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On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension
ARTICLE IN PRESS
Stochastic Processes and their Applications 115 (2005) 1764–1781 www.elsevier.com/locate/spa
On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension Latifa Debbia,b,c,, Marco Dozzia a
Institut Elie Cartan, B.P 239, Universite´ Henri Poincare´ Nancy1, Vandoeuvre-Le`s-Nancy, France b Department of Mathematics, Faculty of Sciences, University Ferhat Abbas, El-Maabouda Se´tif 19000, Algeria c Department of Mathematics, Faculty of Science, University Mohamed Boudiaf, B.P 166 Echbilia, M’sila 28000, Algeria Received 31 December 2004; received in revised form 11 May 2005; accepted 21 May 2005 Available online 27 June 2005
Abstract Existence, uniqueness and regularity of the trajectories of mild solutions of one-dimensional nonlinear stochastic fractional partial differential equations of order a41 containing derivatives of entire order and perturbed by space–time white noise are studied. The fractional derivative operator is defined by means of a generalized Riesz–Feller potential. r 2005 Elsevier B.V. All rights reserved. MSC: primary 26A33; 60H15; 60G60 Keywords: Fractional derivative operator; Stochastic partial differential equation; Space–time white noise; Ho¨lder continuity; Equation of high order
Corresponding author. Institut Elie Cartan, B.P 239, Universite´ Henri Poincare´ Nancy1, VandoeuvreLe`s-Nancy, France. Tel.: +333 83 68 45 15; fax: +333 83 68 45 34. E-mail addresses: [email protected], [email protected] (L. Debbi), [email protected] (M. Dozzi).
0304-4149/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spa.2005.06.001
ARTICLE IN PRESS L. Debbi, M. Dozzi / Stochastic Processes and their Applications 115 (2005) 1764–1781
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1. Introduction Fractional equations, both partial and ordinary ones, have received more attention in recent years. Various phenomena in physics, like diffusion in a disordered or fractal medium, or in image analysis, or in risk management have been modeled by means of fractional equations. Some of them are of high order [13]. We refer to [12] and the part II of [17] for recent surveys on these applications. However, comparatively few publications treat stochastic fractional partial differential equations. Most of them investigate evolution type equations, driven by a fractional power of the Laplacian [11,20,3 and the references therein]. These operators generate symmetric stable semigroups when the order of derivation is less than 2. Mueller [16] and Wu [19] proved the existence of a solution of stochastic fractional heat, respectively Burgers, equation perturbed by a stable noise. The work of Bonaccorsi and Tubaro [2] can be applied to stochastic evolution equations with fractional time derivative. Angulo et al. [1] have considered a linear partial differential equation driven by the composition of the inverses of Riesz and Bessel potentials and perturbed by a space–time white noise. In [3], Dalang and Mueller have studied the stochastic equation which is of second order in time, driven by a power Laplacian and perturbed by colored noise (white in time and homogeneous in space). The stochastic equation studied by Kotelenez [10] is driven by a pseudodifferential operator which covers the power Laplacian (symmetric case). The aim of this work is to generalize, in the framework of the multi-parameter processes, the results of Walsh [18] to nonlinear stochastic fractional partial differential equations (SFPDE) of high order containing also derivatives of entire order. We study the existence, uniqueness, and regularity of the solution of the equation formally given by m X qu qk hk ðt; xÞ ¼ x Dad uðt; xÞ þ gðt; x; uðt; xÞÞ þ ðt; x; uðt; xÞÞ qt qxk k¼1
þ f ðt; x; uðt; xÞÞ
q2 W ðt; xÞ; qtqx
t40; x 2 R,
uð0; xÞ ¼ u0 ðxÞ,
(1)
where ½a is the integer part of a, m 2 N and 1pmp½a. x Dad is the fractional differential operator with respect to the spatial variable, to be defined below. We suppose that a 2 ð1; þ1ÞnN and that the functions f ; g; hk : ½0; þ1Þ R R ! R satisfy Lipschitz and growth conditions: for all T40, there exists a constant K T 40 such that for all t 2 ½0; T and for all x; y; z 2 R, ðjhk ðt; x; yÞ hk ðt; x; zÞj þ jgðt; x; yÞ gðt; x; zÞj þ jf ðt; x; yÞ f ðt; x; zÞjÞ pK T jy zj, ðjhk ðt; x; zÞj þ jgðt; x; zÞj þ jf ðt; x; zÞjÞpK T ð1 þ jzjÞ.
ð2Þ
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Let ðO; F; ðFt ; tX0Þ; PÞ be a complete probability space, endowed with an increasing and right-continuous filtration ðFt ; tX0Þ. Let W ¼ fW ðt; xÞ; tX0; x 2 Rg be a centered Gaussian field which is white noise in time and in space with covariance function given by Kððt; xÞ; ðs; yÞÞ ¼ 14 ðsgnðxÞ þ sgnðyÞÞ2 ðt ^ sÞðjxj ^ jyjÞ, where sgn is the sign function. W is in fact composed of two independent Brownian sheets, one in the positive direction of the spatial variable and the other one in the negative direction. We suppose that W generates a ðFt ; tX0Þ-martingale measure in the sense of Walsh [18]. The initial condition u0 is supposed to be F0 BðRÞ measurable, where BðRÞ is the Borelian s-algebra over R. The fractional differential operator x Dad used in this paper is an extension of the inverse of the generalized Riesz–Feller potential [7,9] when a42. It is given for a40 by Definition 1. The fractional differential Dad jðxÞ is given by Dad jðxÞ ¼ F 1 fd ca ðlÞFfjðxÞ; lg; xg,
(3)
where d ca ðlÞ
p
¼ jlja e id2sgnðlÞ ,
(4)
jdjp minfa ½a2 ; 2 þ ½a2 ag, ½a2 is the largest even integer less or equal to a (even part of a), and d ¼ 0 when a 2 2N þ 1, and F (respectively F 1 ) is the Fourier (respectively Fourier inverse) transform. The operator Dad is a closed, densely defined operator on L2 ðRÞ and it is the infinitesimal generator of a semigroup which is in general not symmetric and not a contraction. This operator generalizes the fractional differential operators in [7,8,15] where 0oap2. It is selfadjoint only when d ¼ 0 and in this case, it coincides with the fractional power of the Laplacian. Evidently, when a ¼ 2 it is the Laplacian itself. Furthermore, it is proven in [5], that when jdj ¼ 2 þ ½a2 a or jdj ¼ a ½a2 , it coincides with the Riemann–Liouville differential operator. By [9], Dad can be represented for 1oao2, by Z þ1 jðx þ yÞ jðxÞ yj0 ðxÞ Dad jðxÞ ¼ ðM d 1ð 1;0Þ þ M dþ 1ð0;þ1Þ Þ dy, jyj1þa
1 and for 0oao1, by Z þ1 jðx þ yÞ jðxÞ Dad jðxÞ ¼ ðM d 1ð 1;0Þ þ M dþ 1ð0;þ1Þ Þ dy, 1þa jyj
1 where M d and M dþ are two nonnegative constants satisfying M d þ M dþ 40 and 1ð 1;0Þ and 1ð0;þ1Þ are the indicator functions of the intervals ð 1; 0Þ, respectively, ð0; þ1Þ, and j is a smooth function for which the integrals exist, and j0 is its derivative. For more details about this operator we refer to [5]. In the following definition we give a rigorous meaning of (1) and of its solution.
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Definition 2. Let 0oTo1. A measurable stochastic field fu ¼ uðt; xÞ; t 2 ½0; T; x 2 Rg is said to be a mild solution of Eq. (1) on the interval ½0; T if, for all x 2 R, the process fuðt; xÞ; t 2 ½0; Tg is Ft -adapted and if u satisfies the integral equation Z þ1 Z t Z þ1 uðt; xÞ ¼ G a ðt; x yÞu0 ðyÞ dy þ gðs; y; uðs; yÞÞGa ðt s; x yÞ dy
1 m X
þ
ð 1Þkþ1
0
1
þ1
hk ðs; y; uðs; yÞÞ 0
k¼1
Z tZ
Z tZ
1
qk G a ðt s; x yÞ dy ds qyk
þ1
f ðs; y; uðs; yÞÞG a ðt s; x yÞW ðdy dsÞ
þ 0
ð5Þ
1
for all t 2 ½0; T and x 2 R, where G a ðt; xÞ is the Green function associated to k
Eq. (1) and qqyGka ðs; yÞ is its partial derivative of order k with respect to the spatial variable. The field fuðt; xÞ; tX0; x 2 Rg is said to be a global mild solution of Eq. (1) if, for all 0oTo1, fuðt; xÞ; t 2 ½0; T; x 2 Rg is a mild solution of (1) on the interval ½0; T. Furthermore, a global mild solution is said to be Lp ðOÞ-bounded for some pX1 if, for all T40, for all t 2 ½0; T and all x 2 R, sup0ptpT supx2R Ejuðt; xÞjp o1. Definition 3. Eq. (1) is said to have a unique mild solution on ½0; T if, for any mild solutions u1 and u2 on ½0; T, we have u1 ðt; xÞ ¼ u2 ðt; xÞ P-a:s: for all t 2 ½0; T and for all x 2 R: In Section 2 we prove the existence and the uniqueness of a global solution, which is Lp ðOÞ-bounded when the initial condition is Lp ðOÞ-bounded (i.e. we take the Lp norm with respect to o 2 O and the supremum over the other variables). In Section 3 we prove the spatial and temporal Ho¨lder regularity of the solution. In [1] the authors show Ho¨lder exponents for linear equations with inverse Bessel–Riesz potential operator, and in [10] the author shows the regularity in weighted norm spaces. For Eq. (1), the regularity both in space and in time is influenced by the derivatives of the nonlinear terms (see Theorem 2). The value of the constants in this paper may change from line to line and some of the standing parameters are not always indicated. In particular, in Section 2 all the constants depend on T.
