Iterated processes and their applications to higher ... - Springer Link

Acta Mathematica Sinica, English Series 1999, April, Vol.15, No.2, p. 173-180

Acta Mathematica Sinica, English Series 9 Springer-Vedag1999

Iterated Processes and Their Applications to Higher Order Differential Equations

Enzo Orsingher Dipartimento di Statistica, Probabilit~ e Statistiche Applicate, Universit~ degli Studi di Roma "La Sapienza", Piazzale Aldo Moro, 5-00185 Roma, Italy E-mail: orsinghe @pow2. sta. uniromal, it Xuelei Zhao Institute of Applied Mathematics, University of Bonn, Bonn 53115, Germany E-mail: [email protected] Institute of Mathematics, Shantou University, Shantou 515063, P. R. China E-mail: [email protected]

A b s t r a c t In this paper we construct models obtained by suitably combining Brownian motions and telegraphs in such a way that their transition functions satisfy higher-order parabolic or hyperbolic equations of different types. Equations with time-varying coefficients are also derived by considering processes endowed either with drift or with suitable modifications of their structure. Finally the distribution of the maximum of the iterated Brownian motion (along with some other properties) is presented. K e y w o r d s Iterated Brownian motion, Telegraph processes, Higher-order parabolic equations, Higherorder hyperbolic equations, Maximum 1 9 9 1 M R S u b j e c t C l a s s i f i c a t i o n 60K99, 60J99, 35G05

1

Introduction

T h e i t e r a t i o n of certain kinds of processes higher order differential equations. F u n a k i two i n d e p e n d e n t B r o w n i a n m o t i o n s in the differential equation. This i t e r a t e d process

has a n i m p o r t a n t a p p l i c a t i o n to the solution of [1], for instance, has c o n s t r u c t e d a n i t e r a t i o n of complex space to solve a fourth order parabolic has now been modified a n d is referred to as the

Received January 31, 1997, Accepted January 8, 1998 This work is partially supported by the Natural Science Foundation of Guangdong Province, National Natural Science Foundation of China grant No. 19501026 and the Alexander yon Humbodlt Foundation

Enzo Orsingher et al.

174

Iterated Brownian motion, which has become a main topic in recent research in probability. On the basis of this idea Hochberg and Orsingher [2] gave another formulation of the iterated processes which is easier to be formally generalized to higher order differential equations, for example, to the 2~th-order equations. Unlike the second order heat equation, the higher order heat type equations cannot be associated with stochastic processes in the usual sense. However, the idea of iteration of some processes provides us with a significant and intuitive approach to the treatment of certain equations. Now, for a given equation a key question is how to find appropriate processes and how to use these processes in order that their transition function satisfies this equation. Notice that we adopt here the idea of a probabilistic approach although the processes associated with the equations cannot be understood in the usual way.

2

Iterated Processes and Parabolic Equations

Consider the fourth-order equation Ou(t,

1 04 . X) = - ~ o z 4 u ( t , z ) ,

.

t > o, z ~ R ,

(2.1)

Bl(t) > 0, Bl(t) < O,

(2.2)

and construct the following process

X(t) = [ B2(iB,(t)), [ iB=(-iBl(t)),

if if

where B1 is a Brownian motion starting at 0, and B2 is defined on the positive imaginary axis and is independent of B1. Hochberg and Orsingher [2] showed that the transition density of X(t), p(t, x), is the fundamental solution of Eq. (2.1). They slightly modified the manner of Funaki [1] for the derivation of a formally different equation. However, (2.2) is somehow difficult to be intuitively justified. Instead of (2.2), we consider the following Y(t) = B2(lSl(t)[). (2.3) Here B1, B2 are mutually independent Brownian motions. Note that Y(t) shares with X(t) similar path properties (see [3-5]), however, Y(t) is much more easy to deal with. Process Y(t) is usually called iterated Brownian motion (IBM, for short) which has become an appealing topic for research in recent years. We claim that Y(t) cannot be associated with an equation in the usual sense, but its density function can be regarded as the solution to the following equation 1 04 ~tu(t, x) = g ~-~ u(t, x) + F ( x ) in the sense of generalized functions, where in x. We now have Theorem 1

F(x) is the "second derivative" of delta function 5o

Y(t) is spatially homogeneous, but it is not time homogeneous and Markovian.

