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Complex Variables and Elliptic EquationsPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713455999

Gradient-type noises I-partial and hybrid integralsMichael Hinz a; Martina Zähle a

a Mathematical Institute, Friedrich-Schiller-University Jena, Jena, Germany

First Published:June2009

To cite this Article Hinz, Michael and Zähle, Martina(2009)'Gradient-type noises I-partial and hybrid integrals',Complex Variables andElliptic Equations,54:6,561 — 583

To link to this Article: DOI: 10.1080/17476930802669652

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Complex Variables and Elliptic EquationsVol. 54, No. 6, June 2009, 561–583

Gradient-type noises I – partial and hybrid integrals

Michael Hinz* and Martina Zahle

Mathematical Institute, Friedrich-Schiller-University Jena, Jena, Germany

Communicated by D.-C. Chang

(Received 11 July 2008; final version received 7 November 2008)

The final objective of our study is to propose a finite-dimensional approach tosystems of parabolic partial differential equations perturbed by low-order noisesof Brownian or fractional Brownian type. The present article is the preparatoryfirst part, where we introduce partial pathwise integrals over D and (a, b)�D,where D is a smooth bounded domain in R

n. Corresponding stochastic versionsappear as limit cases.

Keywords: stochastic partial differential equations; fractional Brownian sheet;fractional calculus; function spaces; Stieltjes integrals

AMS Subject Classifications: 26A33; 26A42; 60H05; 60H15

1. Introduction

The present article is the first part of a paper concerned with a pathwise approach tostochastic partial differential equations. In a finite-dimensional setting, systems of partialdifferential equations will be considered under certain random noises. An example is theparabolic problem with multiplicative noise given by

@u

@t¼

@2u

@x2þ u �

@2Z

@t@x,

uð0, xÞ ¼ f ðxÞ, x 2 ða, bÞ,

uðt, aÞ ¼ uðt, bÞ ¼ 0, t 2 ð0,T Þ,

8>><>>: ð1Þ

where T4 0 and (a, b)�R is a finite interval. Here f : (a, b)!R is a suitable initialcondition and @2Z

@t@x is interpreted as a space-time noise arising from some given random fieldZ¼ (Z(t, x))(t,x)2R2. We will say (1) possesses the (mild) solution u : [0,T ]� (a, b)!R if

uðtÞ ¼ Pt fþ It u,@2Z

@t@x

� �ð2Þ

is well defined as a member of a suitable function space. (Pt)t�0 denotes the transitionsemigroup of Brownian motion killed upon leaving (a, b) and Itðu,

@2Z@t@xÞ is some integral

operator which to give a meaning to is the central matter.

*Corresponding author. Email: [email protected]

ISSN 1747–6933 print/ISSN 1747–6941 online

� 2009 Taylor & Francis

DOI: 10.1080/17476930802669652

http://www.informaworld.com

Downloaded At: 16:09 12 October 2009

Apart from restrictions on � and � in order to solve (1), it will for instance be possibleto consider Z :¼B�,�, where B�,�¼ (B�,�(t, x))(t,x)2R2 is the anisotropic fractional Browniansheet in the sense of [1,2], see also [3]. The special case with �¼ �¼ 1/2 yields the Browniansheet W¼ (W(t, x))(t,x)2R2, and our integral construction will be such that if W is chosen asintegrator, we (essentially) arrive at the stochastic Ito integral used in the spatially one-dimensional examples in [4].

We will also consider space dimensions n4 1. Recall that in the classical approach of [4],the (nþ1)-dimensional Brownian sheet W¼ (W(t, x))(t,x)2Rnþ1 as integrator describes space-time white noise, formally given by the full mixed derivative ð@nþ1WÞ=ð@t@x1 . . . @xnÞ of W.It is known that for space dimension n4 1, the solutions to higher dimensional analoguesof (1) involving space-time white noises cannot be scalar-valued processes, cf. e.g. [4] or [5].

We propose to study lower order noises formally given by mixed partial derivatives@@trZ where rg ¼ ð @g@x1 , . . . , @g

@xnÞ denotes the (spatial) gradient of an R- or R

k-valuedfunction g on [0,T ]�D, interpreted in distributional sense. Z can be replaced by the pathsof a suitable random field, in particular, by some (hybrid) fractional Brownian sheetsor fields, see [2,3,6]. Some interesting motivations will be pointed out in part II of thisstudy, [7], where we will actually solve some problems including (1).

The purpose of the present part I is to survey some background for the definition of thepathwise integral operator Itðu,

@2Z@t@xÞwhich, together with higher dimensional analogues, will

be introduced in part II. We describe basic types of integrals overD or (a, b)�D, whereD isa bounded C1-domain in R

n. We employ some two-parameter Stieltjes-type integration.For the one-parameter case, see [8–16] and the references therein. Since we aim ata construction well suited to PDE theory, we follow [12,14–16] and use a combination offractional calculus and Fourier analysis. We stress that we stay within the ‘case of simpleduality’ which does not require the application of iterated integrals or rough path methods.

Forward integrals of formal typeZ Zða,bÞ�D

f, ð@

@trÞþg

� �dðt, xÞ ð3Þ

for Rn- and R-valued non-random functions f and g, respectively, on (a, b)�D are

introduced. Paths of certain (hybrid) fractional Brownian sheets B�,� can be considered inplace of f and g. In the case where the integrator g is a (hybrid) Brownian sheet W,averaging and convergence in square mean yield a related stochastic integral

ðAÞ

Z Zða,bÞ�D

H, ð@

@trÞþPW

� �dðt, xÞ

for bounded and predictable integrands H, cf. [12,15]. When D¼ (a, b) is an interval, thisstochastic integral then coincides with the two-parameter Ito integral (of the first kind) asconsidered e.g. in [4,17], or [18].

In Section 2, we give a survey on fractional calculus for functions with values inseparable complex Banach spaces. This allows to define Stieltjes-type integrals for suchfunctions in the same way as it was done for scalar-valued in [14], outlined in Section 3.In Section 4, we consider forward integrals for functions on smooth domains and givean existence statement. In Section 5, we ‘clip’ the approaches and obtain integrals forfunctions on (a, b)�D. In Section 6, we consider fractional Brownian sheets and fields and

562 M. Hinz and M. Zahle


integrate one against another in the pathwise sense. In Section 7, where a Brownian sheet

serves as integrator, a stochastic integral is defined as an average limit taken in the mean

square. It is shown to coincide with the two-parameter Ito integral of the first kind.The proofs are shifted to Appendix 1 and sometimes given for special cases only, their

generalizations being a matter of notation. In Appendix 2, we briefly describe Stieltjes

integrals in Banach spaces in the sense of [19], as they are related to the integrals from

Section 3 in the expected manner. Positive constants whose values are not of importance

are denoted by c.

2. Notions from fractional calculus

Let X be a separable complex Banach space with norm k�kX and let I be an interval or R.

In later applications it will be possible to restrict the attention to real subspaces of X.For 1� p51 let Lp(I,X) denote the space of (equivalence classes of) measurable

functions f : I!X such that

f jLpðI,XÞ�� ¼ Z

I

f ðtÞ�� p

Xdt

� �1=p

51 ,

the integrals taken in the sense of Bochner, see e.g. [20]. We write L1(I) in case X¼C.

All proofs of the facts we quote in the following carry over from the scalar-valued case,

we refer the reader to [21].

2.1. Fractional integrals and derivatives on an interval

Let I¼ (a, b) be a bounded interval. Given �4 0 and a function ’2L1((a, b), X), consider

the (forward and backward) Riemann–Liouville fractional integrals of order � by

I�aþ’ðtÞ :¼1

�ð�Þ

Z t

a

’ð�Þ

ðt� �Þ1��d�

and

I�b�’ðtÞ :¼ð�1Þ��

�ð�Þ

Z b

t

’ð�Þ

ð� � tÞ1��d� :

Here for �4 0 the powers are understood as usual in the sense of choosing the main

branch of the analytic function z�, z2C, with the cut along the positive half-axis.