2. Existence and uniqueness of the solution The Green function G a ðt; xÞ associated to Eq. (1) is the fundamental solution of the Cauchy problem q Gðt; xÞ¼x Dad Gðt; xÞ; qt Gð0; xÞ ¼ d0 ðxÞ,
t40; x 2 R,
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where d0 is the Dirac distribution. Using Fourier’s calculus we obtain [4] and [6] Z 1 þ1 p
1 d ca ðlÞt G a ðt; xÞ ¼ F fe ; xg ¼ exp½ ilx tjlja e id2sgnðlÞ dl. 2p 1 The function G a ðt; :Þ has the following properties (see also [4]). Lemma 1. For a 2 ð0; þ1ÞnN R þ1 (i) 1 G a ðt; xÞ dx ¼ 1. (ii) G a ðt; xÞ is real but in general it is not symmetric relatively to x and it is not everywhere positive. (iii) G a ðt; xÞ satisfies the semigroup property, or the Chapman Kolmogorov equation, i.e. for 0osot Z þ1 G a ðt þ s; xÞ ¼ Ga ðt; xÞG a ðs; x xÞ dx.
1
(iv) For 0oap2, the function G a ðt; :Þ is the density of a Le´vy stable process in time t. qb (v) For fixed t, G a ðt; :Þ 2 S 1 ¼ ff 2 C 1 and qx b f is bounded and tends to zero when jxj tends to 1; 8b 2 Rþ g. qn Ga (vi) qn Gna ðt; xÞ ¼ t nþ1 a qx qxn ð1; xÞjx¼t 1a x , for all nX0 (when n ¼ 0, it is called the scaling property), n 1X ð 1Þjþl ða þ dÞ (vii) p G ðlÞ Gðaj þ l þ 1Þ sin j jxj aj ðlþ1Þ a ð1; xÞ ¼ p j¼1 2 j! þ Oðjxj aðnþ1Þ ðlþ1Þ Þ, when jxj is large, where G aðlÞ ð1; :Þ is the derivative of order l of G a ð1; :Þ. Proof. The properties (i)–(vi) are easy to be seen, so we prove (vii). It is sufficient to prove this property for the function G a ð1; xÞ when x40. In fact, when xo0, we use the same calculus taking d in place of d, and the calculus is also available for the derivatives thanks to the representation Z 1 þ1 dp G aðlÞ ð1; xÞ ¼ ð ilÞl exp½ ilx jlja e i 2 sgnðlÞ dl. 2p 1 We are interested in the case a42, the case 0oao2 can be deduced from [14,17]. The function Ga ð1; :Þ can be written as Z þ1 1 dp G a ð1; xÞ ¼ R exp½ ilx la e i 2 dl . p 0 dp
p Let 0or; Ro1 and let the curve C d : ½r; R _ fReidy ; 0pyp 2a g _ flei2a ; rplpRg _ p idy fre ; 0pyp 2a g where the symbol _ means followed by and means that the curve
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is taken in the opposite direction. By the Cauchy Theorem the integral of the dp function exp½ izx za e i 2 over C d vanishes, further the integrals over the two arcs tend to zero when R tends to infinity and r tends to zero, so Z þ1 Z þ1 pd a idp ipd 2 2a exp½ ilx l e dl ¼ e exp½ ilxei2a la dl. 0
0 ipd 2a
pd
By integrating the function e exp½ izxei2a za over the curve C 1 when d is positive and over C 1 when d is negative, we get Z þ1 1 a ip2 ipðd 1Þ ipðaþd 1Þ 2a 2a e exp½ lxe
l e dl . G a ð1; xÞ ¼ R p 0 Making the change of variable x ¼ xl, and then developing the exponential contains x in Taylor series, we find Z þ1 pðaþd 1Þ 1 p a ipðd 1Þ i
a
i R e 2a G a ð1; xÞ ¼ exp½ xe 2a x x e 2 dx px 0 ( ) n j pðd 1Þ X ð 1Þ jp 1 ¼ R ei 2a x aj ei 2 E a ðjÞ px j! j¼0 1 ð 1Þnþ1 aðnþ1Þ iðnþ1Þp ipðd 1Þ R e 2a y þ x e 2 E a ðn þ 1Þ , px ðn þ 1Þ! R þ1 pðaþd 1Þ where E a ðjÞ ¼ 0 exp½ xei 2a xaj dx and jyjo1. By the same technique we find E a ðjÞ ¼ exp½ i pðaþd 1Þj
i pðaþd 1Þ Gðaj þ 1Þ; j 2 1ðn þ 1Þ. Replacing in the formula 2 2a above, we find the series in (vii) for l ¼ 0. & Corollary 1. Let a 2 ð1; þ1ÞnN. Then there exists a constant K a such that jG a ð1; xÞjpK a ð1 þ jxj1þa Þ 1 , jG ðnÞ a ð1; xÞjpK a
1 þ jxjaþn 1 . ð1 þ jxjaþn Þ2
Proof. It is sufficient to prove that the functions ð1 þ jxj1þa ÞGa ð1; xÞ and ð1þjxjaþn Þ2 1þjxjaþn 1
G ðnÞ a ð1; xÞ are bounded on the real axis by using Lemma 1.
Corollary 2. Let a 2 ð1; þ1ÞnN, for any fixed n 2 N, for g such that g Z T Z þ1 n q G a qyn ðs; yÞ dy dso1. 0
1
& 1 aþ1 aþnþ1 ogo nþ1,
We proceed now to the existence and uniqueness of a solution of (1). Let T40 be fixed and pX2 given and suppose that the initial condition u0 is Lp ðOÞ-bounded i.e. supy2R Eju0 ðyÞjp o1. Let us write h0 ðt; x; zÞ instead of gðt; x; zÞ.
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Lemma 2. Let the sequence ðun ; nX0Þ be given by Z þ1 u0 ðt; xÞ ¼ G a ðt; x yÞu0 ðyÞ dy,
1
unþ1 ðt; xÞ ¼ u0 ðt; xÞ þ
m X k¼0
ð 1Þkþ1
Z tZ
þ1
hk ðs; y; un ðs; yÞÞ 0
1
qk G a ðt s; x yÞ dy ds qyk Z t Z þ1 f ðs; y; un ðs; yÞÞG a ðt s; x yÞW ðdy dsÞ. þ
0
ð6Þ
1
Under conditions (2), the sequence ðun ðt; xÞ; nX0Þ is well defined and satisfies sup sup Ejun ðt; xÞjp o1.
(7)
x
0ptpT
Proof. We proceed by recurrence: Using the hypothesis that the initial condition u0 is Lp ðOÞ-bounded, it is easy to see that the functions u0 ðt; xÞ and u1 ðt; xÞ exist and are also Lp ðOÞ-bounded. Suppose now that un exists and is Lp ðOÞ-bounded. A sufficient condition for the existence of the term unþ1 is that 8t40; 8x 2 R and for all k 2 f0; . . . ; mg, Z t Z þ1
E f 2 ðs; y; un ðs; yÞÞG 2a ðt s; x yÞ dy ds o1, 0
1
"Z Z t
þ1
E 0
1
#2 qk G a hk ðs; y; un ðs; yÞÞ k ðt s; x yÞ dy ds o1. qy
In fact under conditions (2) the first integral is bounded 8t40; 8x 2 R by
Z t Z þ1 2 2 Ejun ðs; yÞj G a ðt s; x yÞ dy ds o1, Ka 1 þ 0
1
which is finite by Corollary 2 and the recurrence condition. For the second integral, we apply the Ho¨lder inequality with respect to the k
measure j qqyGka ðt s; x yÞj dy ds, the growth conditions and the hypothesis moa. Now, we prove estimation (7). In fact Z Z " t þ1 m X p p E h ðs; y; un ðs; yÞÞ Ejunþ1 ðt; xÞj pK p Eju0 ðt; xÞj þ 0 1 k k¼0 p qk G a
k ðt s; x yÞ dy ds qy p # Z Z t þ1 þE f ðs; y; un ðs; yÞÞG a ðt s; x yÞW ðdy dsÞ . ð8Þ 0 1
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Applying the Burkholder–Davis–Gundy inequality, the Ho¨lder inequality with respect to the measure G 2a ðt s; x yÞ dy ds and conditions (2) for the last term in the right side of (8), we obtain Z t Z þ1 p f ðs; y; un ðs; yÞÞG a ðt s; x yÞW ðdy dsÞ I n;p ðt; xÞ:¼E 0
1 Z t Z þ1 p2 2 2 pK p E f ðs; y; un ðs; yÞÞG a ðt s; x yÞ dy ds pK p
0
1
p
Z t Z
sup sup Ejf ðs; y; un ðs; yÞÞj y
pK p 1 þ sup sup Ejun ðs; yÞjp o1. 0pspT
0pspT
0
þ1
G2a ðs; yÞ dy ds
p2
1
y
k
Again by Ho¨lder inequalities with respect to the measure j qqyGka ðt s; x yÞj dy ds and the conditions (2), we get the estimation of the other terms of (8). & Theorem 1. Let pX2 be fixed. Under conditions (2) and the assumption that the initial condition u0 is Lp ðOÞ-bounded, Eq. (1) has a unique global solution which is Lp ðOÞbounded. Proof. Let 0oTo1 be fixed and let F n ðt; xÞ ¼ kunþ1 ðt; xÞ un ðt; xÞkp ¼ Ejunþ1 ðt; xÞ un ðt; xÞjp , and H n ðtÞ ¼ sup F n ðt; xÞ, x
