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Equations of Mathematical Physics and Compositions of Brownian and Cauchy Processes a
a
L. Beghin , E. Orsingher & L. Sakhno a
b
Dipartimento di Scienze Statistiche, “Sapienza” Università di Roma, Roma, Italy
b
Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine Published online: 21 Jun 2011.
To cite this article: L. Beghin , E. Orsingher & L. Sakhno (2011): Equations of Mathematical Physics and Compositions of Brownian and Cauchy Processes, Stochastic Analysis and Applications, 29:4, 551-569 To link to this article: http://dx.doi.org/10.1080/07362994.2011.581071
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Stochastic Analysis and Applications, 29: 551–569, 2011 Copyright © Taylor & Francis Group, LLC ISSN 0736-2994 print/1532-9356 online DOI: 10.1080/07362994.2011.581071
Equations of Mathematical Physics and Compositions of Brownian and Cauchy Processes
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L. BEGHIN,1 E. ORSINGHER,1 AND L. SAKHNO2 1
Dipartimento di Scienze Statistiche, “Sapienza” Università di Roma, Roma, Italy 2 Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine We consider different types of processes obtained by composing Brownian motion Bt, fractional Brownian motion BH t and Cauchy processes Ct in different manners. We study also multidimensional iterated processes in d like, for example, B1 Ct Bd Ct and C1 Ct Cd Ct, deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equation, are satisfied by the laws of the iterated processes considered in the work. Similarly, we prove that some processes like CB1 B2 Bn+1 t are governed by fractional diffusion equations. Keywords Cauchy process; Fractional Brownian motion; Fractional partial differential equations; Iterated Brownian motion; Riesz fractional derivative; Vibration of rods; Wave equation. Mathematics Subject Classification 60K99; 35Q99.
1. Introduction The iterated Brownian motion is obtained by the composition of two independent Brownian motions B1 and B2 , as follows: It = B1 B2 t t > 0
(1.1)
Recently this kind of processes has been studied by many authors (see, e.g., [1, 2, 4–6, 11]). As far as the applications are concerned, it has been observed that the iterated Brownian motion I is suitable to describe diffusions in cracks [8] and many other physical phenomena. In particular, fractures on a rectangular slab can be viewed at as trajectories of a Brownian motion (see [7]). The flow of a gas on a Received June 4, 2010; Accepted February 4, 2011 Address correspondence to E. Orsingher, Dipartimento di Scienze Statistiche, “Sapienza” Università di Roma, p.le A.Moro 5, 00185, Roma, Italy; E-mail:
[email protected]
551
552
Beghin et al.
fracture can be represented by a Brownian motion moving on a Brownian sample path and therefore this model produces an iterated Brownian motion (see, e.g., [12]). In Orsingher and Zhao [16] and in De Blassie [8] it is proved that the density of (1.1) satisfies the fourth-order equation 1 4 p 1 p d2 x t = 3 4 + √ x t 2 x 2 2t dx2
(1.2)
for x ∈ and t > 0 and in Orsingher and Beghin [14] it is proved that it coincides with the solution to the fractional equation 1
2 p
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t
1 2
=
1 2 p x ∈ t > 0 x2
(1.3)
23/2
(with initial condition px 0 = x). For general information on fractional time derivatives of the type (1.3), consult Podlubny [17]. Generalizations of this result have been obtained in many directions. First of all, by considering the n-times iterated Brownian motion In t = B1 B2 Bn+1 t
(1.4)