In particular, (�1)�¼ ei��.One has I�aþI

�aþ ¼ I�þ�aþ if �, �� 0 and lim"!0 I

"aþ’ ¼ ’ in Lp((a, b), X), provided

’2Lp((a, b), X). The same is true for I�b�. Let I�aþðLpðða, bÞ,XÞÞ denote the space of

functions f ¼ I�aþ’ with ’2Lp((a, b), X), similarly I�b�ðLpðða, bÞ,XÞÞ. For 05�5 1 and

functions f 2 I�aþðLpðða, bÞ,XÞÞ, respectively f 2 I�b�ðLpðða, bÞ,XÞÞ, consider the (forward,

respectively backward) Weyl–Marchaud fractional derivatives of order �,

D�aþ f ðtÞ :¼

1ða,bÞ

�ð1� �Þ

f ðtÞ

ðt� aÞ�þ �

Z t

a

f ðtÞ � f ð�Þ

ðt� �Þ�þ1d�

� �

Complex Variables and Elliptic Equations 563


and

D�b� f ðtÞ :¼

ð�1Þ�1ða,bÞ

�ð1� �Þ

f ðtÞ

ðb� tÞ�þ �

Z b

t

f ðtÞ � f ð�Þ

ð� � tÞ�þ1d�

� �,

the convergence of the principal values of the hypersingular integrals being pointwise

almost everywhere if p¼ 1 and in Lp((a, b), X) if p� 1. Under these assumptions

I�aþD�aþf ¼ f in Lp((a, b), X), while D�

aþI�aþ’ ¼ ’ is true for any ’2L1((a, b), X). This

may be completed in the case �¼ 1 by putting D1aþf ¼ df=dx and D1

b�f ¼ �df=dxand in the case �¼ 0 by the identity. The space I�aþðLpðða, bÞ,XÞÞ will be endowed with

the norm

f jI�aþðLpðða, bÞ,XÞÞ�� :¼ D�

aþfjLpðða, bÞ,X Þ�� , ð4Þ

similarly for I�b�ðLpðða, bÞ,XÞÞ.

2.2. Integration-by-parts

Now let E and F be two separable real Banach spaces normed by k�kE, and k�kFrespectively, and let L¼L(E,F) denote the Banach space of bounded linear operators

from E into F endowed with the operator norm k�kL. Standard arguments prove the

integration-by-parts rule for derivatives,

ð�1Þ�Z b

a

UðtÞD�aþf ðtÞdt ¼

Z b

a

D�b�UðtÞf ðtÞdt, ð5Þ

provided 0�� 1, p, q� 1, 1/pþ 1/q5 1þ�, f 2 I�aþðLqðða, bÞ,EÞÞ and U 2 I�b�ðLpðða, bÞ,LÞÞ, the limiting cases for � justified by ordinary calculus.

3. Stieltjes-type integrals in Banach spaces

Again, let (E, k�kE) and (F, k�kF) be two separable real Banach spaces and L¼L(E,F) the

space of bounded linear operators normed by k�kL.

3.1. Stieltjes-type integrals

To build up a type of integral as needed in part II, we start with a Stieltjes integral

as introduced [14], now for vector-valued functions. For f : (a, b)!E we suppose

f(aþ)¼ lim�!0 f(aþ �) exists in the strong sense and put faþ¼ 1(a,b)(t)( f(t)� f(aþ )).

For U : (a, b)!L we assume that U(aþ)f¼ lim�!0 U(aþ �)f for any f2E exists as

the limit in the strong sense in F, then consequently U(aþ)2L, too. Set

Uaþ(t)¼ 1(a,b)(t)(U(t)�U(aþ)). The meanings of fb� and Ub� are similar.

Definition 3.1 Suppose 0�� 1, p, q� 1, 1/pþ 1/q� 1. Let f : (a, b)!E be such

that faþ 2 I�aþðLpðða, bÞ,EÞÞ and U : (a, b)!L such that Ub� 2 I1��b� ðLqðða, bÞ,LÞÞ. Define

the forward integral of the E-valued function f with respect to the operator-valued



function U by Z b

a

f ðtÞdUðtÞ :¼ð�1Þ�Z b

a

D1��b� Ub�ðtÞD

�aþfaþðtÞdt

þUðb�Þf ðaþÞ �UðaþÞf ðaþÞ ð6Þ

The integral is directed forward.

Remark 3.2

(i) As in [14], (5) can be used to show that the definition (6) is correct, i.e. does not

depend on the particular choice of 0�� 1.(ii) To justify the notation of the marginal limits in (6), apply the triangle inequality

to kU(aþ ")f(aþ �)�U(aþ)f(aþ)kF for "4 0 and �4 0 and consider

k(U(aþ ")�U(aþ))f(aþ)kF as well as kU(aþ ")( f(aþ �)� f(aþ))kF. For the first

term obviously the order of the limit processes is arbitrary, for the second

term this follows from the existence of U(aþ)2L in the strong sense and since

U(t)2L, t2 (a, b).(iii) In the case that 0��5 1/p, the entire right-hand side in (6) equals the integral

with just f in place of faþ (then without correction terms).(iv) If real Banach spaces are considered, the integrals given by (6) are real-valued,

due to the definition of the fractional derivatives.

The integrals defined this way extend Riemann–Stieltjes integrals in Banach spaces as

studied in [19], see Appendix 2.

Remark 3.3 In part II (U(t)t�0 will be a suitable operator semigroup on E. Under

hypotheses more familiar than those of Definition 3.1, a related type of integral will be

defined, see Remark 5.5 in Section 5.

3.2. A special case

Let E0 denote the dual space of E. For g2E0 let hf, gi denote the dual pairing of f2E and

g2E0. Given a function g : (a, b)!E0.

UðtÞ :¼ �, gðtÞ� �

, t 2 ða, bÞ, ð7Þ

defines a bounded operator-valued function U : (a, b)!E. We assume there is

some g(aþ)2E0 such that hf, g(aþ)i¼ lim�!0 hf, g(aþ �)i for any f2E and put

gaþ(t)¼ 1(a,b)(t)(g(t)� g(aþ)). Specifying Definition 3.1 to this case with

g : (a, b)!E0. gb� 2 I1��b� ðLqðða, bÞ,E0ÞÞ with �, p, q and f as before, the forward integral

according to (6) equals

Z b

a

f ðtÞ, dgðtÞ� �

:¼ ð�1Þ�Z b

a

D�aþfaþðtÞ,D

1��b� gb�ðtÞ

� �dt

þ f ðaþÞ, gðb�Þ� �

� f ðaþÞ, gðaþÞ� �

: ð8Þ

For E¼E0 ¼R we arrive at the integral considered in [14] and [15].



3.3. Average integrals

Finally, we state two limit representations and define an average version.

LEMMA 3.4 Let 1/pþ 1/q� 1.

(i) Under the assumptions of Definition 3.1 we haveZ b

a

f ðtÞdUðtÞ ¼ lim"!0

Z b

a

I"aþf ðtÞdUðtÞ:

(ii) Suppose 05�5 1, 05 "5�, f : (a, b)!E is such that f 2 I��"aþ ðLpðða, bÞ,EÞÞ,�p 6¼ 1, and U: (a, b)!L is such that Ub� 2 I1��b� ðLqðða, bÞ,LÞÞ. Then we haveZ b

a

I"aþf ðtÞdUðtÞ ¼ð1� "Þ

�ð"Þ

Z 10

t"�1Z b

a

Ub�ðsþ tÞ �Ub�ðsÞð Þf ðsÞ dsdt

t,

the integralR10 taken in the sense of principal values.

The proof of the lemma resembles that of the scalar-valued case, we refer to [15],Lemmas 4.1 and 4.2.

In view of the asymptotics of the Gamma function " may replace 1/�(") whenconsidering the limit as "! 0. We set

ðAÞ

Z b

a

f ðtÞdUðtÞ :¼ lim"!0

"

Z 1

0

t"�1Z b

a

Ub�ðsþ tÞ �Ub�ðsÞð Þf ðsÞdsdt

tð9Þ

for measurable functions f : (a, b)!E andU : (a, b)!Lwhenever the right-hand side is welldefined and exists. This is an extension of Definition 3.1. The notation (A) means ‘average’.

4. Gradient-type integrals via duality

This section introduces forward integrals of Rn-valued fields f w.r.t. R-valued fields g over

smooth bounded domains D�Rn. The special case n¼ 1 yields the forward integral as

familiar from [12].First some preliminary facts about function spaces are recalled. Then the forward

integral is defined and using simple Fouriermultiplier arguments, conditions sufficient for itsexistence are proved. Finally, limit representations are shown, leading to an average version.

Note that below we use spaces of complex-valued functions and distributions, but allresults remain valid if the functions are taken to be real-valued.

