ G(u) is linea .
Du ing he las wo decades di e en in eg a ion heo ies ha e been de eloped o
ea mo e gene al noise inpu s, and in pa icula , o ackling he ac ional B ow-
nian mo ion BH. One o hese a emp s is gi en by he Rough Pa h Theo y, and
we e e o Lyons and Qian [18] and F iz and Vic oi [8] o a comp ehensi e p e-
sen a ion o his heo y. Some in e es ing pape s dealing wi h he s udy o SEEs
by using he ough pa h heo y a e [2], [3], [7], [13], [15], [6] and [14] among o h-
e s. In pa icula , in his las pape he au ho s p o ed he exis ence o local mild
solu ions o s ochas ic SEEs d i en by ough pa hs o β-H¨olde –con inuous pa hs
(β∈(1/3,1/2]) wi h a special quad a ic nonlinea i y.
A di e en echnique called F ac ional Calculus was de eloped by Z¨ahle [24], who
conside ed o a ac ional B ownian mo ion wi h H > 1/2 he well-known Young
in eg al. In con as o he I o-o S a ono ich in eg al, ha in eg al can be de-
ined in a pa hwise sense, gi en by ac ional de i a i es, which allows a pa hwise
es ima e o he in eg als in e ms o he in eg and and he in eg a o using spe-
cial no ms. In [20] i is shown he exis ence and uniqueness o he solu ion o a
ini e-dimensional s ochas ic di e en ial equa ion d i en by a ac ional B ownian
mo ion o H > 1/2. These esul s we e ex ended in [19] o show he exis ence o
mild solu ions o SEEs d i en by ac ional B ownian mo ion o H > 1/2.
Recen ly Hu and Nuala [16] ha e p o ed an exis ence and uniqueness esul o
ini e-dimensional s ochas ic di e en ial equa ions ha ing coe icien s which a e
su icien ly smoo h and d i en by a ac ional B ownian mo ion BHwi h H∈
(1/3,1/2], o which hey needed o o mula e a second equa ion o he so-called
a ea in he space o enso s. In ou a icle we adap he echniques in [16] o ob ain a
mild solu ion o (1). Howe e , he e a e signi ican di e ences be ween ou se ing
and he one in [16], as o ins ance ha in o de o de ine he a ea equa ion in he
in ini e-dimensional se ing we ha e o cons uc an a ea objec ω⊗Sω, depending
on he noise pa h ωas well as on he semig oup S, sa is ying use ul p ope ies as
he Chen–equali y.
Unde gene al hypo hesis on he nonlinea i y Gwe de i e he exis ence and unique-
ness o a local pa hwise mild solu ion u o (1). Howe e , global exis ence is missing
in his gene al con ex . Unde some mo e es ic i e condi ions on G, in a o h-
coming pape we will ob ain he exis ence and uniqueness o a global mild pa hwise
solu ion, which in pa icula will gua an ee ha s ochas ic e olu ion equa ions like
(1) and d i en by an Bm BHwi h H∈(1/3,1/2] gene a e andom dynamical
sys ems, a challenging and a he open p oblem o he bes o ou knowledge.
The a icle is o ganized as ollows. In Sec ion 2 we gi e he analy ical backg ound
o p esen ou heo y. In Sec ion 3 we p esen he so called ac ional in eg a ion by
pa s me hod. Using his echnique we can in oduce pa hwise s ochas ic in eg als
allowing us o o mula e pa hwise s ochas ic di e en ial equa ions. In Sec ion 4 we
in oduce mild pa h–a ea solu ions. In addi ion, we o mula e and sol e a ixed-
poin equa ion ha ing wo componen s, a pa h- and an a ea-componen . The ole
o he semig oup Sin he a ea equa ion will be gi en in e ms o a pa icula enso
LOCAL PATHWISE SOLUTIONS TO SEES 3
objec ω⊗Sω. We also p esen an example o show a nonlinea i y G ha ma ches
he abs ac heo y. The appendix sec ion con ains he p oo s o some echnical
esul s.
Finally, we wan o s ess ha wo di e en cons uc ions o he key enso objec
ω⊗Sωby using an app oxima ion o he noise pa h by smoo h pa hs can be ound in
[12]. One cons uc ion conside s as d i ing noise an in ini e-dimensional ac ional
B ownian mo ion BHwi h H∈(1/3,1/2], while, in a less es ic i e se ing, he
second one conside s a Hilbe - alued ace-class B ownian mo ion B1/2.
Fu he mo e, we e e o [10] o a sho and ecen announcemen o ou esul s.
2. P elimina ies
Le V= (V, (·,·),| ·|) be a sepa able Hilbe –space. On Vwe de ine A o be he
nega i e and symme ic gene a o o an analy ic semig oup S. We suppose ha −A
has a poin spec um 0 < λ1≤λ2≤ ··· ending o in ini e whe e he associa ed
eigenelemen s (ei)i∈N o m a comple e o hono mal sys em on V.D((−A)κ) = Vκ
deno es he domain o (−A)κ o κ∈R, and as usual, L(Vκ, Vζ) deno es he space
o con inuous linea ope a o s om Vκin o Vζ, o κ, ζ ∈R. We hen ha e he
ollowing es ima es o he semig oup S:
kS( )kL(Vκ,Vγ)=k(−A)γS( )kL(Vκ,V )≤c κ−γ, o γ≥κ,(3)
kS( )−idkL(Vσ,Vθ)≤c σ−θ, o σ−θ∈[0,1].(4)
F om hese p ope ies we can de i e easily he ollowing esul :
Lemma 1. Fo any ν, η, µ ∈[0,1],κ, γ, ρ ∈Rsuch ha κ≤γ+µ, he e exis s a
cons an c > 0such ha o 0< q < < s < we ha e ha
kS( − )−S( −q)kL(Vκ,Vγ)≤c( −q)µ( − )−µ−γ+κ,
kS( − )−S(s− )−S( −q) + S(s−q)kL(Vρ,Vρ)
≤c( −s)η( −q)ν(s− )−(ν+η).
Th oughou he whole pape we will w i e e y o en a cons an c. This cons an
can change om line o line. Howe e his cons an is always chosen independen
o ime pa ame e s con ained in a ixed in e al [0, T ].
Le V×Vand V⊗Vbe he ca esian p oduc and he enso p oduc o V, see
[17]. The no m o V⊗Vis deno ed by k·k. Fo x, y ∈Vwe deno e by x⊗Vy he
ank-one enso o V⊗V. Then (ei⊗Vej)i,j∈Nis a comple e o hono mal sys em
o V⊗Vwhe e (ei)i∈Ncan be any comple e o hono mal sys em o V, al hough
o he ollowing we choose he o hono mal sys em gi en a he beginning o his
sec ion. Le ( ˆ
V , | · |ˆ
V,(·,·)ˆ
V) be ano he sepa able Hilbe –space. By L2(V, ˆ
V)
(L2(V×V, ˆ
V)) we deno e he Hilbe -Schmid ope a o s om V(V×V) o ˆ
V. In
pa icula G∈L2(V×V, ˆ
V) i and only i
X
i,j |G(ei, ej)|2
ˆ
V<∞.