involving n + 1 independent one-dimensional Brownian motions. It has been shown in Orsingher and Beghin [15] that the law of (1.4) satisfies the following fractional equation 1
2n p 1
t 2n
1
= 2 2n −2
2 p x ∈ t > 0 x2
(1.5)
with px 0 = x For the vector process B1 Bt ··· I d t = ··· Bd Bt
t > 0
(1.6)
where B and B1 Bd are mutually independent Brownian motions, it is proved in Orsingher and Beghin [15] that the joint law pId x1 x2 xd t = 2
0
w2
x2
− k d e 223 4 t e− 2w dw √ 223 4 t k=1 2w
(1.7)
is a solution to the fractional equation 1
2 p t
1 2
= 2
d 2 p j=1
xj2
xj ∈ j = 1 d t > 0
(1.8)
Result (1.7) shows that the components of the d-dimensional vector process (1.6) are no longer independent and the parameter enters into the variance of the “time process” Bt t > 0
Compositions of Brownian and Cauchy Processes
553
For the law of the vector process I d t we can also write the equation which is analogous to (1.2):
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1 p x t = 3 t 2
d 2 2 j=1 xj
2
d d d2 1 p+ √ xk 2 2t j=1 dxj2 k=1
(1.9)
with initial condition px1 xd 0 = dk=1 xk Some further extensions can be found in Baeumer et al. [3], Nane [13], Allouba and Zheng [2], and De Blassie [8]. In D’Ovidio and Orsingher [9] various types of processes obtained by composing independent fractional Brownian motions BH1 t and BH2 t t > 0 (with Hurst parameters respectively equal to H1 H2 ∈ 0 1) have been studied. In particular, for the density pH1 H2 x t of BH1 BH2 t it has been shown that the governing equation has the following structure:
2 2 p p 2 p 2 2 2 p x ∈ t > 0 + t 2 = H1 H2 2x +x 1 + H1 H2 t t t x x2 In this article, we extend some of these results in several directions, by considering different compositions of Brownian motions, fractional Brownian motions and Cauchy processes. We start by studying the iterated Brownian motion defined in (1.1) in the case where B1 is endowed by drift, giving the two governing equations, the fractional and fourth-order one (see (2.4) below). Another direction of our research concerns processes obtained by combining Brownian motions (possibly fractional Brownian motions) and Cauchy processes. In particular, we study the multidimensional vector processes B1 Ct Bd Ct and C1 Bt Cd Bt, which involve independent Brownian motions Bj t t > 0 and Cauchy processes Cj t t > 0 For d = 1 it is has been proved in D’Ovidio and Orsingher [9] that the law of BCt t > 0, given by 2 e− 2s t pBC x t = ds x ∈ t > 0 √ 2 0 2s t + s2 x2
(1.10)
satisfies the following fourth-order equation 1 4 p 1 d2 2 p = − − x x ∈ t > 0 t2 22 x4 t dx2
(1.11)
with initial condition px 0 = x We extend this result to the case where the Brownian motion is substituted by the fractional Brownian motion BH t t > 0, with Hurst parameter H ∈ 0 1. We show that the density function of BH Ct t > 0 resolves the following equation t2
2 2 p 2Ht d2 2 2 x − H p − = − HH − 1 x x1 0 0 has been considered in D’Ovidio and Orsingher [9] and its connection with the wave equation 2 p 1 2 p = − x ∈ t > 0 2 2 t x tx2
(1.12)
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has been established. We will show here that when the process C1 is endowed with a position parameter a = 0, the distribution of the iterated Cauchy process satisfies the following wave equation 2 p 1 2 p = − x ∈ t > 0 2 2 t x tx − a2
(1.13)
Finally, we consider the d-dimensional vector C1 Ct Cd Ct and obtain the equation governed by its density, i.e., d x1 xd t = pCC
2 d+1
d + 0
j=1
s2
s t ds 2 2 + xj t + s 2
(1.14)
where the use of partial Riesz fractional derivatives is required.