4.1. Preliminaries

S(Rn) denotes the space of Schwartz functions and S0(Rn) the space of tempereddistributions on R

n. The Fourier transform of f2S(Rn) or S0(Rn) is denoted by f � f ^,its inverse by f � f _. For 15 p51 and �2R, the Bessel potential spaces of order� are defined by

H�pðR

nÞ :¼ f 2 S0ðRn

Þ : fjH�pðR

nÞ

�� 51o,n

fjH�pðR

nÞ

�� :¼ ð1þ j�j2Þ�=2f^ _

jLpðRnÞ

�� : ð10Þ



Here j � j is used for the Euclidean norm on Rn. For � 2R, the linear operator I�,

I� f ¼ ð1þ j�j2Þ�=2f^

_, ð11Þ

is an isomorphism of H�pðR

nÞ onto H��

p ðRnÞ. For �4 0, (1þ j�j2)��/2 is the Fourier

image of the Bessel kernel G�2L1(Rn) and I��f, f2Lp(R

n), 15 p51, accordingto (11) is realized as the convolution G� f :¼G� * f. The dual space of H�

pðRÞ is H��p0 ðRÞ,

1/pþ 1/p0 ¼ 1. For u 2 H�pðR

nÞ and v 2 H��p0 ðR

nÞ, the dual pairing of u and v is given by

hu, vi and

u, vh ij j � c ujH�pðR

nÞ

�� vjH��p0 ðRnÞ

�� : ð12Þ

For our purposes it seems convenient to introduce also another type of space. Let{e1, . . . , en} denote the standard basis in R

n. For x¼ (x1, . . . , xn) and fixed l¼ 1, . . . , n, writexl0 ¼ (x1, . . . ,xl�1, xlþ1, . . . , xn) for the (n� 1)-vector obtained from x by disposing the

coordinate xl, and identify x with (xl0,xl), x¼ (xl

0, xl). For l¼ 1, . . . , n fixed f � f ^l denotesthe partial Fourier transform of f with respect to xl, i.e.

f^ lð�Þ ¼ ð2�Þ�1=2Z

Rl

eixl�l f ð�0l, xlÞdxl , f 2 SðRnÞ, � 2 R,

where Rl :¼ span {el}. f � f _l denotes its inverse. It follows that for a C1-function m(�l)depending only on �l and being of polynomial growth, we always have(m(�l)f

^l)_l¼ (m(�l)f^)_, where f2S(Rn) or S0(Rn), see [22]. By H�

p,lðRnÞ, 15 p51,

�2R, we denote the space

H �p,lðR

nÞ :¼ f 2 S0ðRn

Þ : f jH�p,lðR

nÞ

�� 51o,nwhere

f jH�p,lðR

nÞ

�� :¼ ð1þ �2l Þ�=2f^

_jLpðR

nÞ

�� : ð13Þ

For �� 0, H�p ðR

nÞ is continuously embeded in H�

p,lðRnÞ, for �5 0 we have the converse

embedding. The space S(R) is dense in both spaces. Given �l2R, l¼ 1, . . . , n, put� ¼ ð�1, . . . ,�nÞ and

H�p ðR

nÞ :¼

\nl¼1

H�jp,lðR

nÞ,

which are Banach spaces if normed by k f jH�p ðR

nÞk :¼

Pnl¼1 k f jH

�lp,lðR

nÞk.

4.2. Forward integrals

Let D�Rn be a bounded domain. Let f¼ ( f1, . . . , fn) be an R

n-valued vector field on Rn

and g an R-valued function on Rn. For fixed l¼ 1, . . . , n denote the ‘forward differences’ of

g in direction el by

@þl,rgðxÞ :¼1

rgðxþ relÞ � gðxÞð Þ , r4 0: ð14Þ



Define the ‘forward pre-gradient’ rþr g, r4 0, of the function g by

rþr gðxÞ :¼ @þ1,rgðxÞ, . . . , @þn,rgðxÞ� �

, r4 0: ð15Þ

Writing h�, �i for the standard scalar product, we have hf ðxÞ,rþr gðxÞi ¼Pn

l¼1 flðxÞ, @þl,rgðxÞ

for f¼ ( f1, . . . , fn).

Definition 4.1 Given an Rn-valued field f¼ ( f1, . . . , fn) on R

n and an R-valued function g

on Rn, the ( partial) forward integral of f w.r.t. to g on D is defined as the limitZ

D

f ðxÞ,rþgðxÞ� �

dx :¼ limr!0

ZD

f ðxÞ,rþr gðxÞ� �

dx, ð16Þ

whenever it exists. We also useRDhf, r

þgidx to denote this integral.

Examples 4.2 Suppose n¼ 1, D¼ (a, b), f and g both are defined on R, g continuous at b.

Let Z b

a

fd�g :¼ limr!0

Z b

a

f ðxÞgbðxþ rÞ � gbðxÞ

rdx

with gb :¼ 1(a,b)(x) (g(x)� g(b)) denote the forward integral of f w.r.t. g as introduced

in [12], whenever the limit exists. ThenZ b

a

fd�g ¼

Zða,bÞ

f,rþg� �

dx:

We prefer þ, indicating the right-sided derivative instead of the traditional ‘�’ referring

to the integration process.

4.3. Existence conditions

As before, let f¼ ( f1, . . . , fn) be an Rn-valued field on R

n and g a R-valued function on Rn.

Usually f and g will not be differentiable, but members of some function spaces. Denoting

by 1D the indicator function of D, we first record the following:

LEMMA 4.3 Let D�Rn be a bounded C1-domain and h 2 H�

pðRnÞ with 15 p51,

05�5 1/p. Then

1DhjH�p ðR

nÞ

�� c hjH�p ðR

nÞ

�� with a constant c4 0 independent of h.

This is proved in [23]. Here it is not necessary that D is C1, C1 would be sufficient.

Consider the gradient rg of g,

rg :¼@g

@x1, . . . ,

@g

@xn

� �,



the partials taken in distributional sense. If g is such that for some �l4 0,@g@xl2 H��lp0, l ðR

nÞ � H��lp0 ðR

nÞ, and if f is such that fl 2 H�l

p ðRnÞ with 05�l5 1/p,

l¼ 1, . . . , n, then h1Dfl,@g@xli may be seen as dual pairing according to (12). We write

1Df,rg� �

:¼Xnl¼1

1Dfl,@g

@xl

� �: ð17Þ

By 1 we denote the n-vector (1, . . . , 1). In this situation we can slightly refine (12) to obtain:

PROPOSITION 4.4 Let 15 p51, 1/pþ 1/p0 ¼ 1 and let � ¼ ð�1, . . . ,�nÞ fulfil 05�l� 1/p,

l¼ 1, . . . , n. Suppose f¼ ( f1, . . . , fn) is such that fl 2 H�lp ðR

nÞ and g 2 H1��

p0 ðRnÞ. Then the

forward integral (16) exists and with notation (17),ZD

f,rþg� �

dx ¼ 1Df,rg� �

:

Moreover, the estimateZD

f,rþg� �

dx

� c

Xnl¼1

fljH�lp ðR

nÞ

�� gjH1��lp0, l ðR

nÞ

�� holds.

The proof can be found in Appendix 1.

Remark 4.5 As in Definition (9), we may also consider the average limit corresponding to

(16): Given two functions f :Rn!R

n and g : Rn!R, set

ðAÞ

ZD

f,rþg� �

dx :¼ lim"!0

"

Z 1

0

r"�1ZD

f ðxÞ,rþr gðxÞ� �

dx dr, ð18Þ

whenever the right-hand side exists. (A) stands for ‘average’. As the existence of (16)

implies that of (18), the latter extends the first. For n¼ 1, D¼ (a, b) and with f, g according

to Examples 4.2, (18) reads

lim"!0

"

Z 1

0

r"�1Z b

a

f ðxÞgðxþ rÞ � gðxÞ

rdxdr:

Recall the definition of the average integral via fractional calculus, (9). Specifying the case

addressed in (7) and (8) further to such f, g : R!R, these special cases of (18) and (9) yield

the same.

5. Hybrid integrals

The aim of this section is to define a Stieltjes-type integral for functions of variables

(t, x)2 (a, b)�D, where (a, b) is a finite interval and D a bounded C1-domain in Rn. We

‘mix’ the above constructions and use the approach via fractional calculus as in Section 3

taken with respect to the variable t2 (a, b) and the approach via forward differences

and duality as in Section 4 for the variable x2D. Though seeming peculiar at first sight,



this construction suits later studies of partial differential equations. There t will denote thetime and x the space parameter.

In the special case where D itself is an interval, Stieltjes-type integrals can also beconstructed using two-parameter fractional calculus. This can be carried through alongthe lines of [14], we refer to [21,24,25] and to [26] for an application to PDEs.