where ðun ; nX0Þ is the sequence given by (6). We have 8t 2 ð0; T; 8x 2 R ! m X ðkÞ F n ðt; xÞ ¼ K p An ðt; xÞ þ Bn ðt; xÞ , k¼0
where Z t Z An ðt; xÞ:¼E 0
þ1
1
p ½f ðs; y; un ðs; yÞÞ f ðs; y; un 1 ðs; yÞÞG a ðt s; x yÞW ðdy dsÞ
and Z Z t
BðkÞ n ðt; xÞ:¼E
0
þ1
1
½hk ðs; y; un ðs; yÞÞ hk ðs; y; un 1 ðs; yÞÞ
p qk G a
k ðt s; x yÞ dy ds . qy
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By Burkholder–Davis–Gundy and Ho¨lder inequality with respect to the measure G 2a ðt s; x yÞ dy ds, and conditions (2), we get Z t Z þ1 p2 2 2 An ðt; xÞpK p E jf ðs; y; un ðs; yÞÞ f ðs; y; un 1 ðs; yÞÞj G a ðt s; x yÞ dy ds 0
1
Z þ1 Z t 2 p pK p;a sup Ejun ðs; yÞ un 1 ðs; yÞj G a ðt s; yÞ dy ds y
0
pK p;a
Z
1
t
1 a
ðt sÞ H n 1 ðsÞ ds, 0
and Z
BðkÞ n ðt; xÞpK a
t
sup Ejun ðs; yÞ un 1 ðs; yÞj y
0
Z
pK a
p
Z
1
t
k q k G a ðt s; yÞ dy qy
þ1
k
ðt sÞ a H n 1 ðsÞ ds. 0
Hence H n ðtÞpK p;a
Z t "X m 0
ðt sÞ
H n 1 ðsÞ ds
1
Z
pK p;a T
#
k a
t
m
ðt sÞ a H n 1 ðsÞ ds. 0
P 1 p By Lemma 3.3 in [18] the series þ1 n¼0 ðH n ðtÞÞ converges uniformly on ½0; T. Hence p the sequence un converges in L ðOÞ uniformly on ½0; T R and the limit satisfies (5). We prove the uniqueness in the space L2 ðOÞ. Let u and v be two mild solutions on ½0; T of Eq. (1). Using the same technique as above, we get Ejuðt; xÞ vðt; xÞj2 0 2 Z Z t þ1 m X qk @ pK E ðhk ðs; y; uðs; yÞÞ hk ðs; y; vðs; yÞÞÞ k G a ðt s; x yÞ dy ds qy 0
1 k¼0 1 Z t Z þ1 2 þE ðf ðs; y; uðs; yÞÞ f ðs; y; vðs; yÞÞÞG a ðt s; x yÞW ðdy dsÞ A 0
m X
pK
1
Z tZ 0
Z tZ
pK
0 t
1
þ1
þE
2 qk juðs; yÞ vðs; yÞj G ðt
s; x
yÞ dy ds a qyk ! 2
E
k¼0
Z
þ1
juðs; yÞ vðs; yÞj2 G2a ðt s; x yÞ dy ds
1
m
ðt sÞ a sup Ejuðs; yÞ vðs; yÞj2 ds. 0
y
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Applying Gronwall Lemma, we get Ejuðt; xÞ vðt; xÞj2 ¼ 0, which ensures the uniqueness in the sense of Definition 3. & Remark 1. In multi-dimensional case we cannot always obtain a field solution. In fact, taking k ¼ 0 and replacing x and a in Eq. (1) by x ¼ ðx1 ; x2 ; . . . ; xd Þ; d41 and a ¼ ða1 ; a2 ; . . . ; ad Þ; 1oai o2, and setting x Dad ¼x1 Dad þx2 Dad þ þxd Dad , it is easy to Q see that the Green function is equal to G a ðt; xÞ ¼ i¼d i¼1 G ai ðt; xi Þ. In order that the 2 stochastic integral is defined in L ðOÞ we need the condition a11 þ a12 þ a1d o1. In the case a1 ¼ a2 ¼ ¼ ad this will be equivalent to doa. This condition coincides also with the nonexistence of the process solution in the one-dimensional case when ao1.
3. Regularity of the solution Lemma 3. For k 2 f0; 1; . . . ; mg, mp½a, we have (i) for
aþ1 aþ1 aþkþ1 ogo kþ1,
Z
þ1 Z þ1
0
(ii) for
1
g qk qk k G a ð1 þ v; zÞ k G a ðv; zÞ dz dvo1, qz qz
aþ1 aþ1 kþ2 ogo kþ1
aþ1 when koa 1 and for 1pgo kþ1 when k ¼ ½a, g Z þ1 Z þ1 k qk q k G a ðv; 1 þ zÞ k G a ðv; zÞ dz dvo1. qz 0
1 qz
Proof. (i) Let us write Z
þ1 0
Z
þ1
ðkÞ g jGðkÞ a ð1 þ v; zÞ G a ðv; zÞj dz dv
1 Z 1 Z þ1 ðkÞ g ¼ jGðkÞ a ð1 þ v; zÞ G a ðv; zÞj dz dv 0
1 Z þ1 Z þ1 ðkÞ g jG ðkÞ þ a ð1 þ v; zÞ G a ðv; zÞj dz dv. 1
1
The first integral in the equality above is finite by Corollary 2. For the second one, 1 we use the property (vi) in Lemma 1 and the change of variable z ¼ ð1 þ vÞa z0 ,
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we obtain Z þ1 Z 1
Z
þ1 ðkÞ g jGðkÞ a ð1 þ v; zÞ G a ðv; zÞj dz dv
1 þ1
v
¼ 1
Z
þ1
1
gðkþ1Þ a
1
ð1 þ vÞa
kþ1
1 !g a v 1 þ v a 0 0 ðkÞ ðkÞ 0 G a ð1; z Þ G a 1; z dz dv. 1þv v
Further, kþ1
1a !g a v 1 þ v ðkÞ G ðkÞ 1; z a ð1; zÞ G a 1þv v ! " g
1 1 þ v a ðkÞ p2g 1 GðkÞ ð1; zÞ
G 1; z a a v # g kþ1
a v ðkÞ þ 1 jG a ð1; zÞjg , 1þv and by Corollary 1
1a !g Z þ1 1 þ v ðkÞ 1; z dz Ga ð1; zÞ GðkÞ a v
1 2 g Z 1 Z z aþk jxj pK a 4 dx dz 2 vþ1 1a ð1 þ jxjaþkþ1 Þ
1 ð v Þ z g Z 1 Z ðvþ1Þa1 z g Z 0 Z z v dx dz þ dx dz þ vþ1 1a
1 ð v Þ z 0 z 3 g Z þ1 Z ðvþ1Þ1a z v xaþk 5 þ dx dz z ð1 þ xaþkþ1 Þ2 1
aþkþ1 g Z þ1 zaþkþ1 g v þ 1 a p2K a 1 þ . ð1 þ zaþkþ1 Þ2 dz 1 v 1 Therefore, kþ1
1a !g Z þ1 a v 1 þ v ðkÞ G ðkÞ 1; z dz a ð1; zÞ G a 1 þ v v
1 ! g
aþkþ1 g
kþ1 a v þ 1 a v pK a;g 1 þ 1
. v vþ1
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By the Lagrange–Taylor formula we obtain for 8vX1, g g
kþ1 v þ 1aþkþ1 a a v
1 þ 1 pK a;g v g . v vþ1 Hence Z
þ1 1
Z
þ1 ðkÞ g jGðkÞ a ð1 þ v; zÞ G a ðv; zÞj dz dv
1
Z
pK a;g
þ1
v
gðaþkþ1Þ a
1
ð1 þ vÞa dv.
1 aþ1 . This last integral converges when g4 aþkþ1
(ii) We have Z
g qk qk k G a ðv; 1 þ zÞ k G a ðv; zÞ dz dv qz 0
1 qz g Z 1 Z þ1 k qk q ¼ k G a ðv; 1 þ zÞ k Ga ðv; zÞ dz dv qz qz 0
1 g Z þ1 Z þ1 k qk q þ k G a ðv; 1 þ zÞ k G a ðv; zÞ dz dv. qz 1
1 qz þ1
Z
þ1
The function x !xg ; xX0 is convex and by Corollary 2, we have Z
g qk qk k Ga ðv; 1 þ zÞ k G a ðv; zÞ dz dv qz 0
1 qz g Z 1 Z þ1 k q p2g k G a ðv; zÞ dz dvo1. qz 0
1 1
Z
þ1
On the other hand, Z
g qk qk k G a ðv; 1 þ zÞ k G a ðv; zÞ dz dv qz qz 1
1 Z þ1 Z þ1
gðkþ1Þ
1
1 ðkÞ g a a ¼ v a jGðkÞ a ð1; v ð1 þ zÞÞ G a ð1; v zÞj dz dv, þ1
Z
1
þ1
1
1775
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1776
1
a where G ðkÞ a ð1; v zÞ is the derivative of order k of the function G a ð1; zÞ, taken in the
1 point v a z. Further
Z
þ1
1
1
ðkÞ g a a jG ðkÞ a ð1; v ð1 þ zÞÞ G a ð1; v zÞj dz
1 g Z þ1 Z v 1a ð1þzÞ ðkþ1Þ jGa ð1; xÞj dx dz p 1
1 v a z g "Z
1 þ1 Z v a ð1þzÞ jxjaþk pK a dx dz aþkþ1 2 1
1 ð1 þ jxj Þ 2va vaz Z 2v1a Z v 1a ð1þzÞ g jxjaþk þ dx dz 1 aþkþ1 2 ð1 þ jxj Þ vaz
1 # Z þ2v1a Z v 1a ð1þzÞ g dx dz , þ 1 1 vaz
2va
ð9Þ
1
By a simple calculus of the primitive and the change of the variable z0 ¼ v a z, we find Z
Z 1 g v a ð1þzÞ xaþk dx dz aþkþ1 2 1
1 ð1 þ x Þ 2va vaz g Z þ1 aþkþ1 aþkþ1 v a ðz þ 1Þaþkþ1 v a zaþkþ1 p dz aþkþ1 aþkþ1 ð1 þ v aþkþ1 a zaþkþ1 Þð1 þ v a ðz þ 1Þ Þ 0 g Z þ1
1 ðz0 þ v a Þaþkþ1 z0 aþkþ1 1 0 pva dz . ð1 þ z0 aþkþ1 Þð1 þ ðz0 þ v 1a Þaþkþ1 Þ 0 þ1
By the Taylor–Lagrange formula, we can obtain the following estimation: 8vX1 and 8z 2 R,
1
1
jjz þ v a jaþkþ1 jzjaþkþ1 jpKv a ðjzjaþk þ jzjaþk 1 þ jz þ 1jaþk 1 Þ, hence g Z 1 v a ð1þzÞ xaþk dx dz aþkþ1 2 1
1 ð1 þ x Þ 2va vaz g Z þ1 aþk 1 g z þ zaþk 1 þ ðz þ 1Þaþk 1 pK a;g v a dz. ð1 þ zaþkþ1 Þ2 0
Z
þ1
ð10Þ
ARTICLE IN PRESS L. Debbi, M. Dozzi / Stochastic Processes and their Applications 115 (2005) 1764–1781
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In the same way we see that Z
1
2va
1
g Z 1 v a ð1þzÞ jxjaþk dx dz 1 aþkþ1 2 vaz ð1 þ jxj Þ
is upper bounded by the right side of (10). Hence Z þ1 Z þ1 Z g jG a ðv; 1 þ zÞ G a ðv; zÞj dz dvpK a;g 1
þ1
v
1 gðkþ2Þ a
dv,
1
1
aþ1 which converges when g4 kþ2 .