We no e ha G∈L2(V×V, ˆ
V) can be ex ended o a linea ope a o ˆ
Gde ined
on V⊗Vsuch ha ˆ
G∈L2(V⊗V, ˆ
V), see [17] Chap e 2.6. Mo e p ecisely,
we can cons uc a weak Hilbe -Schmid mapping p:V×V→V⊗Vwhe e
4 MAR´
IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨
ORN SCHMALFUSS
p(ei, ej) = ei⊗Vej o i, j ∈N. Then ˆ
Gon V⊗Vis de e mined by ac o iza ion
such ha G=ˆ
Gp. In addi ion, we ha e
kˆ
Gk2
L2(V⊗V, ˆ
V):= X
i,j |ˆ
G(ei⊗Vej)|2
ˆ
V=X
i,j |G(ei, ej)|2
ˆ
V=kGk2
L2(V×V, ˆ
V).
In he ollowing we will w i e o ˆ
Galso he symbol G.
Le us now desc ibe he coe icien o he e olu ion equa ion ha we ha e in mind.
Lemma 2. Le ˆ
Vbe a subspace o V. Assume ha he mapping G:V→L2(V, ˆ
V)
is h ee imes con inuously F ´eche –di e en iable wi h bounded i s , second and
hi d de i a i es DG(u),D2G(u)and D3G(u), o u∈V. Le us deno e, e-
spec i ely, by cDG, cD2Gand cD3G he bounds o DG,D2Gand D3G, and le
cG=kG(0)kL2(V, ˆ
V). Then, o u1, u2, 1, 2∈V, we ha e
• kG(u1)kL2(V, ˆ
V)≤cG+cDG|u1|,
• kG(u1)−G( 1)kL2(V, ˆ
V)≤cDG|u1− 1|,
• kDG(u1)−DG( 1)kL2(V×V, ˆ
V)≤cD2G|u1− 1|,
• kG(u1)−G(u2)−DG(u2)(u1−u2)kL2(V, ˆ
V)≤cD2G|u1−u2|2,
• kG(u1)−G( 1)−(G(u2)−G( 2))kL2(V, ˆ
V)≤cDG|u1− 1−(u2− 2)|
+cD2G|u1−u2|(|u1− 1|+|u2− 2|),
• kDG(u1)−DG( 1)−(DG(u2)−DG( 2))kL2(V×V, ˆ
V)
≤cD2G|u1− 1−(u2− 2)|+cD3G|u1−u2|(|u1− 1|+|u2− 2|).
• kG(u1)−G(u2)−DG(u2)(u1−u2)−(G( 1)−G( 2)−DG( 2)( 1− 2))kL2(V, ˆ
V)
≤cD2G(|u1−u2|+| 1− 2|)|u1− 1−(u2− 2)|
+cD3G| 1− 2||u2− 2|(|u1−u2|+|u1− 1−(u2− 2)|).
These es ima es ollow by he mean alue heo em; o a p oo o he las one see
[20].
No ice ha , in pa icula , DG :V→L2(V, L2(V, ˆ
V)) (o equi alen ly, DG :V→
L2(V×V, ˆ
V)) is a bilinea map, ha can be ex ended o DG :V→L2(V⊗V, ˆ
V),
and D2G(u) is a ilinea map.
Nex we in oduce some unc ion spaces. Le T > 0. Fo β∈(0,1], we conside he
Banach–space o β–H¨olde –con inuous unc ions on [0, T ] wi h alues in V, deno ed
by Cβ([0, T ]; V), wi h he semino m
kukβ= sup
0≤ ≤T|u( )|+|||u|||β,|||u|||β= sup
0≤s< ≤T
|u( )−u(s)|
( −s)β.
I β= 1 we call hese unc ions Lipschi z–con inuous. Le Cβ,∼([0, T ]; V) be he
space o unc ions on [0, T ] wi h alues in Vand wi h no m
kukβ,∼= sup
0≤ ≤T|u( )|+ sup
0<s< ≤T
sβ|u( )−u(s)|
( −s)β.
LOCAL PATHWISE SOLUTIONS TO SEES 5
Lemma 3. Cβ,∼([0, T ], V )is a Banach–space.
The p oo o his esul can be ound in Chen e al. [4].
Le ∆0,T be he iangle {(s, ) : 0 < s ≤ ≤T}. Fo β+β′<1, β ≤β′we
in oduce he space Cβ+β′,∼(∆0,T , V ⊗V) o con inuous unc ions de ined on
∆0,T , which a e ze o o 0 < s = , such ha
k kβ+β′,∼= sup
0<s< ≤T
sβk (s, )k
( −s)β+β′<∞.
These unc ions may no be de ined o s= 0 and can ha e a singula i y o
(s, ), s = 0.
Lemma 4. The space Cβ+β′,∼(∆0,T ;V⊗V)is a Banach–space.
The p oo is simila o he p oo o Lemma 3 and he e o e we omi he e.
Le us de ine ¯
∆0,T ={(s, ) : 0 ≤s≤ ≤T}and conside he Banach–space
Cβ+β′(¯
∆0,T ;V⊗V) o con inuous unc ions de ined on ¯
∆0,T , which a e ze o o
s= , equipped wi h he no m
k kβ+β′= sup
0≤s< ≤T
k (s, )k
( −s)β+β′<∞.
We o en use he ollowing in eg al o mula: o e e y s < ,µ, ν > −1
(5) Z
s
( −s)µ( − )νd =c( −s)µ+ν+1
whe e conly depends on µ, ν. This p ope y ollows by he de ini ion o he Be a
unc ion simply by pe o ming a sui able change o a iable.
3. F ac ional Calculus
In his pape he main ins umen o ea (2) is ac ional calculus. In his
sec ion we p esen he main ea u es o his heo y. We a e going o assume
ha o some T > 0 we ha e ha ω∈Cβ′([0, T ]; V), u∈Cβ,∼([0, T ]; V) and
∈Cβ+β′,∼(∆0,T ;V⊗V) o 1/3< β < β′<1/2, and ha his iple o elemen s
sa is ies he Chen–equali y gi en by
(6) (s, ) + ( , ) + (u( )−u(s)) ⊗V(ω( )−ω( )) = (s, )
o 0 < s ≤ ≤ ≤T. We would like o emphasize ha when ωis smoo h an
example o is gi en by
(7) (u⊗ω)( , ) = Z
(u(q)−u( )) ⊗Vdω(q).
This enso a ea is clea ly well de ined and sa is ies, o 0 < < ,
k(u⊗ω)( , )k ≤ c
βkukβ,∼kωkC1( − )1+β≤c
β( − )β+β′,
(8)
and he e o e (u⊗ω)∈Cβ+β′,∼(∆0,T ;V⊗V). Mo eo e , he Chen–equali y easily
ollows in his case.

6 MAR´
IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨
ORN SCHMALFUSS
Le α∈(0,1). We de ine he igh hand side ac ional de i a i e o o de αo u
and he le hand side ac ional de i a i e o o de 1 −αo ω −(·) := ω(·)−ω( ),
gi en o 0 < s ≤ ≤ by he exp essions
Dα
s+u[ ] = 1
Γ(1 −α)u( )
( −s)α+αZ
s
u( )−u(q)
( −q)1+αdq
D1−α
−ω −[ ] =(−1)1−α
Γ(α)ω( )−ω( )
( − )1−α+ (1 −α)Z
ω( )−ω(q)
(q− )2−αdq,
whe e Γ(·) deno es he Gamma unc ion. Fo enso alued elemen s and o
0< < we de ine
D1−α
− [ ] =(−1)1−α
Γ(α) ( , )
( − )1−α+ (1 −α)Z
( , q)
(q− )2−αdq.