Below the hybrid integral is defined, and limit statements are deduced, needed fora representation of Ito-type integrals in Section 7. Functions depending on t and x willrepeatedly be seen as vector-valued functions of t. If not mentioned otherwise, fractionalcalculus always applies to the variable t2 (a, b). For the hybrid function spaces involved,we use the shortcut notation

H�,�aþ,p :¼ I �aþðLpðða, bÞ,H

�pðR

nÞÞ , ð19Þ

and

H1��,1��b�,p0 :¼ I�b�ðLp0 ðða, bÞ,H

1��p0 ðR

nÞÞ : ð20Þ

Assumptions Let f¼ ( f1, . . . , fn): (a, b)�Rn!R

n and g : (a, b)�Rn!R be given. D is

a bounded C1-domain in Rn. We formulate conditions under which we introduce

the hybrid integral. Assume that with some 15 p51, 1/pþ 1/p0 ¼ 1 and 0��, �l� 1,�l5 1/p, l¼ 1, . . . , n the following holds:

(I) fl and g possess the strong limits fl(aþ), g(aþ) and g(b�) in H�lp ðR

nÞ and

H1��p0 ðR

nÞ, respectively.

(II) fl,aþ 2 H�,�laþ,p

and gb� 2 H1��,1��b�,p0

, where fl,aþ(x) :¼ 1(a,b)(x)( fl(x)� fl(aþ)) is as inSection 3, and gb� is defined similarly.

Certain Holder properties imply these assumptions at once, they are known to hold a.s. forsamples of some random fields, see the next section. For (t, x)2 (a, b)�R

n, u2R smallenough in modulus and r2R

n, let

�u,r’ðt, xÞ :¼ ’ðtþ u, xþ rÞ � ’ðtþ u, xÞ � ’ðt, xþ rÞ þ ’ðt, xÞ ð21Þ

denote the ‘rectangular’ increments of a function ’ on (a, b)�Rn. j�jn denotes the

Euclidean norm in Rn, n is suppressed from notation if n¼ 1.

LEMMA 5.1 Let U�R be an open neighbourhood of [a, b], f :U�Rn!R

n and g :U�R

n!R

n, and assume there is a compact set K�Rn such that for any t2U, the supports

of f and g are contained in K. Let 05�005�5�05 1 and 05�l005�l5�l

05 1,l¼ 1, . . . , n. Suppose that both f and g fulfil the multiple Holder conditions

�u,r flðt, xÞj � cjuj�0

jrj�0l

n , ð22Þ

and

�u,selgðt, xÞ � cjuj1��

00

jsj1��00l , ð23Þ

as well as the simple Holder conditions

fl ðt, xþ rÞ � fl ðt, xÞ � cjrj

�0l

n , fl ðtþ u,xÞ � fl ðt, xÞ � cjuj�

0

, ð24Þ



and

gðt, xþ selÞ � gðt, xÞ � cjsj�

00l , gðtþ u, xÞ � gðt, xÞ

� cjuj�00

, ð25Þ

for all (t, x)2 [a, b]�Rn, small s, u2R, r2R

n and with a universal positive constant c. Then

assumptions (I) and (II) are fulfilled.

The proof is sketched in Appendix 1.

Definition 5.2 Suppose the functions f and g are as specified above and fulfil (I) and (II).

We define the hybrid (forward) integral of f w.r.t. g over (a, b)�D byZ Zða,bÞ�D

f,@

@tr

� �þg

� �dðt, xÞ :¼ ð�1Þ�

Z b

a

ZD

D�aþfaþðtÞ,r

þD1��b� gb�ðtÞ

� �dxdt

þ

ZD

f ðaþÞ,rþgðb�Þ� �

dx�

ZD

f ðaþÞ,rþgðaþÞ� �

dx,ð26Þ

the integrals over D with meaning as in (16). Here rþ refers to x only.

Remark 5.3 The definition is correct, i.e. the value of the integral is independent of the

values � and �l. For �l this follows from the proof of Proposition 4.4, for � it is similar to

the one-parameter case [14].

PROPOSITION 5.4 Under the assumptions of Definition 5.2, the integral in (26) exists.

The spatial forward limit may then be replaced by the corresponding distributional

derivative.

In view of the norms (4) and (10), this is an immediate consequence of Proposition 4.4

together with Holder’s inequality.

Remark 5.5 Recall (2). Our definition of the integral operator Itðu,@2Z@t@xÞ in part II, [7], will

be such that in terms of the transition densities p(t, x, y) of the heat semigroup (P(t))t�0,

it could at least formally be written in the above notation asZ Zð0,tÞ�ða,bÞ

,@2

@t@y

� �þZ

* +d ðs, yÞ,

where (s)¼ p(t� s, x, �)u(s, �). Under suitable assumptions, this will be interpreted asZ t

0

Zða,bÞ

D�0þ ðsÞ

@

@yD1��

t� ZtðsÞdy ds

and can be evaluated by means of semigroup theory.

5.1. Average integrals

We wish to obtain an average version of (26), a special case of which will be used to

represent the two parameter Ito integral in Section 7. The average integral is obtained by

means of averaging with respect to the variable t.



LEMMA 5.6 Under the assumptions of Definition 5.2, we haveZ b

a

ZD

D�aþf ðtÞ,r

þD1��b� gb�ðtÞ

� �dxdt ¼ lim

"!0

Z b

a

ZD

D��"aþ f ðtÞ,rþD1��

b� gb�ðtÞ� �

dxdt:

This follows straightforward using Propositions 4.4 and 5.4 together with the triangleinequality and the continuity properties of fractional integrals. We now use the additionalassumption that

(III) For a.e. x2D, g(b�, x) exists and equals g(b�)(x).

Under the hypotheses of Lemma 5.1, (III) is guaranteed.

LEMMA 5.7 Let ", r4 0, 05�5 1 and 05�5 1/p. Suppose (I) and (III) are valid,fl 2 H

��",�laþ,p , l¼ 1, . . . , n, and gb� 2 H

1��,1��b�,p0 . Then

1� "

�ð"Þ

Z 10

u"�1Z b

a

ZD

f ðtÞ,rþrgb�ðtþ uÞ � gb�ðtÞ

u

� �dx dt du

¼ ð�1Þ�Z b

a

ZD

D��"aþ f ðtÞ,rþr D

1��b� gb�ðtÞ

� �dxdt: ð27Þ

For the sake of legibility we formulate the proof only for the case n¼ 1 and D¼ (a0. b0),it can be found in Appendix 1. The case of general n follows by simple modifications.

Similar to the former sections we now define an average integral. For a functionh: (0, 1)� (0, 1)!R we write limj(",r)j!0 h("(t), r(t)) if this limit exists and is independent ofthe particular path on which (", r) tends to the origin. Here j � j denotes the Euclidean normin R

2. Set

ðAÞ

Zða,bÞ�D

f,@

@tr

� �þg

� �dðt, xÞ

:¼ limjð",rÞj!0

"

Z 1

0

u"�1Z b

a

ZD

f ðtÞ,rþrgb�ðtþ uÞ � gb�ðtÞ

u

� �dx dt du: ð28Þ

In view of Lemma 5.6 and 5.7, (28) is an extension of (26).

6. Pathwise integration of fractional Brownian sheets

In the course of this section, we survey some random fields and their path regularityand integrate them against each other in purely pathwise sense. Let (�,F,P) be aprobability space.

6.1. Fractional Brownian fields

An R-valued fractional Brownian field B�¼ {B�(x): x2Rn} of order 05�5 1 on R

n isa random field B�: ��R

n!R such that B� is Gaussian with mean zero, and the

covariance function is given by

E B�ðxÞB�ð yÞ� �

¼1

2xj j2�n þ y

2�n� y� x 2�

n

� �, ð29Þ



x, y2Rn. Recall that j�jn denotes the Euclidean norm on R

n. (29) in particular implies that

a.s. B�(0)¼ 0. We refer to [6]. For �¼ 1/2 we obtain Levy’s n-parameter Brownian motion,

for n¼ 1 the fractional Brownian motion, see [27,28]. From the covariance structure (29)

one easily deduces that a.s. the paths of a suitable modification of B� are �0-Holder

continuous on any compact set K�Rn and for any 05�05�.