2. Iterated Brownian Motion with Drift If the processes composing the iterated Brownian motions possess drift, the fractional equations and the higher-order equations governing the distributions are somewhat different. We start by considering the process I t = B1 B2 t t > 0, with law pI x t
=2
x− s2 2s
e− √
0
s2
e− 2t ds √ 2s 2t
(2.1)
It has been shown in Beghin and Orsingher [5] that (2.1) solves the following fractional equation of order = 1/2: 1
2 p
=
1
t 2
1 2 p p −√ x ∈ t > 0 3/2 2 2 x 2 x
(2.2)
As a check we evaluate the Laplace–Fourier transform of (2.1):
e− t
+ −
0
=2
e 0
eix 2
0
− 21 2 s+i s e
−s
√
s2 e− 2t ds dx dt √ 2s 2t
x− s2 2s
e− √
1
2 −1
2
2
ds =
2 23/2
−
i √ 2
+
√
(2.3)
Compositions of Brownian and Cauchy Processes
555
The previous result coincides with the Fourier–Laplace transform of (2.2). We show in the following theorem that, as in the case where = 0, the density of I t satisfies also a fourth-order equation. Theorem 2.1. The density of the process I t = B1 B2 t t > 0 given in (2.1) is a solution to the following equation 1 px t = t 2
1 2 − 2 2 x x
2 px t + √
1
2t
d 1 d2 − x x ∈ t > 0 2 dx2 dx (2.4)
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with initial condition px 0 = x. Proof. By taking the first-order derivative of (2.1), we have that 2 s2 e− x− s 2s e− 2t p x t = 2 ds √ √ t I 0 2s t 2t 2 s2 e− x− s 2s 2 e− 2t = ds √ √ 0 2s s2 2t = integrating by parts
x− s2
s2
e− 2s e− 2t =− ds √ √ s 2s s 2t 0 2 x− s2 s2 s2 2 e− x− s 2s e− 2s e− 2t e− 2t =− ds √ √ √ √ + s 2s 2t s2 0 2s 2t 0
x− s2 s2 1 2 d e− 2s e− 2t 1 d2 − − x + ds √ √ dx s 2 x2 x 0 2t 2 dx2 2s 2t 2 1 d 1 1 2 1 d2 − pI x t + √ − x (2.5) = 2 2 x2 x dx 2t 2 dx2
= √
1
Remark 2.1. The differential operator appearing in the fourth-order equation (2.4) is the formal square of the operator appearing in the fractional equation (2.2). In the special case where = 0, we obtain again Equation (1.2). We consider now the case where the process representing the “time” possesses drift, then the law of the iterated process I t = B1 B2 t t > 0, reads 1 e− 2s e− 2t = ds √ √ Dt 0 2s 2t s− t2
x2
qI x t where
Dt = Pr
B2 t
>0 =
√ − t
is the normalizing factor for the density of B2 t
y2
e− 2 √ dy 2
(2.6)
556
Beghin et al. The Fourier transform of qI x t becomes
+
−
√
1 e− 2 tw+ t− 2 = dw √ Dt − √t 2w t + t 2
eix qI x tdx
w2
(2.7)
The evaluation of the Laplace transform of (2.7) poses serious problems. For this reason the case where the process representing the “time” possesses drift is not further developed here.
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3. Iterated Processes Involving the Cauchy Process The transition function of a centered Cauchy process Ct t > 0 is given by pC x t =
t2
t x ∈ t > 0 + x2
(3.1)
and it is well known that pC is a solution to the Laplace equation
2 2 + 2 2 x t
pC x t = 0
(3.2)
Furthermore, pC is a solution to the following space-fractional equation p x t = − p x t t C x C
(3.3)
fy 1 d x fy fx = dy − dy x dx − x − y y−x x
(3.4)
where
is a special case (for = 1) of the Riesz fractional space-derivative and possesses Fourier transform equal to
+
−
eix
+ eix fxdx = fxdx = x −
(3.5)
By composing the standard Brownian motion and the Cauchy process, we obtain the following new processes: JBC t = BCt t > 0
(3.6)
JCB t = CBt t > 0
(3.7)
and
where C and B are mutually independent.
Compositions of Brownian and Cauchy Processes
557
An alternative definition of the first-order fractional derivative is given in Saichev and Zaslavsky [18] (see formula (A.39)) and reads 1 + fx − y − 2fx + fx + y d dy fx = − dx 0 y2
(3.8)
This is a special case of + fx − y − 2fx + fx + y d 1 fx = dy dx 2− cos 2 0 y2
(3.9)
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for 0 < ≤ 1. We note that (3.9) reduces to (3.8) for = 1 because, by the reflection formula of the Gamma function, the constant can be rewritten as 1 1 + 1 =− 2− cos 2 2 cos 2 sin =−
1 + sin 2
which, for = 1, yields −1/ We can convert definition (3.4) into (3.8) by means of integrations by parts, after suitable changes of variables. As a further check we can show that the Fourier transform of (3.8) coincides with (3.5): 1 + ix + fx − y − 2fx + fx + y e dy dx − − y2 0 + eiy + 1 1 + 2 + ix =− fweiw dw dy + fxe dx dy + y2 − y2 − 0 0 + e−iy 1 + − fweiw dw dy y2 − 0 + eiy − 2 + e−iy 1 = − dy (3.10) y2 0 The last integral can be evaluated as follows:
+ e−iy − 1 eiy − 1 dy + dy y2 y2 0 y + 1 y + 1 iw dy ie dw − dy ie−iw dw = y2 y2 0 0 0 0 + + 1 + + 1 −iw = i eiw dw dy − i e dw dy y2 y2 0 w 0 w + iw + e−iw e = − 2 dw − 2 dw 2wi 2wi 0 0 + sin w = −2 dw = − w 0
+ 0
which, inserted into (3.10), gives (3.5).