Lemma 5. Suppose ha β > α. Then he e exis s a cons an c > 0such ha o
0≤s < ≤ ≤T
|Dα
s+u(·)[ ]| ≤ ckukβ,∼
( −s)α,|D1−α
−ω −[ ]| ≤ c|||ω|||β′( − )α+β′−1,
and o 0< < q <
kD1−α
− [q]−D1−α
− [ ]k ≤ c
β(kukβ,∼|||ω|||β′+k kβ+β′,∼)(q− )α+β+β′−1.
(9)
The p oo o he wo i s inequali ies ollows s aigh o wa dly, and he p oo o
(9) is simila o he one o Lemma 6.3 in [16] wi h he di e ence ha in ha pape
he au ho s wo k in di e en unc ion spaces. We e e he eade o Lemma 21
and Co olla y 29 in he Appendix sec ion.
As an ex ension o he ac ional de i a i e o o de α, o he mapping G:V7→
L2(V, ˆ
V) and s < we in oduce he so–called compensa ed ac ional de i a i e o
o de αgi en by
ˆ
Dα
s+G(u(·))[ ] = 1
Γ(1 −α)G(u( ))
( −s)α
+αZ
s
G(u( )) −G(u(q)) −DG(u(q))(u( )−u(q))
( −q)1+αdq.
I is immedia e o p o e he ollowing esul :
Lemma 6. Suppose ha α < 2βand Gsa is ies he assump ions o Lemma 2.
Then he e exis s a posi i e cons an csuch ha o e e y 0≤s < ≤T
|ˆ
Dα
s+G(u(·))[ ]| ≤ c(1 + kuk2
β,∼)
( −s)α.
Le us assume o a while ha V, ˆ
V=R. We i s ecall he ollowing use ul
p ope y which is an in eg a ion by pa s o mula
(−1)αZ
s
Dα
s+u[ ]ω( )d =Z
s
u( )Dα
−ω[ ]d ,(10)
see Z¨ahle [24], o mula (21). Fo β > α and α+β′>1 he ac ional in eg al is
gi en by
Z
s
udω := (−1)αZ
s
Dα
s+u[ ]D1−α
−ω −[ ]d ,
LOCAL PATHWISE SOLUTIONS TO SEES 7
see again [24], which is a e sion o he Young in eg al. By Lemma 5 and he
p ope y (5), o he abo e in eg al i is easy o de i e ha
Z
s
udω≤ckukβ,∼|||ω|||β′( −s)β′.
Fo H¨olde –con inuous uand ω his kind o in eg al was de ined by Young [23].
Howe e , ou unc ion uis no H¨olde –con inuous in he s ong sense bu u∈
Cβ,∼([0, T ]; R), in which case ha in eg al is also well de ined in he abo e sense
since, acco ding o [24], wha we need is ha u∈Iα
s+(Lp((s, ); R)), u(s+) bounded
and ω −∈I1−α
−(Lq((s, ); R)), wi h αp < 1, p−1+q−1≤1 ( o he de ini ion o hese
spaces we e e o Samko e al. [22]). In pa icula , unde ou condi ions on α, β and
β′, we know ha ω −∈I1−α
−(Lq((s, ); R)) o any q > 1 and u∈Iα
s+(Lp((s, ); R))
when αp < 1, see Theo em 13.2 o [22].
Nex we in oduce in eg als o ac ional ype wi h alues in a sepa able Hilbe –
space. To do ha , we need a new sepa able Hilbe –space ( ˜
V , |·|˜
V,(·,·)˜
V).
Lemma 7. Assume β > α and α+β′>1. Le ˆ
V , ˜
Vbe wo sepa able Hilbe –
spaces, being (˜ei)i∈Nand ( j)j∈Ncomple e o hono mal basis o ˜
Vand ˆ
V esp., and
le
[s, ]∋ 7→ F( )∈L2(˜
V , ˆ
V),[s, ]∋ 7→ ξ( )∈˜
V
be measu able unc ions such ha F∈Cβ,∼([0, T ]; L2(˜
V , ˆ
V)),ξ∈Cβ′([0, T ]; ˜
V)
and 7→ kDα
s+F[ ]kL2(˜
V , ˆ
V)|D1−α
−ξ[ ]|˜
Vis Lebesgue-in eg able. Then o 0≤s≤
≤ ≤Twe can de ine
Z
s
F( )dξ( ) := (−1)αX
jX
iZ
s
Dα
s+( j, F(·)˜ei)ˆ
V[ ]D1−α
−(˜ei, ξ(·))˜
V[ ]d  j.
Tha his exp ession is well de ined ollows by
Z
s
F( )dξ( )ˆ
V
=X
jX
iZ
s
Dα
s+( j, F(·)˜ei)ˆ
V[ ]D1−α
−(˜ei, ξ(·))˜
V[ ]d 21
2
≤X
jZ
sX
i
(Dα
s+( j, F(·)˜ei)ˆ
V[ ])2X
i
(D1−α
−(˜ei, ξ(·))˜
V[ ])21
2
d 21
2
≤Z
sX
j,i
(Dα
s+( j, F(·)˜ei)ˆ
V[ ])2X
i
(D1−α
−(˜ei, ξ(·))˜
V[ ])21
2
d
≤Z
sX
j,i
(Dα
s+( j, F(·)˜ei)ˆ
V[ ])21
2X
i
(D1−α
−(˜ei, ξ(·))˜
V[ ])21
2
d
=Z
skDα
s+F[ ]kL2(˜
V , ˆ
V)|D1−α
−ξ[ ]|˜
Vd < ∞.(11)
8 MAR´
IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨
ORN SCHMALFUSS
Obse e ha his las in eg al is ini e since
kDα
s+F[ ]kL2(˜
V , ˆ
V)=X
j,i
(Dα
s+( j, F(·)˜ei)ˆ
V[ ])21
2
=X
j,i 1
Γ(1 −α)( j, F( )˜ei)ˆ
V
( −s)α+αZ
s
( j, F( )˜ei)ˆ
V−( j, F(q)˜ei)ˆ
V
( −q)1+αdq21
2
≤√2c(Pj,i( j, F( )˜ei)2
ˆ
V)1
2
( −s)α+X
j,i Z
s
( j, F( )˜ei)ˆ
V−( j, F(q)˜ei)ˆ
V
( −q)1+αdq21
2
≤√2ckF( )kL2(˜
V , ˆ
V)
( −s)α+Z
s
kF( )−F(q)kL2(˜
V , ˆ
V)
( −q)1+αdq
≤qc( −s)−αkFkCβ,∼([0,T ];L2(˜
V , ˆ
V)),
and |D1−α
−ξ[ ]|˜
V≤c|||ξ|||β′( − )α+β′−1(see Lemma 5), so i su ices o apply (5).
Now we can apply Lemma 7 o de ine in eg als o ac ional ype wi h alues in a
sepa able Hilbe –space, as well as o conside in eg a o s wi h alues in Vo V⊗V.
Fo example, conside an in eg and o he ype G(u( )) whe e u∈Cβ,∼([0, T ]; V),
and β < α < 2β,α+β′>1, β + 1 >2α. No e ha i DG is bounded hen
G(u)∈Cβ,∼([0, T ], V ), bu since we do no assume ha β > α, hen Dα
s+G(u(·))
is no well-de ined. Howe e , we can apply Lemma 7 in he ollowing way
Z
s
G(u(·))dω = (−1)αZ
s
Dα
s+DG(u(·))(u(·)−u(s),·)[ ]D1−α
−ω −[ ]d
+(−1)αZ
s
Dα
s+(G(u(·)) −DG(u(·))(u(·)−u(s),·))[ ]D1−α
−ω −[ ]d .