6.2. Anisotropic fractional Brownian sheets

An R-valued anisotropic fractional Brownian sheet B� ¼ fB�ðt; xÞ : ðt; xÞ 2 Rng on R

n of

orders � ¼ ð�1, . . . ,�nÞ, 05�l5 1, l¼ 1, . . . , n, is a random field B� : ��Rn! R such

that B� is Gaussian with mean zero, and the covariance function is given by

E B�ðt, xÞB�ðs, yÞ� �

¼Ynl¼1

1

2xlj j

2�lþ yl 2�l� yl � xl

2�l� �,

s, t2R, x, y2Rn. In particular B�ð0Þ ¼ 0 a.s. We refer to [1–3]. The case �l¼ 1/2,

l¼ 1, . . . , n, yields the Brownian sheet on Rn, see e.g. [4].

6.3. (Hybrid) fractional Brownian sheets

We mix the constructions and obtain Gaussian fields that will serve as standard examples

in part II. An R-valued (hybrid) fractional Brownian sheet B�,�¼ {B�,� (t, x): (t, x)2Rnþ1}

of orders 05�, �5 1 on Rnþ1 is a random field B�,�: ��R

nþ1!R such that B�,� is

Gaussian with mean zero, and the covariance function is given by

E B�,�ðt, xÞB�,�ðs, yÞ� �

¼1

2tj j2�þ sj j2�� t� sj j2� 1

2xj j2�n þ y

2�n� y� x 2�

n

� �, ð30Þ

s, t2R, x, y2Rn. Here j�jn denotes the Euclidean norm on R

n and j�j the absolute value

on R. Again B�,�(0)¼ 0 a.s. For the case n¼ 1 we obtain an anisotropic sheet on R2.

In [1] it was shown that anisotropic sheets B�,� on R2 have stationary rectangular

increments, i.e. the law of the random variables

�u,rB�,�ðt, xÞ :¼ B�,�ðtþ u,xþ rÞ � B�,�ðtþ u, xÞ � B�,�ðt, xþ rÞ þ B�,�ðu, xÞ,

u2R, r2R, does not depend on (t, x)2Rnþ1. Their arguments easily carry over to

anisotropic sheets B� on Rn or B�,� on R

nþ1, as well as to hybrid sheets B�,� on Rnþ1. Using

well-known moment properties of Gaussian random variables, and following the lines of

[1] and [9] we deduce:

LEMMA 6.1 Let K�Rn be compact and [a, b]�R as before.

(1) The hybrid sheet B�,� on Rnþ1, 05�, �5 1, possesses a modification, again denoted

by B�,�, whose paths on [a, b]�K a.s. fulfil the Holder conditions (22) and (24) in place

of fl for all 05�05� and 05�05�. The constant c there possibly depends on !.(2) The anisotropic sheet B�,� on R

nþ1, 05�, �l, l¼ 1, . . . , n, � ¼ ð�1, . . . ,�nÞ, possessesa modification, again denoted by B�,�, whose paths on [a, b]�K a.s. fulfil the Holder

conditions (23) and (25) in place of g for all 05�005� and 05�l005�l.



Lemma 6.1 can be combined with Lemma 5.1 by a simple smooth cut-off w.r.t. the

Rn-variable x: Choose K such that it contains an open neighbourhood of D. Let

’ 2 C10 ðRnÞ be such that it is supported in K, 0� ’(x)� 1, x2R

n and ’(x)¼ 1 for x 2 D.

Obviously the product ’B�,�(!) still satisfies the mentioned Holder conditions for

P-a.e. !2�.This allows to state the following pathwise result.

COROLLARY 6.2 Let the vector B¼ (B�,�1, . . . ,B�,�n) consist of n hybrid fractional Brownian

sheets B�,�l over (�,F,P) with 05�, �l5 1, l¼ 1, . . . , n. Suppose B�0, �0 is an anisotropic

fractional Brownian sheet over the same probability space, such that �04 1� � and

�l04 1� �l, l¼ 1, . . . , n.Then the integral Z Z

ða,bÞ�D

B,@

@tr

� �þB�0, �0

� �dðt, xÞ ð31Þ

exists P-a.s.

Proof Choose a smooth function ’ as above and some p such that �l5 1/p for all

l¼ 1, . . . , n. Then for a.e. !, ’B(!)¼ (’B�,�1 (!), . . . , ’B�,�n(!)) and ’B�0, �0 ð!Þ in place of f,

and g respectively, fulfil the hypotheses of Definition 5.2 by Lemma 5.1 and the above

discussion. For P-a.e. !2�, the existence ofZ Zða,bÞ�D

’Bð!Þ,@

@tr

� �þB�0, �0 ð!Þ

� �dðt,xÞ

follows. The a.s. existence of the integral (31) follows, its independence of the choice of

’ is obvious. g

Similar arguments yield a correspond result for integrals over D only.

7. Stochastic integrals

We consider stochastic versions of the integrals (18) and (28), where the integrators are

given by the n-parameter Brownian motion and the hybrid Brownian sheet, respectively.

For bounded, continuous and predictable integrandsH they will be shown to coincide with

corresponding integrals of (partial) Ito type.Again (�,F,P) is a given probability space. We assume all processes to be real-

valued, the extension to complex-valued is straightforward. Throughout this section,

D denotes a bounded and convex domain in Rn. We remark that under some con-

ditions on the boundary, more general domains can be considered. We use the identi-

fication x¼ (xl0, xl), where xl

0 ¼ (x1, . . . , xl�1, xlþ1, . . . , xn). For any fixed l¼ 1, . . . , n and

xl0 2R

n�1, set

dþl ðx0lÞ :¼ inf r 2 R : ðx0l, relÞ 2 @D

� �,

d�l ðx0lÞ :¼ inf r4 dþl ðx

0lÞ : ðx

0l, relÞ 2 @D

� �



and �l ðx0lÞ :¼ d�l ðx

0lÞel. Finally, put

PlðDÞ :¼ x0l 2 Rn�1 : 9xl 2 R

n such that ðx0l, xlÞ 2 D� �

:

7.1. (Partial) stochastic integrals on D

As mentioned, setting �¼ 1/2 in (29), we arrive at the n-parameter Brownian motion W on

Rn. From the covariance structure it follows that for l¼ 1, . . . , n and xl

0 2Pl fixed,

{WD(xl0, xl) : xl2R} with

WDðx0l, xlÞ :¼Wðx0l, xlÞ �Wðx0l, þl ðx

0lÞÞ

is a Brownian motion on R, ðx0l, þl ðx

0lÞÞ playing the role of the origin.

For l¼ 1, . . . , n and xl0 2Pl fixed set

Fx0l

xl :¼ � WDðx0l, ylÞ : dþl ðx0lÞ � yl � xl

:

Let Px0l denote the �-field on ½dþl ðx

0lÞ,1Þ �� generated by integrands of form

hð!, xlÞ ¼ Xð!Þ1ð yl,zlðxlÞ or h0ð!, xlÞ ¼ X0ð!Þ1 dþlðx0

lÞf gðxlÞ,

where !2�, X is Fx0l

yl -measurable, X0 is Fx0l

dþlðx0

lÞ-measurable and dþl ðx

0lÞ � yl 5 zl. We set

Pl :¼ \x0l2PlPx0

l and call a random field H¼ (H1, . . . ,Hn) on Rn predictable w.r.t. D if Hl is

Pl-measurable for all l¼ 1, . . . , n. For our purposes it seems convenient to assume that

H¼ (H1, . . . ,Hn) is defined on the whole of Rn. H is called square integrable if for all

l¼ 1, . . . , n and all xl0 2Pl, Ef

R d�lðx0

lÞ

dþlðx0

lÞHlðx

0l, xlÞ

2dxlg51. For predictable and square

integrable H, each integralZ½dþ

lðx0

lÞd�

lðx0

lÞ

Hlðx0l, xlÞdW

Dðx0l, xlÞ , l ¼ 1, . . . , n , x0l 2 Pl,

is well defined in the Ito sense and fulfils a corresponding Ito isometry. We obtain

a (partial) Ito-type integral for such integrands on D by settingZD

H, dWh i :¼Xnl¼1

ZPl

Z½dþ

lðx0

lÞ, d�

lðx0

lÞ

Hlðx0l, xlÞdW

Dj ðx0l, xlÞdx

0l: ð32Þ

Assuming in addition that H is bounded and a.s. continuous on @D�Rn, Proposition 1.1

in [12] shows that the forward integral equals the Ito integral, in our situation that meansZ½dþ

lðx0

lÞ, d�

lðx0

lÞ

Hlðx0l, xlÞdW

Dðx0l, xlÞ

¼ limr!0

1

r

Z d�lðx0

lÞ

dþlðx0

lÞ

Hlðx0l,xlÞ W

Dðx0l, xl þ relÞ �WDðx0l,xlÞ

dxl, ð33Þ

where the limit is taken in the mean square. The correction denoted by the superscript

Dmay be omitted in the difference, and with the notation of the former sections we considerZD

H,rþPW

� �dx :¼ lim

r!0

ZD

H,rþr W� �

dx, ð34Þ

the limit taken in square mean.