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Beghin et al.
It has been shown in D’Ovidio and Orsingher [9] that the transition density of the process JBC t = BCt, t > 0, which is given by 2 e− 2s t pBC x t = ds x ∈ t > 0 √ 2 0 2s t + s2 x2
(3.11)
satisfies the following equation 2 p 1 4 p 1 d2 =− 2 4 − x x ∈ t > 0 2 t 2 x t dx2
(3.12)
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with initial condition px 0 = x Remark 3.1. In mathematical physics the equation 2 u 4 u = −K t2 x4
(3.13)
represents the vibrations of rods (see Elmore and Heald [10], p. 116). Equation (3.13) coincides with (3.12) for x = 0 Remark 3.2. The density (3.11) can be worked out in an equivalent form, by applying the subordinating relationship e− 2w te− 2w t = dw √ √ t2 + s2 0 2w 2w3 s2
t2
Thus, the transition density of the process IBC t = BCt, t > 0 can be rewritten as pBC x t = 2
0
x2
e− 2s √ 2s
0
s2 t2 e− 2w te− 2w dw ds √ √ 2w 2w3
(3.14)
Formula (3.14) corresponds to the law of an iterated Brownian motion It = B1 B2 t, taken at a random time Tt which coincides in distribution with the first-passage time of a standard Brownian motion. Thus, (3.14) coincides with the density of the process ITt t > 0, where Tt = infs Bs = t We consider now the composition of a fractional Brownian motion with a Cauchy process, JBH C t = BH Ct t > 0. Theorem 3.1. The density of JBH C t t > 0, which is given by t 2 + e− 2s2H ds pBH C x t = √ 2 0 2s2H t + s2 x2
satisfies the following equation 2 p 2 2Ht d2 t2 2 = − HH − 1 x − H 2 2 x2 p − x1 0 0, which is given by Theorem 3.2. The joint density of JBC xj2
d e− 2s 2 + t d x1 xd t = ds pBC √ 0 j=1 2s t2 + s2
(3.27)
is a solution to the following equation
2 d d 2 p 2 1 1 d2 = − px x t − x xj xd (3.28) 1 d 1 t2 22 j=1 xj2 t j=1 dxj2 with initial condition px1 xd 0 =
d j=1
xj .
Proof. By taking the second order time-derivative of (3.27) we get, by means of two integrations by parts, that x2 d 2 2 − 2sj + e 2 d t ds + p x xd t = − √ t2 BC 1 0 s2 j=1 2s t2 + s2 + x2 d − 2sj 2 e t + √ 2 2 s j=1 2s t + s
d 2 1 2 j=1 xj2
2
0
d +
xj2
e− 2s t ds + √ 2 + s2 t 0 2s j=1 + xj2 d d 2 − e 2s 1 t + √ t2 + s2 j=1 xj2 j=1 2s
2 =−
(3.29)
0
which coincides with (3.28).
Compositions of Brownian and Cauchy Processes
563
d Remark 3.4. In view of Remark 3.2, we note that the vector process JBC t is equivalent in distribution to the following d-dimensional process:
I1 Tt ···
t > 0
··· Id Tt
where Ij , j = 1 n are independent iterated Brownian motions and Tt coincides with the first-passage time of a standard Brownian motion.