(12)
Since u, ω, a e coupled by he Chen–equali y (6), we ge
D1−α
− (s, ·) −[ ] = (−1)1−α
Γ(α) (s, )− (s, )
( − )α+ (1 −α)Z
(s, )− (s, q)
(q− )2−αdq
=(−1)1−α
Γ(α)− ( , )−(u( )−u(s)) ⊗V(ω( )−ω( ))
( − )α
+ (1 −α)Z
− ( , q)−(u( )−u(s)) ⊗V(ω(q)−ω( ))
(q− )2−αdq
=−D1−α
− [ ] + (u( )−u(s)) ⊗VD1−α
−ω −[ ].
(13)
Simila ly, i is easy o de i e ha
Dα
s+(G(u(·)) −DG(u(·))(u(·)−u(s),·))[ ]
=ˆ
Dα
s+G(u(·))[ ]−Dα
s+DG(u(·))[ ](u( )−u(s),·),
and hus, coming back o (12) we ob ain ha
LOCAL PATHWISE SOLUTIONS TO SEES 9
Z
s
G(u(·))dω =(−1)αZ
s
ˆ
Dα
s+G(u(·))[ ]D1−α
−ω −[ ]d
−(−1)αZ
s
Dα
s+DG(u(·))[ ]D1−α
− (·, )[ ]d
=(−1)αZ
s
ˆ
Dα
s+G(u(·))[ ]D1−α
−ω −[ ]d
−(−1)2α−1Z
s
D2α−1
s+DG(u(·))[ ]D1−α
−D1−α
− (·, )[ ]d ,
(14)
whe e he las equali y is ue hanks o he ac ional in eg a ion by pa s o mula
(10). The las in eg al on he igh hand side o (14) is well-de ined due o, o
0≤s≤ ≤ ≤T,
(15) kD1−α
−D1−α
− [ ]k ≤ c(k kβ+β′,∼+kukβ,∼|||ω|||β′)( −s)−β( − )β+β′+2α−2
(see Lemma 21 abo e) and hence his in eg al can be de ined by using Lemma 7.
Rema k 8. I may be checked ha he in eg al gi en in Lemma 7 can be also de ined
o an in eg a o like F( ) = S( − )G(u( )), which is no H¨olde –con inuous a
ze o. The eason is because he j-modes o his exp ession a e H¨olde con inuous.
Fu he mo e, he in eg abili y condi ion (11) is sa is ied by he egula i y p ope ies
o he analy ic semig oup Sand he egula i y o Gand u.
Finally, on accoun o (14) and he es ima es o he di e en ac ional de i a i es
ha ake pa in ha exp ession, we can es ablish he ollowing esul :
Lemma 9. Suppose u, , ω sa is y he assump ions om he beginning o his sec-
ion and Gsa is ies he assump ions o Lemma 2. Le β < α < 2β,α+β′>
1, β + 1 >2α. Then o 0≤s≤ ≤Twe ha e
Z
s
G(u)dω≤c((1 + kuk2
β,∼)|||ω|||β′+kukβ,∼k kβ+β′,∼)( −s)β′.
4. Pa h-A ea-solu ions o s ochas ic e olu ion equa ions
In his sec ion we gi e he de ini ion o a mild solu ion o (2) and es ablish a esul
abou he local exis ence and uniqueness o such solu ion. In o de o unde s and
he no ion o mild solu ion o (2), in a i s s ep we conside he case in which
he d i ing noise is egula , o la e on conside he H¨olde case in which we a e
in e es ed in.
No e ha we could also add a nonlinea di usion e m Fon he igh -hand side o
he equa ion in (1). Ne e heless, o simpli y he whole p esen a ion we ha e no
conside ed i since he d -nonlinea i y is no he in e es ing p oblem o be ea ed
in he pape .
4.1. Sys em (2) d i en by smoo h pa hs ω.In his si ua ion, since Ahas he
p ope ies o Sec ion 2 and DG is bounded, we emind ha o any u0∈V he e
exis s a unique solu ion u∈C([0, T ]; V) o (2) o any T > 0, see [21]. In addi ion,
applying he p ope ies (3), (4) we ob ain ha his solu ion is in Cγ,∼([0, T ]; V) o
any γ∈(0,1). Howe e , we do no ob ain ha he solu ion is in Cγ([0, T ]; V), due o
he ac ha 7→ S( )u0is no H¨olde -con inuous in gene al bu in Cγ,∼([0, T ]; V).
16 MAR´
IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨
ORN SCHMALFUSS
Lemma 17. Suppose ha he condi ions o Lemma 15 hold. Then he e exis s a
c > 0depending on ωsuch ha o U1, U2∈ˆ
W0,T
||T1(U1)−T1(U2)||β,∼≤cT β′(1 + ||U1||2
W+||U2||2
W)||U1−U2||W+c|u1
0−u2
0|.
P oo . Le us deno e ∆u=u1−u2,∆U= (u1−u2, 1− 2). Then o ∈[0, T ]
by Lemma 2 we ha e
kG(u1( )) −G(u2( ))kL2(V, ˆ
V)≤cDGk∆ukβ,∼
and
kG(u1( )) −G(u1(q)) −DG(u1(q))(u1( )−u1(q))
−(G(u2( )) −G(u2(q)) −DG(u2(q))(u2( )−u2(q)))kL2(V, ˆ
V)
≤cD2G(ku1kβ,∼+ku2kβ,∼)k∆ukβ,∼
( −q)2β
q2β
+cD3Gku2kβ,∼
( −q)β
qβsup
∈[0,T ]|∆u( )|(2ku1kβ,∼+ku2kβ,∼)( −q)β
qβ
≤c(ku1kβ,∼+ku2kβ,∼)k∆ukβ,∼
( −q)2β
q2β(1 + ku2kβ,∼).
Now we can ollow he p oo o Lemma 15. 
Up o now we ha e ob ained app op ia e es ima es o he i s componen T1o he
ope a o Tgi en by (26). Now we aim a ge ing he co esponding es ima es o
T2, he second componen o T. To his end, we need he wo ollowing Lemma a,
dealing wi h es ima es o some o he e ms appea ing in he exp ession o T2.
Lemma 18. Unde he Hypo hesis (H) he ollowing s a emen s hold:
(i) Fo he mapping
e∈V7→ ωS(s, )e= (−1)−αZ
s
(S(ξ−s)e)⊗Vdω(ξ)
he ollowing p ope ies hold ue: o 0≤s≤ ≤ ≤T, e ∈Vand 1/3< β′<
β′′ < H,
kωS( , )e−ωS(s, )ek ≤ c( −s)β′(|||ω|||β′+|||ω|||β′′ )|e|,
kωS(s, )ek ≤ c( −s)β′|||ω|||β′|e|,
kωS(s, )(−A)β′ek ≤ c|||ω|||β′′ |e|.
(ii) The mapping
E∈L2(V, ˆ
V)7→ Sω(s, )E=Z
s
S( − )Edω( )
is in L2(L2(V, ˆ
V), V ), wi h no m bounded by c|||ω|||β′( −s)β′kEkL2(V, ˆ
V).

LOCAL PATHWISE SOLUTIONS TO SEES 17
P oo . Le β′< β′′ < H such ha o α < α′<1 we ha e β′+α′<1< β′′ +α.