LEMMA 7.1 Let W denote the n-parameter Brownian motion on Rn and let

H¼ (H1, . . . ,Hn) be a bounded random field on Rn, predictable w.r.t. D and a.s. continuous

on @D�Rn. Then the limit (34) exists and equals the Ito-type integral (32),Z

D

H,rþPW

� �dx ¼

ZD

H, dWh i:

7.2. Stochastic integrals on ½a, b �D

Setting �¼ �¼ 1/2 in (30), we obtain a Gaussian field W¼ {W(t,x): t2R, x2Rn}, which

might be called the hybrid Brownian sheet on Rnþ1. A corresponding (partial) Ito-type

integral on ½a, b �D can be defined following the construction of the Ito integral in the

plane, see e.g. [4,17,18]:For s, t2R and x, y2R

n, we write (t, x) (s, y) if and only if t� s and xl� yl,

l¼ 1, . . . , n. We write (t, x) (s, y) if all inequalities are strict.Fix some l and xl

0 2Pl as above, then {WD(t, xl0.xl): t2R, xl2R} with

WDðt,x0l,xlÞ :¼Wðt, x0l, xlÞ �Wðt, x0l, þl ðx

0lÞÞ

is a Brownian sheet on R2 and ð0, x0l,

þl ðx

0lÞÞ plays the role of the origin. With l and xl

0 2Pl

still fixed we set

Fx0l

t,xl :¼ � WDðt,x0l, ylÞ : ð0, dþl ðx0lÞÞ ðs, ylÞ ðt, xlÞ

:

Now let Px0l denote the �-field on ½a,1Þ � ½dþl ðx

0lÞ,1Þ �� generated by integrands

of form

hð!, t,xlÞ ¼ Xð!Þ1ðc,dðtÞ1ð yl,zlðxlÞ, ð35Þ

where !2�, X is Fx0l

ðc;ylÞ-measurable, ða, dþl ðx

0lÞÞ ðc, ylÞ ðzl, dÞ, together with inte-

grands of a similar form but with 1{0}(t) in place of 1(c,d](t) or 1fdþlðx0

lÞgðxlÞ in place

of 1(yl, zl](xl) or both, considered under the respective measurability assumptions. Put

again Pl :¼ \x0l2PlPx0

l and call a random field H¼ (H1, . . . ,Hn) on Rnþ1 predictable

w.r.t. (a, b)�D if Hl is Pl-measurable for all l¼ 1, . . . , n. H is called square integrable

if for all l¼ 1, . . . , n and all xl0 2Pl, Ef

R Rða,bÞ�ðdþ

lðx0

lÞ, d�

lðx0

lÞÞHlðt, x

0l, xlÞ

2dðt, xlÞg51. For

predictable and square integrable H, each integralZ Z½a,b�½dþ

lðx0

lÞ, d�

lðx0

lÞ

Hlðt, x0l, xlÞdWðt, x

0l, xlÞ ð36Þ

is well defined as two-parameter Ito integral (of the first kind). For integrands of form (35)

it equals X�d�c,zl�ylW(c, xl

0, yl), the difference according to (21) referring to c and yl.

The usual isometry property holds at the level of (36). SettingZ Z½a,b�D

H, dWh i :¼Xnl¼1

ZPl

Z Z½a,b�½dþ

lðx0

lÞ, d�

lðx0

lÞ

Hlðt, x0l, xlÞdWðt, x

0l, xlÞdx

0l , ð37Þ

we obtain an integral of (partial) Ito type.



Now consider the limit

limjðu,rÞj!0

Z b

a

ZD

Hðt, xÞ,rþrWbðtþ u, xÞ �Wbðt,xÞ

ur

� �dxdt , ð38Þ

taken in the mean square and, whenever it exists. The correction Wb(t, x)¼ 1(a,b)(t)�(W(t, x)�W(b, x)) w.r.t. (a, b) is understood pathwise. For u, r4 0 fixed a member of thesequence in (38) equals

Xnl¼1

Z Zða,bÞ�D

Hlðt, xÞ�u,relWbðt; xÞ

urdt dx ,

with �u,relWb(t, x) according to (21). Consider its average version

ðAÞ

Zða,bÞ�D

H,@

@tr

� �þP

W

� �dðt, xÞ

:¼ limjð",rÞj!0

"

Z 1

0

u"�1Z b

a

ZD

HðtÞ,rþrWbðtþ uÞ �WbðtÞ

u

� �dxdt du, ð39Þ

taken in square mean, whenever it exists. This is the stochastic variant of the pathwise(hybrid) average integral (28). Adapting the proofs known from the one-parameter case,[12,16], we see that for suitable integrands, it equals the Ito-type integral:

LEMMA 7.2 Let H¼ (H1, . . . ,Hn) be a random field on Rnþ1, predictable w.r.t. ½a, b �D,

a.s. bounded and continuous. Then:

(i) If limit (38) exists, so does the average limit (39), and both agree.(ii) The average limit (38) exists and equals the Ito-type integral (37).

Consequently also

ðAÞ

Zða,bÞ�D

H,@

@tr

� �þP

W

� �dðt, xÞ ¼

Z Z½a,b�D

H, dWh i:

(i) is an obvious modification of arguments from [16, p. 3]. For convenience, the proofof (ii) is sketched in Appendix 1 for the case n¼ 1, D¼ (a0, b0), where WD is the Browniansheet on R

2.We finally remark that in the case n¼ 1, [4] used a weakly adapted two-parameter

integral to study partial differential equations, contained in a construction similar tothe above using the filtrations F

x0l

t :¼ _dþlðx0

lÞ�xlF

x0l

t,xl . The above can easily be adaptedto this case.

References

[1] A. Ayache, S. Leger, andM. Pontier,Drap brownien fractionnaire, Pot. Anal. 17 (2002), pp. 31–43.[2] A. Kamont, On the fractional anisotropic Wiener field, Probab. Math. Stat. 16 (1996), pp. 85–98.[3] Y. Xiao, Sample path properties of anisotropic Gaussian random fields, in A minicourse on

stochastic partial differential equations, Lecture Notes in Math. Vol 1962, Springer, Berlin, 2009,

pp. 145–212.



[4] J.B.Walsh,An introduction to stochastic partial differential equations, in Ecole d’ete de probabilites

de Saint-Flour, XIV-1984, Lecture Notes inMath. Vol. 1180, Springer, Berlin, 1986, pp. 265–439.

[5] R.C. Dalang and N.E. Frangos, The stochastic wave equation in two spatial dimensions, Ann.

Probab. 26 (1998), pp. 187–212.

[6] T. Lindstrøm, Fractional Brownian fields as integrals of white noise, Bull. London Math. Soc. 25

(1993), pp. 83–88.

[7] M. Hinz and M. Zahle, Gradient type noises II – systems of stochastic partial differential

equations, preprint, University of Jena, 2008.[8] Z. Ciesielski, G. Kerkyacharian, and R. Roynette, Quelques espaces fonctionnels assccies a des

processus gaussiens, Studia Math. 107 (1993), pp. 171–204.[9] D. Feyel and A. De La Pradelle, Fractional integrals and Brownian processes, Pot. Anal. 10

(1999), pp. 273–288.[10] T.J. Lyons, Differential equations driven by Rough signals I: An extension of an inequality by L.C.

Young, Math. Res. Letters 1 (1994), pp. 451–464.[11] D. Nualart, Stochastic integration with respect to fractional Brownian motion and applications,

Contemp. Math. 336 (2003), pp. 3–39.[12] F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration, Probab.

Th. Relat. Fields 97 (1993), pp. 403–421.[13] L.C. Young, An inequality of Holder type, connected with Stieltjes integration, Acta Math. 67

(1936), pp. 251–282.[14] M. Zahle, Integration with respect to fractal functions and stochastic calculus I, Probab. Th. and

Relat. Fields 111 (1998), pp. 333–374.[15] M. Zahle, Integration with respect to fractal functions and stochastic calculus II, Math. Nachr.