&
The following theorem gives the spatial and the temporal regularity of the solution of (5). Theorem 2. Under conditions (2) and the condition that u0 is Lp -bounded for all pX2, we have
For fixed x 2 R the process fuðt; xÞ; t40g has Ho¨lder continuous trajectories with a m exponent minfa 1 2a ; a g , for any 40, P-a.s. For ao3 and for fixed t the process fuðt; xÞ; x 2 Rg has Ho¨lder continuous trajectories with exponent minfa 1 2 ; a ½ag , for any 40, P-a.s.
Proof. (i) It is easy to see that u0 ðt; xÞ is smooth function with respect to t and to x. Let x 2 R be fixed and let t40, y40, and n41, and let Z t Z þ1 ðkÞ J 1;n ðt; x; yÞ:¼E hk ðs; y; uðs; yÞÞ 0
1 n ! qk qk
G ðt þ y
s; x
yÞ
G ðt
s; x
yÞ dy ds , a a qyk qyk n Z tþy Z þ1 k q LðkÞ hk ðs; y; uðs; yÞÞ k G a ðt þ y s; x yÞ dy ds , n ðt; x; yÞ:¼E qy t
1 Z t Z I 1;n ðt; x; yÞ:¼E 0
þ1
1
Z Ln ðt; x; yÞ:¼E
t
tþy
Z
n f ðs; y; uðs; yÞÞðGa ðt þ y s; x yÞ G a ðt s; x yÞÞW ðdy dsÞ , þ1
1
n f ðs; y; uðs; yÞÞG a ðt þ y s; x yÞW ðdy dsÞ .
Applying Burkholder–Davis–Gundy inequality, Ho¨lder inequality, the fact that the solution u is Lp -bounded for all p41 and using conditions (2), it is easy
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to see that
Z t Z
þ1
jf ðs; y; uðs; yÞÞjn ðG a ðt þ y s; x yÞ 2
G a ðt s; x yÞÞ dy ds
I 1;n ðt; x; yÞpK n E
0
Z t Z
1
n2 1
þ1 2
ðG a ðs þ y; yÞ G a ðs; yÞÞ dy ds
0
1
pK n sup Ejf ðs; y; uðs; yÞÞj
n
Z t Z
½0;T R
n2
þ1 2
ðG a ðs þ y; yÞ G a ðs; yÞÞ dy ds 0
,
1
and Ln ðt; x; yÞpK n sup Ejf ðs; y; uðs; yÞÞj
n
Z
½0;T R
y
0
Z
n2
þ1
G2a ðs
þ y; yÞ dy ds
.
1
1
By changes of variables s ¼ yv; y ¼ ya z and by Lemma 3 and Corollary 2, we get I 1;n ðt; x; yÞ þ Ln ðt; x; yÞ " Z þ1 Z a 1 n 2a pK n y 0
Z
2
Z
ðGa ðs þ 1; yÞ Ga ðs; yÞÞ dy ds
1
G 2a ðs; yÞ dy ds
0
1
na 1 2a
.
pK n;a y
2
þ1
þ
n2
þ1
n2 #
ð11Þ
Using Ho¨lder inequality and by following the same technique as above, we find n J ðkÞ 1;n ðt; x; yÞp sup Ejhk ðs; y; uðs; yÞÞj ½0;T R
!n k qk q k G a ðs þ y; yÞ k G a ðs; yÞdy ds qy
1 qy !n Z þ1 Z þ1 k qk q k G a ðs þ 1; yÞ k Ga ðs; yÞ dy ds , qy 0
1 qy
Z tZ
0
pK n yn
a k a
þ1
and LðkÞ n ðt; x; yÞp
sup Ejhk ðs; y; uðs; yÞÞj ½0;T R
pK n yn
a k a
n
Z 0
Z 0
y
Z
!n k q k Ga ðs þ y; yÞdy ds qy !n
þ1
1
þ1 k q k Ga ðs; yÞdy ds qy
1
2Z
,
ARTICLE IN PRESS L. Debbi, M. Dozzi / Stochastic Processes and their Applications 115 (2005) 1764–1781
1779
hence a k
ðkÞ n a J ðkÞ . 1;n ðt; x; yÞ þ L1;n ðt; x; yÞpK n;a y
(12)
From (11) and (12), we obtain the result. (ii) Let ao3, t40 be fixed. For x 2 R, h40, and for n41, we have " Ejuðt; x þ hÞ uðt; xÞj pK n L0;n ðt; x; hÞ þ n
m X
# J ðkÞ 2;n ðt; x; hÞ
þ I 2;n ðt; x; hÞ ,
k¼0
where Z L0;n ðt; x; hÞ:¼E
þ1
1
n ðG a ðt; x þ h yÞ G a ðt; x yÞÞu0 ðyÞ dy dy ds ,
Z t Z
J ðkÞ 2;n ðt; x; hÞ:¼E
0
þ1
hk ðs; y; uðs; yÞÞ
1
n ! qk qk
G ðt
s; x þ h
yÞ
G ðt
s; x
yÞ dy ds , a a qyk qyk
and Z t Z I 2;n ðt; x; hÞ ¼ E
þ1
f ðs; y; uðs; yÞÞðG a ðt s; x þ h yÞ n
Ga ðt s; x yÞÞW ðdy dsÞ . 0
1
The integrals in the terms L0;n ðt; x; hÞ and J ðkÞ 2;n ðt; x; hÞ for kpmo½a are smooth functions, so it is sufficient to estimate the regularity of the terms J ð½aÞ 2;n ðt; x; hÞ and I 2;n ðt; x; hÞ. We apply the Burkholder–Davis–Gundy inequality, the Ho¨lder inequality, conditions (2) and the fact that the solution is Lp -bounded, to get I 2;n ðt; x; hÞpK n sup Ejf ðs; y; uðs; yÞÞjn ½0;T R
pK n h
na 1 2
Z
Z t Z 0
þ1
Z
þ1
n2 ðG a ðs; y þ hÞ Ga ðs; yÞÞ2 dy ds
1
þ1 2
ðG a ðs; y þ 1Þ G a ðs; yÞÞ dy ds 0
1
n2 ð13Þ
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and n J ð½aÞ 2;n ðt; x; hÞp sup Ejh½a ðs; y; uðs; yÞÞj ½0;T R
½a n q q½a
½a G a ðs; y þ hÞ qy½a G a ðs; yÞdy ds 0
1 qy
Z þ1 Z þ1 ½a n q q½a dy ds . pK n hnða ½aÞ G ðs; y þ 1Þ
G ðs; yÞ a a ½a qy½a 0
1 qy
Z t Z
þ1
ð14Þ By Lemma 3 the integrals on the right side of (13) and (14) converge, hence 1
a 1 2 ;a ½ag
ðEjuðt; x þ hÞ uðt; xÞjn Þn pK n;a hminf
:
&
Acknowledgements The first author wishes to thank Professor Bernard Roynette for the invitation to the Institut Elie Cartan de Nancy and Professor Rudolf Gorenflo for bringing to her attention several important articles on the fractional calculus. References [1] J.M. Angulo, M.D. Ruiz-Medina, V.V. Anh, W. Grecksch, Fractional diffusion and fractional heat equation, Adv. Appl. Probab. 32 (2000) 1077–1099. [2] S. Bonaccorsi, L. Tubaro, Mittag-Leffler’s function and stochastic linear Volterra equations of convolution type, Stochast. Anal. Appl. 21 (1) (2003) 61–78. [3] R. Dalang, C. Mueller, Some nonlinear s.p.d.e.’s that are second order in time, Electron. J. Probab. 8 (1) (2003) 1–21. [4] L. Debbi, Explicit solutions of some fractional equations via stable subordinators, preprint. [5] L. Debbi, On some properties of a high fractional differential operator which is not in general selfadjoint, preprint. [6] L. Debbi, L. Abbaoui, Explicit solution of some fractional heat equations via Le´vy motion, Maghreb Math. Rev., to appear. [7] W. Feller, Generalization of Marcel Riesz’ potentials and the semi-groups generated by them, Meddelanden Fran Lunds Universitets Matematiska Seminarium Supplementband 1952, pp. 73–81. [8] R. Gorenflo, F. Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calculus Appl. Anal. 1 (2) (1998) 167–191. [9] T. Komatsu, On the martingale problem for generators of stable processes with perturbations, Osaka J. Math. 21 (1984) 113–132. [10] P. Kotelenez, Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations, Stochast. Stochast. Reports 41 (1992) 177–199. [11] V.Yu. Kylov, Some properties of the distribution corresponding to the equation qu=qt ¼ ð 1Þqþ1 q2q u=qt2q , Soviet Math. Dokl. 1 (1960) 260–263. [12] A. Le Mehaute, T. Machado, J.C. Trigeassou, J. Sabatier, Fractional differentiation and its applications, FDA’04, Proceedings of the first IFAC Workshop, vol. 2004-1, International Federation of Automatic Control, ENSEIRB, Bordeaux, France, July 19–21, 2004.