Then
ωS( , )e−ωS(s, )e=(−1)αZ
(S(ξ− )e−S(ξ−s)e)⊗Vdω(ξ)
−(−1)αZ
s
S(ξ−s)e⊗Vdω(ξ).
(29)
In e p e ing hese in eg als in a ac ional sense and using Lemma 1 we ob ain
|Dα
+(S(·− )e−S(·−s)e)[ξ]| ≤ c( −s)β′
(ξ− )α+β′+αZξ
( −s)β′(ξ−q)α′
(ξ−q)1+α(q− )α′+β′dq|e|
≤c( −s)β′(ξ− )−α−β′|e|.
Mo eo e , since β′′ < H, due o (H3) ω∈Cβ′′ ([0, T ]; V) (and in pa icula ω∈
Cβ′([0, T ], V )), hen by Lemma 5
|D1−α
−ω −[ξ]| ≤ c|||ω|||β′′ ( −ξ)α+β′′−1.
Hence, by applying (5), he i s in eg al on he igh -hand side o (29) is bounded
in pa icula by c( −s)β′|||ω|||β′′ |e|.
Fu he mo e, o he las e m in (29), hanks o (H1) and he egula i y p ope ies
o he semig oup we ob ain
Z
skDα
s+S(·−s)e[ξ]⊗VD1−α
−ω −[ξ]kdξ ≤c|||ω|||β′( −s)β′|e|.
The second s a emen o (i) ollows di ec ly om he i s one aking = . Fo he
las conclusion o (i), o he pa ame e s chosen a he beginning o he p oo ,
|Dα
s+(S(·−s)(−A)β′e)[ξ]| ≤ c|e|
(ξ−s)α+β′
+Zξ
s
(ξ−q)α′|e|
(ξ−q)1+α(q−s)α′+β′dq≤c|e|(ξ−s)−α−β′,
and he e o e, since in pa icula β′+α < 1,
Z
skDα
s+(S(·−s)(−A)β′e)[ξ]⊗VD1−α
−ω −[ξ]kdξ
≤c|||ω|||β′′ |e|Z
s
(ξ−s)−α−β′( −ξ)α+β′′−1dξ
≤c|||ω|||β′′ ( −s)β′′−β′|e| ≤ c|||ω|||β′′ |e|.
Now we p o e (ii). Fi s o all, no e ha Sωis well-de ined since, hanks o Rema k
8, i su ices he H¨olde –con inui y o 7→ S( − )Eon any in e al [ǫ, ]. In addi ion
by he embedding ˆ
V⊂V he mapping
L2(V, ˆ
V)∋E7→ ED1−α
−ω −[ ]∈V
is in L2(L2(V, ˆ
V), V ), and he no m can be es ima ed cV, ˆ
V|D1−α
−ω −[ ]|, see Lemma
28 (ii). Fu he mo e, he in eg and Dα
s+S( −·)[ ]·D1−α
−ω −[ ] is weakly measu -
able wi h espec o L2(L2(V, ˆ
V), V ) such ha by Pe is’ heo em he in eg and is
18 MAR´
IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨
ORN SCHMALFUSS
measu able. Because o D1−α
−Eω −[ ] = ED1−α
−ω −[ ], we ge
kSω(s, )·kL2(L2(V, ˆ
V),V )=


Z
s
S( − )·dω( )


L2(L2(V, ˆ
V),V )
≤Z
skDα
s+S( −·)[ ]kL(V)k·D1−α
−ω −[ ]kL2(L2(V, ˆ
V),V )d
≤c|||ω|||β′( −s)β′.

Co olla y 19. Le (ωn)n∈Nbe a sequence con e ging o ωin Cβ′([0, T ]; V). Then
lim
n→∞ sup
0≤s < ≤T
|e|= 1
k(ω−ωn)S(s, )ek
( −s)β′= 0,
lim
n→∞ sup
0≤s< ≤T
kSω−ωn(s, )·kL2(L2(V, ˆ
V),V )
( −s)β′= 0.
F om he p e ious esul we also ob ain ha
Dα
s+(S(·−s)(Sω(q, s)·))[ξ]⊗VD1−α
−ω −[ξ]
is weakly L2(L2(V, ˆ
V), V ⊗V)–measu able.
Lemma 20. Suppose ha he hypo hesis (H) holds. Fo 0< s ≤q≤ ≤T,
˜
E∈L2(V⊗V, ˆ
V)and U= (u, )∈ˆ
W0,T conside he mapping
w( , s, q) : L2(V⊗V, ˆ
V)∋˜
E7→ ˜
Ew( , s, q)∈V⊗V
gi en by (27) and (28). Then i is well-de ined and sa is ies, o β′< β′′ he
es ima e
kw( , s, q)kL2(L2(V⊗V, ˆ
V),V ⊗V)≤c||U||Ws−β(q−s)β+β′( −s)β′,
whe e he cons an cdepends on |||ω|||β′′ and k(ω⊗Sω)k2β′. In pa icula
c∼(1 + |||ω|||2
β′′ )k(ω⊗Sω)k2β′.
Fo he p oo , see he Appendix.
We would like o poin ou ha wsa is ies he gene alized Chen–equali y
˜
Ew( , s, ) + ˜
Ew( , , q)−˜
E(u( )−u(s),·)(ω⊗Sω)( , q)
=˜
Ew( , s, q) + ωS(q, )Sω(q, )˜
E(u( )−u(s),·),
(30)
o 0 < s ≤ ≤q≤ ≤T, and he Chen–equali y
˜
Ew( , s, ) + ˜
Ew( , , )−˜
E(u( )−u(s),·)(ω⊗Sω)( , ) = ˜
Ew( , s, )(31)
whe e he la e one is ob ained om (30) aking q= . In o de o p o e hese
wo p ope ies we only need o ollow an app oxima ion a gumen , conside ing
(ωn,(ωn⊗Sωn)) sa is ying (H3), and he e o e con e ging o (ω, (ω⊗Sω)) in
Cβ′([0, T ], V )×C2β′(¯
∆0,T ;L2(L2(V, ˆ
V), V ⊗V)) and ake in o accoun ha he
app oxima ing elemen s wn:= (u⊗(ωn⊗Sωn)) gi en in Lemma 10 sa is y he
Chen–equali ies (23) and (24). In pa icula , he e ms (ωn
S⊗ωn) con e ge o he
LOCAL PATHWISE SOLUTIONS TO SEES 19
co esponding e m (ωS⊗ω), see he p oo o Lemma 20 in he Appendix sec ion.
The ollowing esul gi es es ima es o he ac ional de i a i es o and w.
Lemma 21. Le U= (u, )∈ˆ
W0,T . Then o 0< < ≤Twe ha e
kD1−α
−D1−α
− [ ]kβ+β′,∼≤c||U||W −β( − )2α+β+β′−2,
kD1−α
−D1−α
−w( , ·,·)[ ]k ≤ c||U||W −β( − )2α+β+2β′−2,
whe e he i s cons an cdepends on |||ω|||β′′ , and he second one on |||ω|||β′′ and
k(ω⊗Sω)k2β′.
We ha e also shi ed he p oo o his esul o he Appendix sec ion.