225 (2001), pp. 145–183.[16] M. Zahle, Forward integrals and stochastic differential equations, in Seminar on Stochastic

analysis, Random fields and Applications III, Progress in Probability, R.C. Dalang, M. Dozzi,

and F. Russo, eds., Birkhauser, Basel, 2002, pp. 293–302.[17] R. Cairoli and J.B. Walsh, Stochastic integrals in the plane, Acta Math. 134 (1975), pp. 111–183.

[18] E. Wong and M. Zakai, Martingales and stochastic integrals for processes with

a multidimensional parameter, Z. Wahrsch. verw. Geb. 29 (1974), pp. 109–122.

[19] M. Gowurin, Uber die Stieltjessche Integration abstrakter Funktionen, Fund. Math. 27 (1936),

pp. 255–268.

[20] E. Hille and R.S. Phillips, Functional Analysis and Semigroups, AMS, Providence, R.I., 1957.[21] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives. Theory and

Applications, Gordon and Breach, London, 1993.[22] S.M. Nikolskij, Approximation of Functions of Several Variables and Embedding Theorems,

Springer, New York, 1975.[23] H. Triebel, Theory of function spaces, Geest and Portig, Leipzig, and Birkhauser, Basel, 1983.[24] Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, LNM

1929, Springer, New York, 2008.[25] C. Tudor and M. Tudor, On the two-parameter fractional Brownian motion and Stieltjes integrals

for Holder functions, J. Math. Anal. Appl. 286 (2003), pp. 765–781.

[26] M. Erraoui, D. Nualart, and Y. Ouknine,Hyperbolic stochastic partial differential equations with

additive fractional Brownian sheet, Stoch. Dyn. 3 (2003), pp. 121–139.

[27] J.-P. Kahane, Some random series of functions, 2nd ed., Cambridge University Press,

Cambridge, 1985.

[28] B.B. Mandelbrot and J.W. van Ness, Fractional Brownian motion, fractional Brownian noise and

applications, SIAM Rev. 10 (1968), pp. 422–437.

[29] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University

Press, Princeton, NJ, 1970.



Appendix 1. Proofs

Proof of Proposition 4.4 Given a2R, the translation Ta is defined by ( f �Ta)(x) :¼ f(xþ a).

Proof The estimate for the pairing is obvious, we need to verify existence and value of theforward integral. It suffices to consider a single summand in (17). We fix an integer 1� l� n andwrite � for �l. For r4 0 set

Ilð f, g, rÞ ¼

ZR

n1Dfl@

þl,rgdx ¼

ZR

n1DflðxÞ

g � Trel � g

rðxÞ dx:

First assume g2S(Rn). Then

Ilð f, g, rÞ ¼

Z 1

0

ZR

n1Dflð Þ

^ð�Þ eirt�l i�l gð�Þ d� dt:

For fixed t4 0, the inner integral in the last line equalsZR

nð1þ �2l Þ

�=2 1Dflð Þ^

_ðxÞ i�lð1þ �

2l Þ��=2 eirt�lel g

_ðxÞ dx

and by Holder’s inequality this is (componentwise) bounded above by

ð1þ �2l Þ�=2 1Dflð Þ

^ _

jLpðRnÞ

�� i�lð1þ �2l Þ��=2eirt�l g

_jLp0 ðR

nÞ

�� c ð1þ �2Þ�=2 1Dflð Þ

^ _

jLpðRnÞ

�� ð1þ �2l Þð1��Þ=2g _�Trtel jLp0 ðR

nÞ

�� ,note that ð1þ �2l Þ

�=2ð1þ �2Þ��=2 and �lð1þ �

2l Þ�1=2 are Fourier multipliers. Hence by Lemma 4.3 and

translation invariance of the Lp0(Rn)-norm,

Ilð f, g, rÞ � c fljH

�pðR

nÞkkgjH1��

p0 , l ðRnÞ

�� : ðA1Þ

For fixed 05 r5 r0 we similarly obtain

Ilð f, g, rÞ � Ilð f, g, r0Þ

� c fljH�pðR

nÞ

��

Z 1

0

ð1þ �2l Þð1��Þ=2g

_�Ttrel � ð1þ �

2l Þð1��Þ=2g

_�Ttr0el jLp0 ðR

nÞ

�� dt: ðA2Þ

Approximating g 2 H1��p0, l ðR

nÞ by functions ’j2S(R

n) in H1��p0 , l ðR

nÞ, (A1) and (A2) carry over, notice

that by Holder

Ilð f, g� ’j, r,wÞ � fljLpðR

nÞ

�� g � Trel � g

r�’j � Trel � ’j

rjLp0 ðR

nÞ

��

and ’j tends to g in Lp0(Rn).

Finally, the right-hand side of (A2) tends to zero as r and r0 do, since for any u2Lp0(Rn)

and h2R, limr!0ku� u(� þ rh)jLp0(Rn)k¼ 0. Therefore (16) exists. On the other hand, for f 2 H�

pðRnÞ

and g2S(Rn)

1Dfl,@g

@xl

� �� Ilð f,g, rÞ

� c fljH

�pðR

nÞk

Z 1

0

i�lð1þ �2l Þ��=2g

_� i�lð1þ �

2l Þ��=2g

_�Ttrel jLp0 ðR

nÞ

�� dt��tends to zero as r does by similar arguments, and completion yields the desired equality. g



Proof of Lemma 5.7 For n¼ 1 and D¼ (a0, b0). In this case the assertion reads

1� "

�ð"Þ

Z 10

u"�1Z b

a

Z b0

a0f ðt, xÞ,

�u,rgb�ðt, xÞ

urdxdt du

¼ ð�1Þ�Z b

a

Z b0

a0D��"

aþ f ðt, xÞ,D1��

b� gb�ðt, xþ rÞ �D1��b� gb�ðt, xÞ

rdx dt: ðA3Þ

Proof For u and r fixed, we may rewrite the inner integrals on the left-hand side of (A3) asZ b

a

Z b0

a0f ðt, xÞ

�u,rgb�ðt, xÞ

urdx dt

¼ ð�1Þ��"Z b

a

Z b0

a0D��"

aþ f ðt, xÞI ��"b� �u,rgb�ðt, xÞ

urdx dt, ðA4Þ

as Fubini’s theorem and the integration-by-parts formula for fractional derivatives (5) show. Next,note that for r4 0 fixed,

ð1� "Þð�1Þ1�"

�ð"Þ

Z 1�

I��"b� �u,rgb�ðt, xÞ

u2�"rdu,

seen as Marchaud derivative, converges to

�D1��


rðA5Þ

as � decreases to zero. By Fubini’s theorem and Holder’s inequality we may therefore rewrite theleft-hand side of (A3) as

ð�1Þ�Z b

a

Z b0

a0D��"

aþ f ðt, xÞ,D1��


rdxdt,

as desired. g

Proof of Lemma 5.1

Proof We verify (II) for f. In the latter case multiply g(b�) by a smooth cut-off function of x, whichequals one on a neighbourhood of D.

By standard embedding theorems, khjH�0

p ðRÞk is bounded by a constant times

hjLpðRÞ�� þ Z

R

ZR

hðxþ yÞ � hðxÞ p

y 1þ�0p dx dy

!1=p

ðA6Þ

in case p� 2. For p4 2 this has to be replaced by

hjLpðRÞ�� þ Z

R

ZR

hðxþ yÞ � hðxÞ pdx� �2=p

dy

y 1þ2�0

!1=2

, ðA7Þ

see e.g. [29]. For compactly supported h, Holder’s inequality quickly shows that (A7) is bounded bya constant times (A6) with �0 þ � in place of �0 for any �4 0. Hence it suffices to show that theLp((a, b))-norm of (A6), with

D�0

aþf ðt, xÞ ¼ c�01ða,bÞðtÞf ðt, xÞ

ðt� aÞ�0 þ �

0

Z t

a

f ðt, xÞ � f ð�, xÞ

ðt� �Þ�0þ1

d�

� �ðA8Þ



in place of h, is finite. An estimate of the first summand of (A8) together with the difference part of(A6) in Lp((a, b)) is given by

Z b

a

Z Zjx�yj5r0

Z a

t

jfðt; xÞ � fð�; xÞ � fðt; yÞ þ fð�; yÞ

ðt; �Þ1þ�0 d�

� �p

�1

jx� yj1þ�0pdx dy dt

�

Z b

a

Z Zjx�yj5r0

Z t

a

jt� �jð��0 Þ�1d�

� �p

jx� yjð��0 Þp�1dxdydt51,

we have used (22). The terms arising from combinations of the remaining summands of(A8) and (A6) obey similar estimates obtained from (22) and (24) and the fact that each f(t, �)has compact support.