Proof The density of Y(t), qt(x, y) = f + ~ Plsl (x, y)pt(O, s)ds, where pt(x, y) is the density of a Brownian motion starting at x. The spatial homogeneity of Y(t) is a straightforward consequence of that of Brownian motion. Since B(t) is not linear in t we cannot expect the time homogeneity.


175

In order to prove the non-Markovian p r o p e r t y it is sufficient to prove t h a t the transition density of Y(t) does not satisfy the C h a p m a n - K o l m o g o r o v equation, i.e.

qt+~(x, y) r

z)q~(z, y)dz.

(2.4)

Since

Ps(x,Y)-- ~ s e x p { ~ qt+~(x, y) = 2

/o

s>O,

}, 2v/~~ exp

ox {

1

{

u2

/

du,

and

/; =

qt(x, z)q~(z, y)dz //~//

=4 =4

1

dz

1~-"12 21ul

du=e-

1

~2 /_"~

-~

v~2~l~l

dz

du

e

~0~176 f ~

2.

t;-~,4 2

21uI

1 e 2~ f0 ~ d ~ 1e 2v~-i

I~-~,12

1

1

dv~e-

1

~2

1

1

v 2

- - -2-s

I.~, ~ 2 v1~ e ~2 ~. .2

du Jo dv X/27r( u + v) e 2(.+.) __2v/~te- 2~ 2v/_~se 2~

( by C h a p m a n - K o l m o g o r o v equation)

= 4 f0 ~ dv f ~ d r = e1 J~ ~/2~ 2

1

1

1

2v/~

. _ l~-~I__.___.~ ~

fo~ dr 2v/~e 0~

,~-~12 2~

2r

,~-~I 2

1

e (~_.)2 2,

f ff dv 2,/~e

(r-:) 2

- -

OO

I

1

e .2 2.

1 2X/~S

.2 e

2~

~

( by C h a p m a n - K o l m o g o r o v equation again )

= qt+~(x,y). T h a t is, (2.4) holds. This completes the proof. Now we construct the n-iterated Brownian motion. Let Bi, i = 2 , . . . , n be m u t u a l l y independent Brownian motions defined on the positive imaginary axis and let B1 be the s t a n d a r d Brownian m o t i o n starting at 0. We will construct the multiply iterated process inductively. W h e n n = 2 the related process is given by Formula (2.2), and for convenience we shall write X2(t) instead of X(t). For n = 3 we define

Xa(t) =

B3(iB2(iB1)), iB3(-iB2(iB1)), B3(iB2(-iB1)), iB3(-iB~(-iB1)),

if if if if

Bl(t) > 0 and B2(iBl(t)) > O, Bl(t) > 0 and B2(iBl(t)) < O, B l ( t ) < 0 and B2(-iBl(t)) > O, Bl(t) < 0 and B2(-iBl(t)) < O.

Now suppose we have constructed X,~-l(t), i.e. the (n - 1)th iteration, n > 3. We then define

176

E n z o Orsingher et al.

X,~(t) as follows (considering t h a t Xn-l(t) can either be real-valued or be complex-valued): X~(t) =

{

B,~(iX~_l(t)),

if X,~_l(t) > O,

iBn(-iX,~_l(t)), iBn(-Xn_l(t)), B,~(Xn_l(t)),

if X~-l(t) < O,

(2.5)

if iX~_~(t) > O, if iX,~_l(t) < O.