Lemma 22. Assume ha u0∈V. The e exis s posi i e cons an s ˜cand csuch
ha o U∈ˆ
W0,T
kT2(U, ω,(ω⊗Sω), u0)kβ+β′,∼≤˜c|u0|+cT β′(1 + kUk2
W)
and, in addi ion, o wo elemen s U1, U2∈ˆ
W0,T :
kT2(U1, ω, (ω⊗Sω), u0)−T2(U2, ω, (ω⊗Sω), u0)kβ+β′,∼
≤˜c|u1
0−u2
0|+cT β′(1 + kU1k2
W+kU2k2
W)kU1−U2kW.
The cons an cdepends on |||ω|||β′′ ,|||ω|||β′and k(ω⊗Sω)k2β′, while ˜con |||ω|||β′.
P oo . Le us deno e T2(U)(s, ) =: B1(s, ) + B2(s, ) + B3(s, ), co esponding o
he h ee di e en addends o T2.
Fo B1we can conside he ollowing spli ing:
B1(s, ) = Z
s
(S(ξ−s)−id)u(s)⊗Vdω(ξ)
=Z
s
(S(ξ)−S(s))u0⊗Vdω(ξ)
+Z
s
(S(ξ−s)−id)Zs
0
S(s− )G(u( ))dω( )⊗Vdω(ξ)
=:B11(s, ) + B12(s, ).
B1can be in e p e ed in he ac ional sense hanks o he egula i y o i s in eg and,
which means ha
B11(s, ) = (−1)αZ
s
Dα
s+((S(·)−S(s))u0)[ξ]⊗VD1−α
−ω −[ξ]dξ.
Fo α < α′whe e α′is su icien ly close o αand s > 0, applying (3) and (4),
|Dα
s+((S(·)−S(s))u0)[ξ]| ≤ c|(S(ξ)−S(s))u0|
(ξ−s)α+Zξ
s
|(S(ξ)−S(q))u0|
(ξ−q)1+αdq
≤c(ξ−s)β−α
sβ+Zξ
s
(ξ−q)α′q−β
(ξ−q)1+αqα′−βdq|u0|
≤c(ξ−s)β−α
sβ+1
sβZξ
s
(ξ−q)α′
(ξ−q)1+α(q−s)α′−βdq|u0| ≤ c(ξ−s)β−α
sβ|u0|,
20 MAR´
IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨
ORN SCHMALFUSS
hence, o s > 0,
|B11(s, )| ≤ c|||ω|||β′|u0|
sβZ
s
(ξ−s)β−α( −ξ)β′+α−1dξ ≤c|||ω|||β′|u0|s−β( −s)β′+β,
which implies kB11kβ+β′,∼≤˜c|u0|. Mo eo e , no e ha

Dα
s+(S(·−s)−id) Zs
0
S(s− )G(u( ))dω( )[ξ]
≤c|S(ξ−s)−id) Rs
0S(s− )G(u( ))dω( )|
(ξ−s)α
+Zξ
s
|Rs
0S(ξ− )−S(q− ))G(u( ))dω( )|
(ξ−q)1+αdq.
To deal wi h B12 on accoun o Co olla y 16 and hanks o he ac ha β′> β we
ha e
Zξ
s
|Rs
0(S(ξ−q)−id)S(q− )G(u( ))dω( )|
(ξ−q)1+αdq
≤Zξ
s
(ξ−q)α′
(q−s)α′−β(ξ−q)1+α
(−A)βZs
0
S(s− )G(u( ))dω( )
dq
≤c(1 + |||ω|||β′)(1 + kUk2
W)(ξ−s)β−αsβ′−β
and by (5)
kB12kβ+β′,∼≤cT β′|||ω|||2
β′(1 + kUk2
W).
Finally, a simila es ima e ollows o B2and B3. In o de o see his, no e ha
B2and B3can be conside ed in a simila way o C11 and C12 in he p oo o
Lemma 15, wi h he di e ence ha now we ha e o es ima e D1−α
−(ω⊗Sω)(·, ) −[ ]
and D1−α
−D1−α
−w( , ·,·)[ ], o which we use he 2β′-H¨olde con inui y o (ω⊗Sω)
oge he wi h Lemma 21, a i ing a
kB2kβ+β′,∼≤cT β′k(ω⊗Sω)k2β′(1 + kUk2
W),
kB3kβ+β′,∼≤cT β′(1 + |||ω|||2
β′′ )k(ω⊗Sω)k2β′(1 + kUk2
W).
The second pa o his lemma can be p o en simila ly and hus we omi i he e. 
Now we app oxima e T(U, ω, (ω⊗Sω), u0) by piecewise linea noise.
Lemma 23. Le U∈ˆ
W0,T and assume ha (ωn,(ωn⊗Sωn)) sa is ies (H3). Then
lim
n→∞ kT(U, ωn,(ωn⊗Sωn), u0)−T(U, ω, (ω⊗Sω), u0)kW= 0.
In addi ion, T(U, ω, (ω⊗Sω), u0)∈ˆ
W0,T .
P oo . No e ha i a e m o Tcon ains ωo (ω⊗Sω) hen his e m depends
on hese exp essions linea ly o bi-linea ly (see he de ini ion o T1,T2 oge he
wi h (27)) which jus gi es he con e gence conclusion. To see he second pa
o he s a emen we ake U= (u, )∈ˆ
W0,T . Fo his elemen we choose an
app oxima ing sequence (un,(un⊗ωn)) om (25). We no e ha T1((un,(un⊗
ωn)), ωn, un
0)∈Cγ([0, T ], V ) and T1((un,(un⊗ωn)), ωn, un
0)(0) ∈D((−A)) o
any γ∈(0,1), see Pazy [21] Theo em 4.3.1 and (3)-(4). The e o e T2((un,(un⊗
ωn)), ωn,(ωn⊗Sωn), un
0) can be de ined as (T1((un,(un, ωn)), ωn, un
0)⊗ωn) gi en
LOCAL PATHWISE SOLUTIONS TO SEES 21
by (7). By de ini ion o he space ˆ
W0,T , by Lemmas 17, 22, and he i s pa o
his lemma we ha e ha ha T(U, ω, (ω⊗Sω), u0)∈ˆ
W0,T .
Le us now p o e he uniqueness o he pa h-a ea-solu ion in ˆ
W0,T i such a solu ion
exis s.
Theo em 24. Suppose ha U1= (u1, 1), U2= (u2, 2)∈ˆ
W0,T a e wo pa h-a ea
solu ions ela ed o he ini ial condi ion u0∈V. Then we ha e U1=U2.
P oo . Le T1∈[0, T ) such ha [0, T1] ep esen s he maximal in e al o unique-
ness. He e T1= 0 means ha wo di e en solu ions a e jus b anching om
0. No e ha he es ic ions ˜
Ui= (ui
[T1,˜
T], i
∆[T1,˜
T]) a e pa h-a ea solu ions oo
wi h espec o he in e al [T1,˜
T],˜
T≤Twi h ini ial condi ion u(T1). This ol-
lows because o T1we can apply Lemma 12. On he o he hand, T2 es ic ed o
T1< s < ≤˜
Tkeeps he same s uc u e as he o iginal T2. Then, i he cons an C
is an es ima e o k˜
Uik2
WT1,˜
T, simila o he es ima es ha we ob ained in Lemmas
17 and 22, we ob ain
06=k˜
U1−˜
U2kWT1,˜
T≤c(˜
T−T1)β′(1 + 2C)k˜
U1−˜
U2kWT1,˜
T
o equi alen ly
1≤c(˜
T−T1)β′(1 + 2C)
o any ˜
T > T1, which is a con adic ion i ˜
T−T1is su icien ly small. 
Now we p esen he main heo em o he pape .