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[13] X. Leoncini, G. Zaslavsky, Ets, stickiness, and anomalous transport, Phys. Rev. E 65 (2002) 046216–1–046216–16. [14] E. Lukacs, Characteristic Functions, second ed., 1970, Griffin, 1960. [15] F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space–time fractional diffusion equation, Fract. Calculus Appl. Anal. 4 (2) (2001) 153–192. [16] C. Mueller, The heat equation with Le´vy noise, Stochast. Proc. Appl. 74 (1998) 67–82. [17] V.V. Uchaikin, V.M. Zolotarev, Chance and stability, stable distributions and their applications, Mod. Probab. Statist. VSP. 1999. [18] J.B. Walsh, An Introduction To Stochastic Partial Differential Equations, Lectures Notes in Mathematics, vol. 1180, Springer, Berlin, 1986, Ecole d’e´te´ de Probabilite´s de Saint-Flour XIV-1984. [19] J.L. Wu, Fractal Burgers equation with stable Le´vy noise, International Conference SPDE and Applications–VII, January 4–10, 2004. [20] J. Zabczyk, Symmetric solutions of semilinear stochastic equations, Lecture Notes in Mathematics, vol. 1390, Springer, Berlin, 1988, pp. 237–256.
Stochastic Processes and their Applications 115 (2005) 1764–1781 www.elsevier.com/locate/spa
On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension Latifa Debbia,b,c,, Marco Dozzia a
Institut Elie Cartan, B.P 239, Universite´ Henri Poincare´ Nancy1, Vandoeuvre-Le`s-Nancy, France b Department of Mathematics, Faculty of Sciences, University Ferhat Abbas, El-Maabouda Se´tif 19000, Algeria c Department of Mathematics, Faculty of Science, University Mohamed Boudiaf, B.P 166 Echbilia, M’sila 28000, Algeria Received 31 December 2004; received in revised form 11 May 2005; accepted 21 May 2005 Available online 27 June 2005
Abstract Existence, uniqueness and regularity of the trajectories of mild solutions of one-dimensional nonlinear stochastic fractional partial differential equations of order a41 containing derivatives of entire order and perturbed by space–time white noise are studied. The fractional derivative operator is defined by means of a generalized Riesz–Feller potential. r 2005 Elsevier B.V. All rights reserved. MSC: primary 26A33; 60H15; 60G60 Keywords: Fractional derivative operator; Stochastic partial differential equation; Space–time white noise; Ho¨lder continuity; Equation of high order
Corresponding author. Institut Elie Cartan, B.P 239, Universite´ Henri Poincare´ Nancy1, VandoeuvreLe`s-Nancy, France. Tel.: +333 83 68 45 15; fax: +333 83 68 45 34. E-mail addresses: [email protected], [email protected] (L. Debbi), [email protected] (M. Dozzi).
0304-4149/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spa.2005.06.001
ARTICLE IN PRESS L. Debbi, M. Dozzi / Stochastic Processes and their Applications 115 (2005) 1764–1781
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1. Introduction Fractional equations, both partial and ordinary ones, have received more attention in recent years. Various phenomena in physics, like diffusion in a disordered or fractal medium, or in image analysis, or in risk management have been modeled by means of fractional equations. Some of them are of high order [13]. We refer to [12] and the part II of [17] for recent surveys on these applications. However, comparatively few publications treat stochastic fractional partial differential equations. Most of them investigate evolution type equations, driven by a fractional power of the Laplacian [11,20,3 and the references therein]. These operators generate symmetric stable semigroups when the order of derivation is less than 2. Mueller [16] and Wu [19] proved the existence of a solution of stochastic fractional heat, respectively Burgers, equation perturbed by a stable noise. The work of Bonaccorsi and Tubaro [2] can be applied to stochastic evolution equations with fractional time derivative. Angulo et al. [1] have considered a linear partial differential equation driven by the composition of the inverses of Riesz and Bessel potentials and perturbed by a space–time white noise. In [3], Dalang and Mueller have studied the stochastic equation which is of second order in time, driven by a power Laplacian and perturbed by colored noise (white in time and homogeneous in space). The stochastic equation studied by Kotelenez [10] is driven by a pseudodifferential operator which covers the power Laplacian (symmetric case). The aim of this work is to generalize, in the framework of the multi-parameter processes, the results of Walsh [18] to nonlinear stochastic fractional partial differential equations (SFPDE) of high order containing also derivatives of entire order. We study the existence, uniqueness, and regularity of the solution of the equation formally given by m X qu qk hk ðt; xÞ ¼ x Dad uðt; xÞ þ gðt; x; uðt; xÞÞ þ ðt; x; uðt; xÞÞ qt qxk k¼1
þ f ðt; x; uðt; xÞÞ
q2 W ðt; xÞ; qtqx
t40; x 2 R,
uð0; xÞ ¼ u0 ðxÞ,
(1)
where ½a is the integer part of a, m 2 N and 1pmp½a. x Dad is the fractional differential operator with respect to the spatial variable, to be defined below. We suppose that a 2 ð1; þ1ÞnN and that the functions f ; g; hk : ½0; þ1Þ R R ! R satisfy Lipschitz and growth conditions: for all T40, there exists a constant K T 40 such that for all t 2 ½0; T and for all x; y; z 2 R, ðjhk ðt; x; yÞ hk ðt; x; zÞj þ jgðt; x; yÞ gðt; x; zÞj þ jf ðt; x; yÞ f ðt; x; zÞjÞ pK T jy zj, ðjhk ðt; x; zÞj þ jgðt; x; zÞj þ jf ðt; x; zÞjÞpK T ð1 þ jzjÞ.
ð2Þ
ARTICLE IN PRESS L. Debbi, M. Dozzi / Stochastic Processes and their Applications 115 (2005) 1764–1781
1766
Let ðO; F; ðFt ; tX0Þ; PÞ be a complete probability space, endowed with an increasing and right-continuous filtration ðFt ; tX0Þ. Let W ¼ fW ðt; xÞ; tX0; x 2 Rg be a centered Gaussian field which is white noise in time and in space with covariance function given by Kððt; xÞ; ðs; yÞÞ ¼ 14 ðsgnðxÞ þ sgnðyÞÞ2 ðt ^ sÞðjxj ^ jyjÞ, where sgn is the sign function. W is in fact composed of two independent Brownian sheets, one in the positive direction of the spatial variable and the other one in the negative direction. We suppose that W generates a ðFt ; tX0Þ-martingale measure in the sense of Walsh [18]. The initial condition u0 is supposed to be F0 BðRÞ measurable, where BðRÞ is the Borelian s-algebra over R. The fractional differential operator x Dad used in this paper is an extension of the inverse of the generalized Riesz–Feller potential [7,9] when a42. It is given for a40 by Definition 1. The fractional differential Dad jðxÞ is given by Dad jðxÞ ¼ F 1 fd ca ðlÞFfjðxÞ; lg; xg,
(3)
where d ca ðlÞ
p
¼ jlja e id2sgnðlÞ ,
(4)
jdjp minfa ½a2 ; 2 þ ½a2 ag, ½a2 is the largest even integer less or equal to a (even part of a), and d ¼ 0 when a 2 2N þ 1, and F (respectively F 1 ) is the Fourier (respectively Fourier inverse) transform. The operator Dad is a closed, densely defined operator on L2 ðRÞ and it is the infinitesimal generator of a semigroup which is in general not symmetric and not a contraction. This operator generalizes the fractional differential operators in [7,8,15] where 0oap2. It is selfadjoint only when d ¼ 0 and in this case, it coincides with the fractional power of the Laplacian. Evidently, when a ¼ 2 it is the Laplacian itself. Furthermore, it is proven in [5], that when jdj ¼ 2 þ ½a2 a or jdj ¼ a ½a2 , it coincides with the Riemann–Liouville differential operator. By [9], Dad can be represented for 1oao2, by Z þ1 jðx þ yÞ jðxÞ yj0 ðxÞ Dad jðxÞ ¼ ðM d 1ð 1;0Þ þ M dþ 1ð0;þ1Þ Þ dy, jyj1þa
1 and for 0oao1, by Z þ1 jðx þ yÞ jðxÞ Dad jðxÞ ¼ ðM d 1ð 1;0Þ þ M dþ 1ð0;þ1Þ Þ dy, 1þa jyj
1 where M d and M dþ are two nonnegative constants satisfying M d þ M dþ 40 and 1ð 1;0Þ and 1ð0;þ1Þ are the indicator functions of the intervals ð 1; 0Þ, respectively, ð0; þ1Þ, and j is a smooth function for which the integrals exist, and j0 is its derivative. For more details about this operator we refer to [5]. In the following definition we give a rigorous meaning of (1) and of its solution.