(I) follows from the continuity of f at a, seen as H�0

p ðRÞ-valued function, which is shown byarguments similar to the above.

For g one can proceed similarly, it suffices to note that with l¼ 1, . . . , n fixed, Rl¼ span {el}, andR?l � R

n denoting its orthogonal complement,

gjH�lp0 , lðR

nÞ

�� ¼ ð1þ �2l Þ�l=2g

_jLp0 ðR

nÞ

�� ¼

ZR?l

gð�0l, �ÞjH�lp0 ðRÞ

�� p0d�0l !1=p0

,

where �l0 ¼ (�1, . . . , �l�1, �lþ1, . . . , �n) as before. g

Proof of Lemma 7.2 (ii) Assume n¼ 1, D¼ (a0, b0) in Lemma 7.2, then WD is the Brownian sheet.

Proof Notice first that if (38) exists with Wb,b0(t, x)¼W(t, x)�W(b, x)�W(t, b0)þW(b, b0) in placeof Wb(t, x), it exists in its original form. This follows using bounded convergence. Now follow [12]:For any u, u4 0 and any (t, x)2 [a, b]� [a0. b0],

�u,rWb,b0 ðt, xÞ ¼Wððtþ uÞ ^ b, ðxþ rÞ ^ b0Þ �Wðt, ðxþ rÞ ^ b0Þ �Wððtþ uÞ ^ b, xÞ þWðt, xÞ:

Now it follows that with Y(t, x) :¼H(t, x)1{(t,x)(s,y)(tþu,xþr)},Z Z½a,b�½a0 , b0

YdW ¼ Hðt, xÞ�u,rWb,b0 ðt, xÞ,

the right-hand side in the Ito sense. By Lemma A.1 below then

1

ur

Z½a,b�½a0, b0

Hðt, xÞ�u,rWb,b0 ðt, xÞdðt, xÞ ¼

Z Z½a,b�½a0, b0

�1

ur

Z½s�u,s�½y�r,y

Hðt, xÞ1½a,b�½a0 , b0 ðt, xÞdðt, xÞ

� �1½a,b�½a0 , b0 ðs, yÞdWðs, yÞ:

By the isometry, the expectation of the square of

Z Z½a,b�½a0, b0

1

ur

Z Z½s�u,s�½y�r,y

Hðt, xÞ1½a,b�½a0 , b0 ðt, xÞd ðt, xÞ �Hðs, yÞ

� �dWðs, yÞ

equals the expectation of

Z Z½a,b�½a0, b0

1

ur

Z Z½s�u,s�½y�r,y

Hðt, xÞ1½a,b�½a0, b0 ðt, xÞd ðt, xÞ �Hðs, yÞ

� �2

dðs, yÞ:



Now take into account that along any increasing path (u, r),

limjðu,rÞj!0

1

ur

Z½s�u,s�½y�r,y

Hðt, xÞ1½a,b�½a0, b0 ðt, xÞ d ðt, xÞ ¼ Hðs, yÞ a.s.

by the a.s. continuity of H. g

LEMMA A.1 Let H be a bounded and P�B([a, b]� [a0, b0])-measurable mapping on�� [a, b]� [a0, b0]� [a, b]� [a0, b0]. ThenZ

½a,b�½a0 , b0

Z½a,b�½a0, b0

Hðu, vÞdWðuÞdv ¼

Z½a,b�½a0 , b0

Z½a,b�½a0, b0

Hðu, vÞdvdWðuÞ:

Here B([a, b]� [a0, b0]) denotes the Borel �-field over ([a, b]� [a0, b0]). For integrands of form (35)this is obvious, for arbitrary H it follows by a monotone class argument, use the isometry togetherwith the bounded convergence theorem.

Appendix 2. Riemann–Stieltjes integrals in the sense of Gowurin

The following construction had been introduced in [19]. Let E and F be real Banach spaces normedby k�kE and k�kF, respectively, and L(E,F) the space of bounded linear operators from E into F,endowed with the operator norm. Let (a, b)�R be a bounded interval and consider twofunctions f : (a, b)!E and U : (a, b)!L(E,F). We assume throughout that the limits f(aþ),f(b�), U(aþ) and U(b�) exist in the respective spaces and set f(a) :¼ f(aþ), etc. below. LetP�¼ {ti: i¼ 0, . . . , k, a¼ t05 t15 � � �5 tk¼ b} with some k2N be a partition of (a, b) withmaxi jti� ti�1j5�. If the Riemann–Stieltjes sums

Sð f,U,P�Þ ¼Xki¼1

ðUðtiÞ �Uðti�1ÞÞf ð�iÞ½ ,

where �i2 [ti�1, ti], converge to a limit along a sequence of refining partitions P� of the abovetype as � goes to zero, this limit is called Riemann–Stieltjes integral (in the sense of Gowurin) anddenoted by

ðRS Þ

Z b

a

f dU :¼ lim�!0

Sð f,U,P�Þ :

A function U : (a, b)!L(E,F) is said to have the !-property on (a, b), if there exists someM4 0 suchthat for any partition P of the above type and any xi2E, i¼ 0, . . . , k� 1,kPk

i¼1ðUðtiÞ �Uðti�1ÞÞxikF �Mmaxi kxikE. Similar to the scalar-valued case, one can show that

ðRS ÞR b

a fdU exists if f is strongly continuous and U has the !-property. It also exists if U is strongly

continuous and f is of bounded variation on (a, b), i.e. there exists some M4 0 such that

supPPk

i¼1 k f ðtiÞ � f ðti�1ÞkE �M, the supremum taken over all partitions of (a, b).The concept extends to complex Banach spaces E and F in the coordinatewise sense.

The following simple lemma holds.

LEMMA A.2 Suppose U: (a, b)!L(E,F) is uniformly bounded on (a, b) and has the !-property.Assume �: (a, b)� (a, b)!E is a function such that for a.e. � 2 (a, b), �(�, �) is strongly continuous on(a, b)n{�} and

supt2ða,bÞ

Z b

a

�ðt, �Þ��

Ed�51 : ðA9Þ

Then both integrals below exist and



ðRS Þ

Z b

a

Z t

a

�ðt, �Þd� dUðtÞ ¼

Z b

a

ðRS Þ

Z b

�

�ðt, �ÞdUðtÞd�:

Proof By (A9), �ðtÞ ¼R ta �ðt, �Þd� is strongly continuous on (a, b) and with �i2 [ti�1, ti], the

right-hand side rewrites

Z b

a

�ðtÞdUðtÞ ¼ lim�!0

Xki¼1

Z �i

a

ðUðtiÞ �Uðti�1ÞÞ�ð�i, �Þd�

¼ lim�!0

Z b

a

Xki¼1

ðUðtiÞ �Uðti�1ÞÞ1½�þ",bÞð�iÞ�ð�i, �Þd�

þ lim�!0

Z b

a

Xki¼1

ðUðtiÞ �Uðti�1ÞÞ1ð�,�þ"Þð�iÞ�ð�i, �Þd�

ðA10Þ

for any "4 0. We have used the uniform boundedness of U in the first equality. By the !-propertyof U, for any a� c5 d� b,

Xki¼1

ðUðtiÞ �Uðti�1ÞÞ1ðc,dÞð�iÞ�ð�i, �Þ

��F

�Mmaxi

1ðc,dÞð�iÞ �ð�i, �Þ��

F

and by (A9),

Z b

a

Xki¼1

ðUðtiÞ �Uðti�1ÞÞ1ðc,dÞð�iÞ�ð�i, �Þ

��F

d�

�Mmaxi

Z b

a

1ðc,dÞð�iÞ �ð�i, �Þ��

Ed�51 :

Therefore the first summand in (A10) isR b

a

R b�þ" �ðt, �ÞdUðtÞd� , while the second is bounded by

M supt2ða,bÞR tðt�"Þ_a k�ðt, �ÞkEd�, which tends to zero as " does. g

With the aid of this lemma it can be seen that for suitable functions f and U the integralaccording to Definition 3.1 coincides with the Riemann–Stieltjes integral. Using Lemma A.2 one canfollow the lines of [14, Theorem 2.4.(i)] to obtain:

PROPOSITION A.3 Suppose U: (a, b)!L(E,F) has the !-property and f : (a, b)!E is stronglycontinuous. Assume further, f and U satisfy the hypotheses of Definition 3.1 and with some 0�� 1,

supt2ða,bÞ

Z b

a

D�aþfaþð�Þ

�� Eðt� �Þ��1d�51 :

ThenR b

a f dU ¼ ðRS ÞR b

a f dU.



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