Theo em 25. Suppose ha he s anding condi ions (H) a e sa is ied and suppose
ha T > 0is chosen su icien ly small depending on he pa ame e s o he p oblem.
Then Thas a ixed poin in ˆ
W0,T ⊂W0,T ha de ines a mild pa h-a ea solu ion o
(2) which is unique.
P oo . Lemmas 15, 22 and 23 p o e ha Tmaps a closed cen e ed ball om ˆ
W0,T
in o i sel i T > 0 is su icien ly small and by Lemma 17 and Lemma 22 we ob ain
ha his mapping is a con ac ion i T > 0 is chosen su icien ly small. 
Co olla y 26. Le U∈ˆ
W0,T be a mild pa h-a ea solu ion gi en by Theo em 25
and le Unbe he pa h-a ea solu ion co esponding o he equa ions (2) and (21)
and d i en by a smoo h noise ωn. Then we ha e
lim
n→∞ kUn−UkW= 0.
In ac , o la ge enough nwe can ind a ball in ˆ
W0,T which is mapped in o i sel
by T(·, ωn,(ωn⊗Sωn), u0) and T(·, ω, (ω⊗Sω), u0), and whe e in addi ion hese
mappings a e con ac ions wi h uni o m con ac ion condi ion. Then he conclusion
ollows by he pa ame e e sion o he Banach- ixed poin Theo em oge he wi h
Lemma 23.
Rema k 27. An applica ion o he main esul s o his a icle is o p o e ha he
(pa hwise) mild pa h–a ea solu ions a e global, see he o hcoming pape [11]. In
addi ion, he esul s p esen ed in his pape a e he basemen o p o e ha SEEs
wi h non- i ial di usion coe icien s Gand d i en by ac ional B ownian–mo ions
BHwi h H∈(1/3,1/2] gene a e a andom dynamical sys em, see also [11]. Tha
p ope y has been es ablished ecen ly by he same au ho s when dealing wi h a mo e
egula Bm, namely BHwi h H∈(1/2,1), see [9].

22 MAR´
IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨
ORN SCHMALFUSS
5. Example
Le V=l2be he space o squa e addi i e sequences wi h alues in R. In addi ion,
le Abe a nega i e symme ic ope a o de ined on D(−A)⊂l2wi h compac
in e se. In pa icula , we can assume ha −Ahas a disc e e spec um 0 < λ1≤
λ2≤ ··· ≤ λi≤ ··· → ∞ whe e he associa ed eigenelemen s (ei)i∈N o m a
comple e o hono mal sys em in l2. The spaces D((−A)ν) = Vνa e hen de ined
by
{u= (ui)i∈N∈l2:X
i
λ2ν
iu2
i=: |u|2
Vν<∞}.
Assume ha he e exis s κ > 0 such ha he Hilbe -Schmid embedding Vκ⊂V
holds ue, and ake ˆ
V=Vκ. Conside a sequence o unc ions (gij )i,j∈N, wi h
gij :V→R, and de ine G(u) o u∈Vby
G(u) =X
j
gij(u) ji∈N
o all ∈V.
We assume ha
kG(u)k2
L2(V,Vκ)=X
j|G(u)ej|2
Vκ=X
jX
i
λ2κ
i(G(u)ej)2
i=X
i,j
λ2κ
ig2
ij(u)≤c
uni o mly wi h espec o u∈V.
In addi ion, assume ha gij a e ou imes di e en iable and hei de i a i es a e
uni o mly bounded in he ollowing way
|Dgij(u)(ek)| ≤ cijk
g,1,|D2gij(u)(ek, h1)| ≤ cijk
g,2|h1|,|D3gij(u)(ek, h1, h2)| ≤ cijk
g,3|h1||h2|,
|D4gij(u)(ek, h1, h2, h3)| ≤ cijk
g,4|h1||h2||h3| o any u∈V,
and hese bounds sa is y
X
ijk
λ2κ
i(cijk
g,1)2<∞,X
ijk
λ2κ
i(cijk
g,2)2<∞,X
ijk
λ2κ
i(cijk
g,3)2<∞,X
ijk
λ2κ
i(cijk
g,4)2<∞.
To see o ins ance ha DG exis s, no e ha by Taylo expansion
|gij(u+h)−gij(u)−Dgij(u)(h)|2≤1
2|D2gij(u+ηh)(h, h)|2≤(cijk
g,2)2|h|4
whe e u, h ∈Vand η∈[0,1]. In pa icula , we also no e ha
X
jk 
DG(u)(ek, ej)
2
Vκ
=X
ijk
λ2κ
i|Dgi,j(u)(ek)|2≤X
ijk
λ2κ
i(cijk
g,1)2=: c2
DG <∞.
This condi ion ensu es he Lipschi z con inui y o Gas well as he Hilbe -Schmid
p ope y o DG. Simila ly, we ob ain ha DG is also Lipschi z wi h espec o
he Hilbe -Schmid no m. We also ob ain he exis ence o he second and hi d
de i a i e. Hence he condi ions on Gin Hypo hesis (H) hold.
The de eloped heo y can be also applied o o he examples o G, like ke nel in e-
g als, see [11].
LOCAL PATHWISE SOLUTIONS TO SEES 23
6. Appendix
In he appendix we p o e some echnical es ima es ela ed o w= (u⊗(ω⊗Sω)).
P oo o Lemma 10
P oo . Fo ˜
E∈L2(V⊗V, ˆ
V) we de ine ˜
E:L2(V, V ⊗V)×V→L2(V, V ⊗V)
gi en by
˜
E(Q, u) = Q(˜
E(u, ·)).
F om (17) o smoo h ωwe ha e ha
˜
E(u⊗(ω⊗Sω)( ))(s, q) = −(−1)αZq
s
ωS( , )˜
E(u( )−u(s), ω′( ))d
=−(−1)αZq
s
˜
E(ωS( , ), u( )−u(s))ω′( )d .
Following Theo em 3.3 in [16], we ha e
Zq
s
˜
E(ωS( , ), u( )−u(s))ω′( )d
=(−1)αZq
s
ˆ
Dα
s+ ˜
E(ωS(·, ), u(·)−u(s))[ ]D1−α
q−ωq−[ ]d
−(−1)αZq
s
(ωS( , )−ωS(s, ))Dα
s+˜
E(u(·)−u(s),·)[ ]D1−α
q−ωq−[ ]d
−(−1)αZq
s
Dα
s+ωS(·, )[ ]˜
E(u( )−u(s),·)D1−α
q−ωq−[ ]d
+Zq
s
D ˜
E(ωS( , ), u( )−u(s))(ωS( , )−ωS(s, ), u( )−u(s))ω′( )d .
(32)
Now we calcula e he de i a i e o ˜
E:
D ˜
E(ωS( , ), u( )−u(s))(ωS( , )−ωS(s, ), u( )−u(s))ω′( )
=(ωS( , )−ωS(s, )) ˜
E(u( )−u(s), ω′( )) + ωS( , )˜
E(u( )−u(s), ω′( ))
=˜
E(u( )−u(s),·)D2(ωS( )⊗ω)(s, ) + ωS( , )˜
ED2(u⊗ω)(s, ).