ARTICLE IN PRESS L. Debbi, M. Dozzi / Stochastic Processes and their Applications 115 (2005) 1764–1781
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Definition 2. Let 0oTo1. A measurable stochastic field fu ¼ uðt; xÞ; t 2 ½0; T; x 2 Rg is said to be a mild solution of Eq. (1) on the interval ½0; T if, for all x 2 R, the process fuðt; xÞ; t 2 ½0; Tg is Ft -adapted and if u satisfies the integral equation Z þ1 Z t Z þ1 uðt; xÞ ¼ G a ðt; x yÞu0 ðyÞ dy þ gðs; y; uðs; yÞÞGa ðt s; x yÞ dy
1 m X
þ
ð 1Þkþ1
0
1
þ1
hk ðs; y; uðs; yÞÞ 0
k¼1
Z tZ
Z tZ
1
qk G a ðt s; x yÞ dy ds qyk
þ1
f ðs; y; uðs; yÞÞG a ðt s; x yÞW ðdy dsÞ
þ 0
ð5Þ
1
for all t 2 ½0; T and x 2 R, where G a ðt; xÞ is the Green function associated to k
Eq. (1) and qqyGka ðs; yÞ is its partial derivative of order k with respect to the spatial variable. The field fuðt; xÞ; tX0; x 2 Rg is said to be a global mild solution of Eq. (1) if, for all 0oTo1, fuðt; xÞ; t 2 ½0; T; x 2 Rg is a mild solution of (1) on the interval ½0; T. Furthermore, a global mild solution is said to be Lp ðOÞ-bounded for some pX1 if, for all T40, for all t 2 ½0; T and all x 2 R, sup0ptpT supx2R Ejuðt; xÞjp o1. Definition 3. Eq. (1) is said to have a unique mild solution on ½0; T if, for any mild solutions u1 and u2 on ½0; T, we have u1 ðt; xÞ ¼ u2 ðt; xÞ P-a:s: for all t 2 ½0; T and for all x 2 R: In Section 2 we prove the existence and the uniqueness of a global solution, which is Lp ðOÞ-bounded when the initial condition is Lp ðOÞ-bounded (i.e. we take the Lp norm with respect to o 2 O and the supremum over the other variables). In Section 3 we prove the spatial and temporal Ho¨lder regularity of the solution. In [1] the authors show Ho¨lder exponents for linear equations with inverse Bessel–Riesz potential operator, and in [10] the author shows the regularity in weighted norm spaces. For Eq. (1), the regularity both in space and in time is influenced by the derivatives of the nonlinear terms (see Theorem 2). The value of the constants in this paper may change from line to line and some of the standing parameters are not always indicated. In particular, in Section 2 all the constants depend on T.
2. Existence and uniqueness of the solution The Green function G a ðt; xÞ associated to Eq. (1) is the fundamental solution of the Cauchy problem q Gðt; xÞ¼x Dad Gðt; xÞ; qt Gð0; xÞ ¼ d0 ðxÞ,
t40; x 2 R,
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where d0 is the Dirac distribution. Using Fourier’s calculus we obtain [4] and [6] Z 1 þ1 p
1 d ca ðlÞt G a ðt; xÞ ¼ F fe ; xg ¼ exp½ ilx tjlja e id2sgnðlÞ dl. 2p 1 The function G a ðt; :Þ has the following properties (see also [4]). Lemma 1. For a 2 ð0; þ1ÞnN R þ1 (i) 1 G a ðt; xÞ dx ¼ 1. (ii) G a ðt; xÞ is real but in general it is not symmetric relatively to x and it is not everywhere positive. (iii) G a ðt; xÞ satisfies the semigroup property, or the Chapman Kolmogorov equation, i.e. for 0osot Z þ1 G a ðt þ s; xÞ ¼ Ga ðt; xÞG a ðs; x xÞ dx.
1
(iv) For 0oap2, the function G a ðt; :Þ is the density of a Le´vy stable process in time t. qb (v) For fixed t, G a ðt; :Þ 2 S 1 ¼ ff 2 C 1 and qx b f is bounded and tends to zero when jxj tends to 1; 8b 2 Rþ g. qn Ga (vi) qn Gna ðt; xÞ ¼ t nþ1 a qx qxn ð1; xÞjx¼t 1a x , for all nX0 (when n ¼ 0, it is called the scaling property), n 1X ð 1Þjþl ða þ dÞ (vii) p G ðlÞ Gðaj þ l þ 1Þ sin j jxj aj ðlþ1Þ a ð1; xÞ ¼ p j¼1 2 j! þ Oðjxj aðnþ1Þ ðlþ1Þ Þ, when jxj is large, where G aðlÞ ð1; :Þ is the derivative of order l of G a ð1; :Þ. Proof. The properties (i)–(vi) are easy to be seen, so we prove (vii). It is sufficient to prove this property for the function G a ð1; xÞ when x40. In fact, when xo0, we use the same calculus taking d in place of d, and the calculus is also available for the derivatives thanks to the representation Z 1 þ1 dp G aðlÞ ð1; xÞ ¼ ð ilÞl exp½ ilx jlja e i 2 sgnðlÞ dl. 2p 1 We are interested in the case a42, the case 0oao2 can be deduced from [14,17]. The function Ga ð1; :Þ can be written as Z þ1 1 dp G a ð1; xÞ ¼ R exp½ ilx la e i 2 dl . p 0 dp
p Let 0or; Ro1 and let the curve C d : ½r; R _ fReidy ; 0pyp 2a g _ flei2a ; rplpRg _ p idy fre ; 0pyp 2a g where the symbol _ means followed by and means that the curve
ARTICLE IN PRESS L. Debbi, M. Dozzi / Stochastic Processes and their Applications 115 (2005) 1764–1781
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is taken in the opposite direction. By the Cauchy Theorem the integral of the dp function exp½ izx za e i 2 over C d vanishes, further the integrals over the two arcs tend to zero when R tends to infinity and r tends to zero, so Z þ1 Z þ1 pd a idp ipd 2 2a exp½ ilx l e dl ¼ e exp½ ilxei2a la dl. 0
0 ipd 2a
pd
By integrating the function e exp½ izxei2a za over the curve C 1 when d is positive and over C 1 when d is negative, we get Z þ1 1 a ip2 ipðd 1Þ ipðaþd 1Þ 2a 2a e exp½ lxe
l e dl . G a ð1; xÞ ¼ R p 0 Making the change of variable x ¼ xl, and then developing the exponential contains x in Taylor series, we find Z þ1 pðaþd 1Þ 1 p a ipðd 1Þ i
a
i R e 2a G a ð1; xÞ ¼ exp½ xe 2a x x e 2 dx px 0 ( ) n j pðd 1Þ X ð 1Þ jp 1 ¼ R ei 2a x aj ei 2 E a ðjÞ px j! j¼0 1 ð 1Þnþ1 aðnþ1Þ iðnþ1Þp ipðd 1Þ R e 2a y þ x e 2 E a ðn þ 1Þ , px ðn þ 1Þ! R þ1 pðaþd 1Þ where E a ðjÞ ¼ 0 exp½ xei 2a xaj dx and jyjo1. By the same technique we find E a ðjÞ ¼ exp½ i pðaþd 1Þj
i pðaþd 1Þ Gðaj þ 1Þ; j 2 1ðn þ 1Þ. Replacing in the formula 2 2a above, we find the series in (vii) for l ¼ 0. & Corollary 1. Let a 2 ð1; þ1ÞnN. Then there exists a constant K a such that jG a ð1; xÞjpK a ð1 þ jxj1þa Þ 1 , jG ðnÞ a ð1; xÞjpK a
1 þ jxjaþn 1 . ð1 þ jxjaþn Þ2
Proof. It is sufficient to prove that the functions ð1 þ jxj1þa ÞGa ð1; xÞ and ð1þjxjaþn Þ2 1þjxjaþn 1
G ðnÞ a ð1; xÞ are bounded on the real axis by using Lemma 1.
Corollary 2. Let a 2 ð1; þ1ÞnN, for any fixed n 2 N, for g such that g Z T Z þ1 n q G a qyn ðs; yÞ dy dso1. 0
1
& 1 aþ1 aþnþ1 ogo nþ1,
We proceed now to the existence and uniqueness of a solution of (1). Let T40 be fixed and pX2 given and suppose that the initial condition u0 is Lp ðOÞ-bounded i.e. supy2R Eju0 ðyÞjp o1. Let us write h0 ðt; x; zÞ instead of gðt; x; zÞ.
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Lemma 2. Let the sequence ðun ; nX0Þ be given by Z þ1 u0 ðt; xÞ ¼ G a ðt; x yÞu0 ðyÞ dy,
1
unþ1 ðt; xÞ ¼ u0 ðt; xÞ þ
m X k¼0
ð 1Þkþ1
Z tZ
þ1
hk ðs; y; un ðs; yÞÞ 0
1
qk G a ðt s; x yÞ dy ds qyk Z t Z þ1 f ðs; y; un ðs; yÞÞG a ðt s; x yÞW ðdy dsÞ. þ
0
ð6Þ
1
Under conditions (2), the sequence ðun ðt; xÞ; nX0Þ is well defined and satisfies sup sup Ejun ðt; xÞjp o1.