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We focus now our attention on the process JCB t = CBt t > 0 and prove that its density s e− 2t 2 + pCB x t = ds √ 0 s2 + x2 2t s2
(3.30)
satisfies a nonhomogeneous backward heat equation. Theorem 3.3. The density (3.30) is a solution to the following equation 1 2 p 1 p =− + x ∈ t > 0 √ 2 2 t 2 x x 2t
(3.31)
with initial condition px 0 = x. Proof. By taking the time-derivative of pCB = pCB x t, we have that
s2 e− 2t ds √ 2t s2
e− 2t s 2 1 + = ds √ 0 s2 + x2 s2 2t − s2 + e 2t s 1 + = − √ + 0 s s2 + x2 2t 0
s p 2 + = 2 2 t 0 s + x t
1 + 2 + 0 s2 which coincides with (3.31).
s 2 s + x2
s2
e− 2t ds √ 2t
The previous results can be further extended to the case where the “time process” is represented by a n-times iterated, instead of standard, Brownian motion.
564
Beghin et al.
We prove that the density of the process CIn t = CB1 B2 Bn+1 t t > 0 is a solution to a non-homogeneous fractional equation. Theorem 3.4. The density of the process CIn t t > 0, which is given by
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pCI x t =
s 2 + p s tds 0 s 2 + x2 n
(3.32)
where
pn x t = 2
n
+ 0
···
−
x2
−
w2 1
2 wn
e 2w1 e 2w2 e− 2t dw1 dw2 √ dwn 2w1 2w2 2t
+ 0
satisfies the following equation 1
2n p 1
t 2n
= −2
1 2n
1 1 p 2n−1+ 2n+1 − 21 t− 2n+1 1 2 x ∈ t > 0 + x2 3/2 − 2n+1 x
−2
2
(3.33)
with initial condition px 0 = x Proof. We start by taking the fractional time-derivative of order 1/2n of the density (3.32): 1
2n t
1 2n
s 2n 2 + p s tds 0 s2 + x2 t 21n n 1 s 2 2 2n −1 + p s tds = s2 + x2 s2 n 0 + 1 2 2n −1 s = s t p + s2 + x2 s n 1
pCI x t =
0
s − p s tds 2 2 s + x s n 0 + 1 2 2n −1 s =− p s t + n s s2 + x2 +
2
1 2n
−1
+
s
2
1 2n
−1
+
2
0
s2
s s 2 + x2
0
pn s tds
1
2 2n −1 1 2 1 2n −2 = p 0 t − 2 p x t n x2 x2 CI
(3.34)
Compositions of Brownian and Cauchy Processes
565
We concentrate now on the first term and evaluate +
+
w2 1
w2 2
+ e 2w3 + e 2w2 e− 2t pn 0 t = 2n dw1 dw2 · · · dwn √ 0 0 0 2w1 2w2 2wn 2t 3 + e− w2tn2 2n− 4 41 + − 41 − 2ww22 dw = √ n+1 w2 e 3 dw2 · · · √ wn t n 0 0 2 1 2n 1 −1−1 3 1 1 1 − − 21 − 41 − n 2 2 8 2 2 2 − = √ n+1 2 4 8 2 2n 2
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−
−
1 n
−
2 wn
2 wn
wn 2 e− 2t × dwn √ 0 t 1 1 1 2n 1 −1−1 3 1 − 21 − 41 − 21 − 21n 2 8 2 2 − 2− 2 − 2n+1 = √ n+1 2 4 8 2 2n 2 1 1 1 × − n+1 t− 2n+1 2 2 1 1 1 1 1 1 1 1 1 2−1+ 2n+1 − n+1 1 − − − − = t 2 n+1 2 22 2 23 2 2n 2 2n+1 2 = by the duplication property of the Gamma function 1 − 21 1 2−1+ 2n+1 − n+1 n 1 n+1− n 2 1 = t 2 22 n+1 − 2n+1 2 1 − 21 1 2n− 2n+1 − n+1 1 = t 2 1 − 2n+1 2 which, multiplied by the constant appearing in (3.34), gives the final form of Equation (3.33). In the d-dimensional case we consider the vector process C1 Bt ··· Cd Bt
t>0
(3.35)
and obtain the governing fractional equation in the following theorem. Theorem 3.5. The joint probability law of the process defined in (3.35) reads d pCB x1 xd t
s2 d s e− 2t 2 + = d ds √ 0 j=1 s2 + xj2 2t
(3.36)
566
Beghin et al.
and satisfies, for d > 1, the following fractional equation d k−1 d p 2 p 2 p xj ∈ j = 1 d t > 0 =2 − t xk xj k=1 xk2 k=2 j=1
with initial condition px1 xd 0 =
d j=1
(3.37)
xj .