Subs i u ing he abo e exp ession in (32), a e applying ac ional in eg a ion (10)
o he las wo e ms, we ha e o calcula e ED1−α
q−(ωS( )⊗ω)q−(s, ·)[ ], o E∈
L2(V, ˆ
V), and ˜
ED1−α
q−(u⊗ω)( )(s, ·)q−[ ]. Fi s , we ha e
ED1−α
q−(ωS( )⊗ω)q−(s, ·)[ ] = D1−α
q−E(ωS( )⊗ω)q−(s, ·)[ ]
=(−1)1−α
Γ(α)E(ωS( )⊗ω)(s, )−E(ωS( )⊗ω)(s, q)
(q− )1−α
+ (1 −α)Zq
E(ωS( )⊗ω)(s, )−E(ωS( )⊗ω)(s, θ)
(θ− )2−αdθ
=−(−1)1−α
Γ(α)E(ωS( )⊗ω)( , q) + (ωS( , )−ωS(s, ))E(ω(q)−ω( ))
(q− )1−α
+ (1 −α)Zq
E(ωS( )⊗ω)( , θ) + (ωS( , )−ωS(s, ))E(ω(θ)−ω( ))
(θ− )2−αdθ
=−ED1−α
q−(ωS( )⊗ω))[ ] + (ωS( , )−ωS(s, ))ED1−α
q−ωq−( ).
24 MAR´
IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨
ORN SCHMALFUSS
Secondly, simila o (13) we ob ain ha
˜
ED1−α
q−(u⊗ω)( )(s, ·)q−[ ] = −˜
ED1−α
q−(u⊗ω)( )[ ] + ˜
E(u( )−u(s), D1−α
q−ωq−( )),
and subs i u ing he las wo exp essions in o (32) we ob ain he conclusion. 
In he ollowing we wan o p o e ha wis well–de ined o U∈ˆ
W0,T . To his end
we will conside a mapping
w( , s, q) = w(U, ω, (ω⊗Sω))( , s, q)
which coincides o smoo h ωwi h he exp ession in oduced in Lemma 10.
In o de o p o e he egula i y o ws a ed in Lemma 20 we need se e al p ope ies
ha we collec and p o e in he ollowing esul .
Lemma 28. (i) Le ˜
E∈L2(V⊗V, ˆ
V)and le ∈V⊗Vbe ixed. Then he
mapping
˜
E7→ ˜
E
is in L2(L2(V⊗V, ˆ
V), V ).
(ii) Le E∈L2(V, ˆ
V)and le u∈Vbe ixed. Then he mapping
E7→ Eu
is in L2(L2(V, ˆ
V), V ).
(iii) Le ˜
E∈L2(V⊗V, ˆ
V)and u∈V. Then
k˜
E(u, ·)kL2(V, ˆ
V)=k˜
E(u⊗V·)kL2(V, ˆ
V)≤ |u|k˜
EkL2(V⊗V, ˆ
V).
P oo . Conside he sepa able Hilbe –space L2(V⊗V, ˆ
V) equipped wi h he com-
ple e o hono mal basis ( ˜
Eijk)i,j,k∈Ngi en by
˜
Eijk(el⊗Vem) = ˜
Eijk(el, em) = 0 : j6=lo k6=m
i:j=land k=m.
We emind ha (ei)i∈Nand ( i)i∈Na e, espec i ely, comple e o hono mal basis
o Vand ˆ
V. Then o ∈V⊗V,
sX
ijk
(˜
Eijk )2=sX
i| i|2X
jk
2
jk =cV, ˆ
Vk k
whe e jk is he mode o wi h espec o ej⊗Vek. This comple es (i).
The second s a emen can be p o en simila ly and he e o e we omi i s p oo .
Finally, we ha e
k˜
E(u, ·)k2
L2(V, ˆ
V)=X
i|˜
E(u, ei)|2
ˆ
V
=X
i|X
k
uk˜
E(ek, ei)|2
ˆ
V≤X
iX
k|uk||˜
E(ek, ei)|ˆ
V2
≤X
iX
k|uk|21
2X
k|˜
E(ek, ei)|2
ˆ
V1
22
=|u|2k˜
Ek2
L2(V⊗V, ˆ
V).

LOCAL PATHWISE SOLUTIONS TO SEES 25
In wha ollows we abb e ia e he no a ion in he ollowing way: le us deno e
L2,⊗=L2(L2(V, ˆ
V), V ⊗V), L2,⊗,⊗=L2(L2(V⊗V, ˆ
V), V ⊗V).
P oo o Lemma 20.
P oo . Le us conside sepa a ely he h ee e ms o w( , s, q) gi en by (27). P e-
cisely, we s a es ima ing he hi d e m
I3(˜
E) := Zq
s
D2α−1
s+ωS(·, )[ ]D1−α
q−D1−α
q−˜
E [ ]d .
As we ha e seen in Lemma 28 (i), o a ixed ∈V⊗V he mapping L2(V⊗V, ˆ
V)∋
˜
E7→ ˜
E is in L2(L2(V⊗V, ˆ
V), V ) whe e an es ima e o he no m o his ope a o is
gi en by cV, ˆ
Vk k. Then, since Lemma 18 (i) in pa icula implies ha D2α−1
s+ωS( , )
is in L(V, V ⊗V), he mapping ˜
E7→ I3(˜
E) is in L2,⊗,⊗. We ha e
kI3(·)kL2,⊗,⊗≤Zq
skD2α−1
s+ωS(·, )[ ]·D1−α
q−D1−α
q− [ ]kL2,⊗,⊗d
≤Zq
skD2α−1
s+ωS(·, )[ ]kL(V,V ⊗V)k·D1−α
q−D1−α
q− [ ]kL2(L2(V⊗V, ˆ
V),V )d .
In o de o es ima e he second ac o in he in eg and o I3, no e ha due o
Lemma 21, o ∈(s, q) and U∈ˆ
W0,T , we ob ain
kD1−α
q−D1−α
q− [ ]k ≤ ckUkW −β(q− )β+β′+2α−2
and as we ha e said a he beginning o his p oo
kD1−α
q−D1−α
q−· [ ]kL2(L2(V⊗V, ˆ
V),V )≤ccV, ˆ
VkUkW −β(q− )β+β′+2α−2.
On he o he hand, by Lemma 18 (i),
kD2α−1
s+ωS(·, )[ ]kL(V,V ⊗V)≤c( − )β′
( −s)2α−1+Z
s
( −ξ)β′
( −ξ)2αdξ(|||ω|||β′+|||ω|||β′′ ).
Combining he p e ious es ima es we can conclude
kI3(·)kL2,⊗,⊗≤ckUkWs−β( −s)β′(q−s)β+β′,
whe e cdepends on |||ω|||β′′ and |||ω|||β′. Nex we deal wi h
I2(˜
E) := Zq
s
ˆ
Dα
s+ωS(·, )˜
E(u(·)−u(s),·)[ ]D1−α
q−ωq−[ ]d .
Obse e ha
kˆ
Dα
s+ωS(·, )˜
E(u(·)−u(s),·)[ ]D1−α
q−ωq−[ ]kL2,⊗,⊗
≤kωS( , )˜
E(u( )−u(s), D1−α
q−ωq−[ ])kL2,⊗,⊗
( −s)α
+cZ
s
k(ωS( , )−ωS(θ, )) ˜
E(u( )−u(θ), D1−α
q−ωq−[ ])kL2,⊗,⊗
( −θ)1+αdθ
≤ccV, ˆ
V(q− )α+β′−1s−βk˜
EkL2(V⊗V, ˆ
V)kukβ,∼|||ω|||β′
×(|||ω|||β′+|||ω|||β′′ )(( − )β′( −s)β−α+ ( −s)β′+β−α),