(7)
x
0ptpT
Proof. We proceed by recurrence: Using the hypothesis that the initial condition u0 is Lp ðOÞ-bounded, it is easy to see that the functions u0 ðt; xÞ and u1 ðt; xÞ exist and are also Lp ðOÞ-bounded. Suppose now that un exists and is Lp ðOÞ-bounded. A sufficient condition for the existence of the term unþ1 is that 8t40; 8x 2 R and for all k 2 f0; . . . ; mg, Z t Z þ1
E f 2 ðs; y; un ðs; yÞÞG 2a ðt s; x yÞ dy ds o1, 0
1
"Z Z t
þ1
E 0
1
#2 qk G a hk ðs; y; un ðs; yÞÞ k ðt s; x yÞ dy ds o1. qy
In fact under conditions (2) the first integral is bounded 8t40; 8x 2 R by
Z t Z þ1 2 2 Ejun ðs; yÞj G a ðt s; x yÞ dy ds o1, Ka 1 þ 0
1
which is finite by Corollary 2 and the recurrence condition. For the second integral, we apply the Ho¨lder inequality with respect to the k
measure j qqyGka ðt s; x yÞj dy ds, the growth conditions and the hypothesis moa. Now, we prove estimation (7). In fact Z Z " t þ1 m X p p E h ðs; y; un ðs; yÞÞ Ejunþ1 ðt; xÞj pK p Eju0 ðt; xÞj þ 0 1 k k¼0 p qk G a
k ðt s; x yÞ dy ds qy p # Z Z t þ1 þE f ðs; y; un ðs; yÞÞG a ðt s; x yÞW ðdy dsÞ . ð8Þ 0 1
ARTICLE IN PRESS L. Debbi, M. Dozzi / Stochastic Processes and their Applications 115 (2005) 1764–1781
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Applying the Burkholder–Davis–Gundy inequality, the Ho¨lder inequality with respect to the measure G 2a ðt s; x yÞ dy ds and conditions (2) for the last term in the right side of (8), we obtain Z t Z þ1 p f ðs; y; un ðs; yÞÞG a ðt s; x yÞW ðdy dsÞ I n;p ðt; xÞ:¼E 0
1 Z t Z þ1 p2 2 2 pK p E f ðs; y; un ðs; yÞÞG a ðt s; x yÞ dy ds pK p
0
1
p
Z t Z
sup sup Ejf ðs; y; un ðs; yÞÞj y
pK p 1 þ sup sup Ejun ðs; yÞjp o1. 0pspT
0pspT
0
þ1
G2a ðs; yÞ dy ds
p2
1
y
k
Again by Ho¨lder inequalities with respect to the measure j qqyGka ðt s; x yÞj dy ds and the conditions (2), we get the estimation of the other terms of (8). & Theorem 1. Let pX2 be fixed. Under conditions (2) and the assumption that the initial condition u0 is Lp ðOÞ-bounded, Eq. (1) has a unique global solution which is Lp ðOÞbounded. Proof. Let 0oTo1 be fixed and let F n ðt; xÞ ¼ kunþ1 ðt; xÞ un ðt; xÞkp ¼ Ejunþ1 ðt; xÞ un ðt; xÞjp , and H n ðtÞ ¼ sup F n ðt; xÞ, x
where ðun ; nX0Þ is the sequence given by (6). We have 8t 2 ð0; T; 8x 2 R ! m X ðkÞ F n ðt; xÞ ¼ K p An ðt; xÞ þ Bn ðt; xÞ , k¼0
where Z t Z An ðt; xÞ:¼E 0
þ1
1
p ½f ðs; y; un ðs; yÞÞ f ðs; y; un 1 ðs; yÞÞG a ðt s; x yÞW ðdy dsÞ
and Z Z t
BðkÞ n ðt; xÞ:¼E
0
þ1
1
½hk ðs; y; un ðs; yÞÞ hk ðs; y; un 1 ðs; yÞÞ
p qk G a
k ðt s; x yÞ dy ds . qy
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By Burkholder–Davis–Gundy and Ho¨lder inequality with respect to the measure G 2a ðt s; x yÞ dy ds, and conditions (2), we get Z t Z þ1 p2 2 2 An ðt; xÞpK p E jf ðs; y; un ðs; yÞÞ f ðs; y; un 1 ðs; yÞÞj G a ðt s; x yÞ dy ds 0
1
Z þ1 Z t 2 p pK p;a sup Ejun ðs; yÞ un 1 ðs; yÞj G a ðt s; yÞ dy ds y
0
pK p;a
Z
1
t
1 a
ðt sÞ H n 1 ðsÞ ds, 0
and Z
BðkÞ n ðt; xÞpK a
t
sup Ejun ðs; yÞ un 1 ðs; yÞj y
0
Z
pK a
p
Z
1
t
k q k G a ðt s; yÞ dy qy
þ1
k
ðt sÞ a H n 1 ðsÞ ds. 0
Hence H n ðtÞpK p;a
Z t "X m 0
ðt sÞ
H n 1 ðsÞ ds
1
Z
pK p;a T
#
k a
t
m
ðt sÞ a H n 1 ðsÞ ds. 0
P 1 p By Lemma 3.3 in [18] the series þ1 n¼0 ðH n ðtÞÞ converges uniformly on ½0; T. Hence p the sequence un converges in L ðOÞ uniformly on ½0; T R and the limit satisfies (5). We prove the uniqueness in the space L2 ðOÞ. Let u and v be two mild solutions on ½0; T of Eq. (1). Using the same technique as above, we get Ejuðt; xÞ vðt; xÞj2 0 2 Z Z t þ1 m X qk @ pK E ðhk ðs; y; uðs; yÞÞ hk ðs; y; vðs; yÞÞÞ k G a ðt s; x yÞ dy ds qy 0
1 k¼0 1 Z t Z þ1 2 þE ðf ðs; y; uðs; yÞÞ f ðs; y; vðs; yÞÞÞG a ðt s; x yÞW ðdy dsÞ A 0
m X
pK
1
Z tZ 0
Z tZ
pK
0 t
1
þ1
þE
2 qk juðs; yÞ vðs; yÞj G ðt
s; x
yÞ dy ds a qyk ! 2
E
k¼0
Z
þ1
juðs; yÞ vðs; yÞj2 G2a ðt s; x yÞ dy ds
1
m
ðt sÞ a sup Ejuðs; yÞ vðs; yÞj2 ds. 0
y
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Applying Gronwall Lemma, we get Ejuðt; xÞ vðt; xÞj2 ¼ 0, which ensures the uniqueness in the sense of Definition 3. & Remark 1. In multi-dimensional case we cannot always obtain a field solution. In fact, taking k ¼ 0 and replacing x and a in Eq. (1) by x ¼ ðx1 ; x2 ; . . . ; xd Þ; d41 and a ¼ ða1 ; a2 ; . . . ; ad Þ; 1oai o2, and setting x Dad ¼x1 Dad þx2 Dad þ þxd Dad , it is easy to Q see that the Green function is equal to G a ðt; xÞ ¼ i¼d i¼1 G ai ðt; xi Þ. In order that the 2 stochastic integral is defined in L ðOÞ we need the condition a11 þ a12 þ a1d o1. In the case a1 ¼ a2 ¼ ¼ ad this will be equivalent to doa. This condition coincides also with the nonexistence of the process solution in the one-dimensional case when ao1.
3. Regularity of the solution Lemma 3. For k 2 f0; 1; . . . ; mg, mp½a, we have (i) for
aþ1 aþ1 aþkþ1 ogo kþ1,
Z
þ1 Z þ1
0
(ii) for
1
g qk qk k G a ð1 þ v; zÞ k G a ðv; zÞ dz dvo1, qz qz
aþ1 aþ1 kþ2 ogo kþ1
aþ1 when koa 1 and for 1pgo kþ1 when k ¼ ½a, g Z þ1 Z þ1 k qk q k G a ðv; 1 þ zÞ k G a ðv; zÞ dz dvo1. qz 0
1 qz
Proof. (i) Let us write Z
þ1 0
Z
þ1
ðkÞ g jGðkÞ a ð1 þ v; zÞ G a ðv; zÞj dz dv
1 Z 1 Z þ1 ðkÞ g ¼ jGðkÞ a ð1 þ v; zÞ G a ðv; zÞj dz dv 0
1 Z þ1 Z þ1 ðkÞ g jG ðkÞ þ a ð1 þ v; zÞ G a ðv; zÞj dz dv. 1
1
The first integral in the equality above is finite by Corollary 2. For the second one, 1 we use the property (vi) in Lemma 1 and the change of variable z ¼ ð1 þ vÞa z0 ,
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we obtain Z þ1 Z 1
Z
þ1 ðkÞ g jGðkÞ a ð1 þ v; zÞ G a ðv; zÞj dz dv
1 þ1
v
¼ 1
Z
þ1
1
gðkþ1Þ a
1
ð1 þ vÞa
kþ1
1 !g a v 1 þ v a 0 0 ðkÞ ðkÞ 0 G a ð1; z Þ G a 1; z dz dv. 1þv v
Further, kþ1
1a !g a v 1 þ v ðkÞ G ðkÞ 1; z a ð1; zÞ G a 1þv v ! " g
1 1 þ v a ðkÞ p2g 1 GðkÞ ð1; zÞ
G 1; z a a v # g kþ1
a v ðkÞ þ 1 jG a ð1; zÞjg , 1þv and by Corollary 1
1a !g Z þ1 1 þ v ðkÞ 1; z dz Ga ð1; zÞ GðkÞ a v
1 2 g Z 1 Z z aþk jxj pK a 4 dx dz 2 vþ1 1a ð1 þ jxjaþkþ1 Þ
1 ð v Þ z g Z 1 Z ðvþ1Þa1 z g Z 0 Z z v dx dz þ dx dz þ vþ1 1a
1 ð v Þ z 0 z 3 g Z þ1 Z ðvþ1Þ1a z v xaþk 5 þ dx dz z ð1 þ xaþkþ1 Þ2 1
aþkþ1 g Z þ1 zaþkþ1 g v þ 1 a p2K a 1 þ . ð1 þ zaþkþ1 Þ2 dz 1 v 1 Therefore, kþ1
1a !g Z þ1 a v 1 þ v ðkÞ G ðkÞ 1; z dz a ð1; zÞ G a 1 þ v v
1 ! g
aþkþ1 g
kþ1 a v þ 1 a v pK a;g 1 þ 1
. v vþ1
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By the Lagrange–Taylor formula we obtain for 8vX1, g g
kþ1 v þ 1aþkþ1 a a v
1 þ 1 pK a;g v g . v vþ1 Hence Z
þ1 1
Z
þ1 ðkÞ g jGðkÞ a ð1 þ v; zÞ G a ðv; zÞj dz dv
1
Z
pK a;g
þ1
v
gðaþkþ1Þ a
1
ð1 þ vÞa dv.