Proof. The time-derivative of (3.36) can be evaluated as follows:
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d p x xd t t CB 1
s2 e− 2t ds √ 2t s2
d e− 2t s 2 1 + ds = d √ 2 2 2 0 j=1 s + xj s 2t
d 2 + s = d 0 j=1 s2 + xj2 t
1 =− d s
d
s 2 2 j=1 s + xj
+
s2 s2 d s e− 2t e− 2t 1 + 2 ds √ √ + d 0 s2 j=1 s2 + xj2 2t 0 2t
+
s2 s2 d d d e− 2t e− 2t 1 + 2 s s s 1 =− d ds √ √ + d k=1 j=1 s2 + xj2 s s2 + xj2 0 s2 j=1 s2 + xj2 2t 2t 0
j=k
(3.38)
The first term in the last member is equal to zero (unlike the one-dimensional case) and this makes Equation (3.37) homogeneous. Since s
d
s 2 2 j=1 s + xj
=
d d
s 2 2 s k=1 j=1 s + xj
s 2 s + xk2
j=k
the second-order derivative becomes 2 s2
d
s 2 2 j=1 s + xj
=
d d s
k=1
j=1 j=k
s 2 s + xk2
s
s 2 s + xj2
d l=1 l=kj
s 2 + s2 s2 + xl2
s 2 s + xk2
d l=1 l=k
s2
s + xl2 (3.39)
Compositions of Brownian and Cauchy Processes
567
In view of (3.2) and (3.3), we can rewrite (3.39) as 2 s2
d
s 2 + x2 s j j=1
d d = x k k=1 j=1
s s2 + xk2
xj
j=k
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−
2
xk2
s2
s + xk2
s s2 + xj2
d l=1 l=k
s2
d l=1 l=kj
s + s2 + xl2
s + xl2
d d d d s 2 = − xk xj l=1 s2 + xl2 k=1 xk2 k=1 j=1
d
s 2 2 l=1 s + xl
(3.40)
j=k
By inserting (3.40) in (3.38), we arrive at Equation (3.37), by applying the commutative property of the fractional derivative (3.8). Therefore, we show that 2 = = xj xk xk xj xk xj The second-order fractional derivative reads +
1 fx − t y − 2fx y + fx + t y 2 fx y = − dt yx y 0 t2 1 + + fx − t y − z − 2fx − t y + fx − t y + z = 2 dz dt + 0 z2 t 2 0 2 + + fx y − z − 2fx y + fx y + z − 2 dz dt + 0 z2 t 2 0 2 + + fx + t y − z − 2fx + t y + fx + t y + z + 2 dz dt 0 z2 t 2 0 (3.41) 2
It is easy to check that xy fx y produces the same result and thus the commutativity of the second-order fractional derivative holds.
We consider now the last case of composition of Cauchy processes: it can be seen that the density of the process C1a C2 t t > 0 where the external process is endowed with a position parameter a ∈ , is a solution to a non-homogeneous wave equation. Indeed, by suitably adapting the proof of Theorem 4.1 in D’Ovidio and Orsingher [9], it is easy to check that pCC x t =
s t 2 + ds 2 2 2 2 0 s + x − a t + s2
568
Beghin et al.
is a solution to 2 p 2 p 1 = − x ∈ t > 0 t2 x2 tx − a2
(3.42)
In the d-dimensional case the iterated Cauchy process can be defined as follows
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C1 Ct d JCC t = ··· Cd Ct
t>0
(3.43)
where C1 Cd and C are mutually independent, standard Cauchy processes. d Theorem 3.6. The density of JCC t t > 0, which can be expressed as
d pCC x1 xd t =
2 d+1
d + 0
j=1
s2
s t ds 2 2 t + s2 + xj
(3.44)
and satisfies, for d > 1, the following fractional equation d d k−1 2 p 2 p 2 p = −2 xj ∈ j = 1 d t > 0 2 2 t xk xj k=1 xk k=2 j=1
with initial condition px1 xd 0 =
d j=1
(3.45)
xj
Proof. The second-time derivative of (3.44) can be evaluated by adapting the proof of the previous theorem, as follows d 2 d s 2 t 2 + ds p x xd t = d+1 2 2 t2 CC 1 t2 t2 + s2 0 j=1 s + xj d 2 + t s 2 = − d+1 ds 2 2 s2 t2 + s2 0 j=1 s + xj
+ d 2 t s = d+1 2 + t + s2 s j=1 s2 + xj2 0
d 2 + 2 t s − d+1 ds t2 + s2 s2 j=1 s2 + xj2 0 = by (3.39) =
d d k−1 2 d 2 d p x x t − 2 x1 xd t pCC 1 d 2 CC x x x k j k k=1 k=2 j=1
Compositions of Brownian and Cauchy Processes
569
d Remark 3.5. The density of JCC t t > 0 can be expressed in the following alternative form
d pCC x1 xd t =
d j=1
2
t 2t ln 2 xj + xj
t2
(3.46)
as can be inferred from the calculations leading to Theorem 4.1 of D’Ovidio and Orsingher [9].
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References 1. Allouba, H. 2002. Brownian-time processes: The p.d.e.connection II and the corresponding Feynman–Kac formula. Trans. of the Amer. Math. Soc. 354:4617–4637. 2. Allouba, H., and Zheng, W. 2001. Brownian-time processes: The p.d.e.connection and half-derivative generator. Ann. Prob. 29(4):1780–1795. 3. Baeumer, B., Meerschaert, M., and Nane, E. 2009. Space-time duality for fractional diffusions. J. Appl. Probab. 46(4):1100–1115. 4. Beghin, L., and Orsingher, E. 2003. The telegraph process stopped at stable-distributed times and its connection with the fractional telegraph equation. Fract. Calc. Appl. Anal. 6(2):187–204. 5. Beghin, L., and Orsingher, E. 2009. Iterated elastic Brownian motions and fractional diffusion equations. Stoch. Proc. Appl. 119(6):1975–2003. 6. Burdzy, K. 1994. Variation of iterated Brownian motion, Lecture Notes, Workshop and Conference on Measure-Valued Processes, Stoch. Partial Diff. Eq. and Interacting Syst., Vol. 5, American Mathematical Society, Providence, RI, pp. 35–53. 7. Chudnovsky, A., and Kunin, B. 1987. A probabilistic model of a brittle crack formation. J. Appl. Phys. 62(10):4124–4129. 8. De Blassie, R. D. 2004. Iterated Brownian motion in an open set. Ann. Appl. Prob. 14(3):1529–1558. 9. D’Ovidio, M., and Orsingher, E. 2011. Composition of processes and related partial differential equations. J. Theoret. Probab. 24:342–375. 10. Elmore, W. C., and Heald, M. A. 1969. Physics of Waves. Dover, New York. 11. Khoshnevisan, D., and Lewis, T. M. 1996. A uniform modulus result for iterated Brownian motion. Ann. Inst. Henri Poincaré 32(3):349–359. 12. Khoshnevisan, D., and Lewis, T. M. 1999. Stochastic calculus for Brownian motion on a Brownian fracture. Ann. Appl. Prob. 9(3):629–667. 13. Nane, E. 2010. Stochastic solution of a class of higher-order Cauchy problems in d . Stochastics and Dynamics 10(3):341–366. 14. Orsingher, E., and Beghin, L. 2004. Time-fractional equations and telegraph processes with Brownian time. Probab. Theory Relat. Fields 128:141–160. 15. Orsingher, E., and Beghin, L. 2009. Fractional diffusion equations and processes with randomly-varying time. Ann. Prob. 37(1):206–249. 16. Orsingher, E., and Zhao, X. 1999. Iterated processes and their applications to higherorder differential equations. Acta Math. Sinica 15(2):173–180. 17. Podlubny, I. 1999. Fractional Differential Equations. Academic Press, San Diego, CA. 18. Saichev, A., and Zaslavsky, G. 1997. Fractional kinetic equations: solutions and applications. Chaos 7(4):753–764.