1 aþ1 . This last integral converges when g4 aþkþ1
(ii) We have Z
g qk qk k G a ðv; 1 þ zÞ k G a ðv; zÞ dz dv qz 0
1 qz g Z 1 Z þ1 k qk q ¼ k G a ðv; 1 þ zÞ k Ga ðv; zÞ dz dv qz qz 0
1 g Z þ1 Z þ1 k qk q þ k G a ðv; 1 þ zÞ k G a ðv; zÞ dz dv. qz 1
1 qz þ1
Z
þ1
The function x !xg ; xX0 is convex and by Corollary 2, we have Z
g qk qk k Ga ðv; 1 þ zÞ k G a ðv; zÞ dz dv qz 0
1 qz g Z 1 Z þ1 k q p2g k G a ðv; zÞ dz dvo1. qz 0
1 1
Z
þ1
On the other hand, Z
g qk qk k G a ðv; 1 þ zÞ k G a ðv; zÞ dz dv qz qz 1
1 Z þ1 Z þ1
gðkþ1Þ
1
1 ðkÞ g a a ¼ v a jGðkÞ a ð1; v ð1 þ zÞÞ G a ð1; v zÞj dz dv, þ1
Z
1
þ1
1
1775
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1
a where G ðkÞ a ð1; v zÞ is the derivative of order k of the function G a ð1; zÞ, taken in the
1 point v a z. Further
Z
þ1
1
1
ðkÞ g a a jG ðkÞ a ð1; v ð1 þ zÞÞ G a ð1; v zÞj dz
1 g Z þ1 Z v 1a ð1þzÞ ðkþ1Þ jGa ð1; xÞj dx dz p 1
1 v a z g "Z
1 þ1 Z v a ð1þzÞ jxjaþk pK a dx dz aþkþ1 2 1
1 ð1 þ jxj Þ 2va vaz Z 2v1a Z v 1a ð1þzÞ g jxjaþk þ dx dz 1 aþkþ1 2 ð1 þ jxj Þ vaz
1 # Z þ2v1a Z v 1a ð1þzÞ g dx dz , þ 1 1 vaz
2va
ð9Þ
1
By a simple calculus of the primitive and the change of the variable z0 ¼ v a z, we find Z
Z 1 g v a ð1þzÞ xaþk dx dz aþkþ1 2 1
1 ð1 þ x Þ 2va vaz g Z þ1 aþkþ1 aþkþ1 v a ðz þ 1Þaþkþ1 v a zaþkþ1 p dz aþkþ1 aþkþ1 ð1 þ v aþkþ1 a zaþkþ1 Þð1 þ v a ðz þ 1Þ Þ 0 g Z þ1
1 ðz0 þ v a Þaþkþ1 z0 aþkþ1 1 0 pva dz . ð1 þ z0 aþkþ1 Þð1 þ ðz0 þ v 1a Þaþkþ1 Þ 0 þ1
By the Taylor–Lagrange formula, we can obtain the following estimation: 8vX1 and 8z 2 R,
1
1
jjz þ v a jaþkþ1 jzjaþkþ1 jpKv a ðjzjaþk þ jzjaþk 1 þ jz þ 1jaþk 1 Þ, hence g Z 1 v a ð1þzÞ xaþk dx dz aþkþ1 2 1
1 ð1 þ x Þ 2va vaz g Z þ1 aþk 1 g z þ zaþk 1 þ ðz þ 1Þaþk 1 pK a;g v a dz. ð1 þ zaþkþ1 Þ2 0
Z
þ1
ð10Þ
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In the same way we see that Z
1
2va
1
g Z 1 v a ð1þzÞ jxjaþk dx dz 1 aþkþ1 2 vaz ð1 þ jxj Þ
is upper bounded by the right side of (10). Hence Z þ1 Z þ1 Z g jG a ðv; 1 þ zÞ G a ðv; zÞj dz dvpK a;g 1
þ1
v
1 gðkþ2Þ a
dv,
1
1
aþ1 which converges when g4 kþ2 .
&
The following theorem gives the spatial and the temporal regularity of the solution of (5). Theorem 2. Under conditions (2) and the condition that u0 is Lp -bounded for all pX2, we have
For fixed x 2 R the process fuðt; xÞ; t40g has Ho¨lder continuous trajectories with a m exponent minfa 1 2a ; a g , for any 40, P-a.s. For ao3 and for fixed t the process fuðt; xÞ; x 2 Rg has Ho¨lder continuous trajectories with exponent minfa 1 2 ; a ½ag , for any 40, P-a.s.
Proof. (i) It is easy to see that u0 ðt; xÞ is smooth function with respect to t and to x. Let x 2 R be fixed and let t40, y40, and n41, and let Z t Z þ1 ðkÞ J 1;n ðt; x; yÞ:¼E hk ðs; y; uðs; yÞÞ 0
1 n ! qk qk
G ðt þ y
s; x
yÞ
G ðt
s; x
yÞ dy ds , a a qyk qyk n Z tþy Z þ1 k q LðkÞ hk ðs; y; uðs; yÞÞ k G a ðt þ y s; x yÞ dy ds , n ðt; x; yÞ:¼E qy t
1 Z t Z I 1;n ðt; x; yÞ:¼E 0
þ1
1
Z Ln ðt; x; yÞ:¼E
t
tþy
Z
n f ðs; y; uðs; yÞÞðGa ðt þ y s; x yÞ G a ðt s; x yÞÞW ðdy dsÞ , þ1
1
n f ðs; y; uðs; yÞÞG a ðt þ y s; x yÞW ðdy dsÞ .
Applying Burkholder–Davis–Gundy inequality, Ho¨lder inequality, the fact that the solution u is Lp -bounded for all p41 and using conditions (2), it is easy
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to see that
Z t Z
þ1
jf ðs; y; uðs; yÞÞjn ðG a ðt þ y s; x yÞ 2
G a ðt s; x yÞÞ dy ds
I 1;n ðt; x; yÞpK n E
0
Z t Z
1
n2 1
þ1 2
ðG a ðs þ y; yÞ G a ðs; yÞÞ dy ds
0
1
pK n sup Ejf ðs; y; uðs; yÞÞj
n
Z t Z
½0;T R
n2
þ1 2
ðG a ðs þ y; yÞ G a ðs; yÞÞ dy ds 0
,
1
and Ln ðt; x; yÞpK n sup Ejf ðs; y; uðs; yÞÞj
n
Z
½0;T R
y
0
Z
n2
þ1
G2a ðs
þ y; yÞ dy ds
.
1
1
By changes of variables s ¼ yv; y ¼ ya z and by Lemma 3 and Corollary 2, we get I 1;n ðt; x; yÞ þ Ln ðt; x; yÞ " Z þ1 Z a 1 n 2a pK n y 0
Z
2
Z
ðGa ðs þ 1; yÞ Ga ðs; yÞÞ dy ds
1
G 2a ðs; yÞ dy ds
0
1
na 1 2a
.
pK n;a y
2
þ1
þ
n2
þ1
n2 #
ð11Þ
Using Ho¨lder inequality and by following the same technique as above, we find n J ðkÞ 1;n ðt; x; yÞp sup Ejhk ðs; y; uðs; yÞÞj ½0;T R
!n k qk q k G a ðs þ y; yÞ k G a ðs; yÞdy ds qy
1 qy !n Z þ1 Z þ1 k qk q k G a ðs þ 1; yÞ k Ga ðs; yÞ dy ds , qy 0
1 qy
Z tZ
0
pK n yn
a k a
þ1
and LðkÞ n ðt; x; yÞp
sup Ejhk ðs; y; uðs; yÞÞj ½0;T R
pK n yn
a k a
n
Z 0
Z 0
y
Z
!n k q k Ga ðs þ y; yÞdy ds qy !n
þ1
1
þ1 k q k Ga ðs; yÞdy ds qy
1
2Z
,
ARTICLE IN PRESS L. Debbi, M. Dozzi / Stochastic Processes and their Applications 115 (2005) 1764–1781
1779
hence a k
ðkÞ n a J ðkÞ . 1;n ðt; x; yÞ þ L1;n ðt; x; yÞpK n;a y
(12)
From (11) and (12), we obtain the result. (ii) Let ao3, t40 be fixed. For x 2 R, h40, and for n41, we have " Ejuðt; x þ hÞ uðt; xÞj pK n L0;n ðt; x; hÞ þ n
m X
# J ðkÞ 2;n ðt; x; hÞ
þ I 2;n ðt; x; hÞ ,
k¼0
where Z L0;n ðt; x; hÞ:¼E
þ1
1
n ðG a ðt; x þ h yÞ G a ðt; x yÞÞu0 ðyÞ dy dy ds ,
Z t Z
J ðkÞ 2;n ðt; x; hÞ:¼E
0
þ1
hk ðs; y; uðs; yÞÞ
1
n ! qk qk
G ðt
s; x þ h
yÞ
G ðt
s; x
yÞ dy ds , a a qyk qyk
and Z t Z I 2;n ðt; x; hÞ ¼ E
þ1
f ðs; y; uðs; yÞÞðG a ðt s; x þ h yÞ n
Ga ðt s; x yÞÞW ðdy dsÞ . 0
1
The integrals in the terms L0;n ðt; x; hÞ and J ðkÞ 2;n ðt; x; hÞ for kpmo½a are smooth functions, so it is sufficient to estimate the regularity of the terms J ð½aÞ 2;n ðt; x; hÞ and I 2;n ðt; x; hÞ. We apply the Burkholder–Davis–Gundy inequality, the Ho¨lder inequality, conditions (2) and the fact that the solution is Lp -bounded, to get I 2;n ðt; x; hÞpK n sup Ejf ðs; y; uðs; yÞÞjn ½0;T R
pK n h
na 1 2
Z
Z t Z 0
þ1
Z
þ1
n2 ðG a ðs; y þ hÞ Ga ðs; yÞÞ2 dy ds
1
þ1 2
ðG a ðs; y þ 1Þ G a ðs; yÞÞ dy ds 0
1
n2 ð13Þ
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and n J ð½aÞ 2;n ðt; x; hÞp sup Ejh½a ðs; y; uðs; yÞÞj ½0;T R
½a n q q½a
½a G a ðs; y þ hÞ qy½a G a ðs; yÞdy ds 0
1 qy
Z þ1 Z þ1 ½a n q q½a dy ds . pK n hnða ½aÞ G ðs; y þ 1Þ
G ðs; yÞ a a ½a qy½a 0
1 qy
Z t Z
þ1
ð14Þ By Lemma 3 the integrals on the right side of (13) and (14) converge, hence 1
a 1 2 ;a ½ag
ðEjuðt; x þ hÞ uðt; xÞjn Þn pK n;a hminf
:
&
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