The last two lines are derived from the fact t h a t - i X ~ _ l plays a role similar to t h a t of the term of B2(iBl(t)) or B2(-iBl(t)). Using arguments similar to those in H o c h b e r g and Orsingher [2] and inductive m e t h o d s we can prove t h a t the density of X~(t) satisfies the 2nth parabolic equation: 02" --~u(t,x) = (--1)n+12 - c " u(t,x), t > 0 (2.6)

Ox2n

where cn = 2c,~-1 + 1, co = 0. T h e t r e a t m e n t of the above equations is mainly based on the following idea. We first find or construct a process such t h a t the Fourier transform of its transition density satisfies the Fourier transform of the equation we consider. We note t h a t this m e t h o d can be used to solve only equations with coefficients (perhaps) depending on t but independent of x. In fact, we have

Let I(t) be an It5 process defined as I(t) = fo g(s)dBs, where g(s) is a Borel measurable function, and Bt is a standard Brownian motion. Let Bl(t) be a Brownian motion defined on the positive imaginary axis, independent of I(t). Set

Theorem

2

X(t) = { Bl(iI(t)), iBl(-if(t)),

if I(t) > 0; if I(t) < O.

(2.7)

Then the density of X(t) satisfies the equation

(2.8)

u(0, z) = ~x.

Proof Conditional on I(t) = s, B1(iI(t)) possesses a Gaussian distribution with variance is, and iB1 (-iI(t) ) has variance i2(-if(t) ) = is. T h u s the distribution of X(t) has density function exp{--x2 /2is }

q(t, x) = 1 j_

re(t, s)ds

(2.9)

where re(t, s) is the density function of I(t). It is well known (cf. [6, T h e o r e m 1.6]) that [(t) is equal in distribution to the Brownian m o t i o n starting at 0 at time fo g2(s) ds" Therefore,

re(t, x) =

1

exp

-

2 fo g~(s)ds

So (2.9) can now be rewritten as

q(t,x) = f ~ J _

exp

{-x2/2is} x/27ris

1 ~/27~ fo g2(u)du exp

{

s2} 2 f0t 92(u)a~

ds,

(2.10)


and its Fourier transform reads: 1

/_~ e'aXq(t, x)dx = /_:~ exp { --~is A2 } = exp {-/~4 ~ t g 2 ( s ) d s / 8 }

exp

177

{

s2 } 2 fog2(u) du ds

(2.11)

.

Obviously, the inverse Fourier transform 1 F ~ q(t,x)= ~-~

e -iAx exp { _A 4 ~0 t g2(s)ds/8 }

dA

(2.12)

is a solution of Eq. (2.8), and, at t = 0, also coincides with the Dirac delta function.

3

Telegraph Equations and the Delayed Iterated Brownian Motions

In the previous section we considered only the density of a process associated with a parabolic equation. Obviously, this is not true in general. For example, the solution to the telegraph equation 02 02 O

ot2u(t,x) =

c

with initial conditions u(0, x) = 5(x), the so-called telegraph process

-

t > 0

(3.1)

ut(O, x) = 0 can be given directly as the distribution of

T(t) = Vo ~ot(--1)N(S)ds.

(3.2)

Here V0 is a random variable taking two values c or - c with probability one half, and N(s), s > 0 is a Poisson process with parameter A > 0 independent of V0 (see Orsingher [7]). T(t) can be interpreted as the current position of a randomly moving particle starting at 0 and with velocity

v(t) Combine

= v0(-1)

T(t) with a standard Brownian motion B(t) starting at 0, and define Y(t) = { B(T(t)), iB(-T(t)),

The density function of

T(t) > 0, T(t) < O.

(3.3)

Y(t) satisfies the equation 02

O

C 2 C~4

~-~u(t, x) + 2A~-~u(t, x) - 4 ~xx~u(t' x).

(3.4)

See Hochberg and Orsingher [2] for details. A natural question is what happens if we consider higher order hyperbolic equations in a similar way. In fact, if we replace B(t) by the nth iterated Brownian motions (in the Funaki sense) X~(t) given by (2.5), that is, if we define

Y~(t) then we have

f Xn(T(t)), l iX,(-T(t)),

T(t) > O, T(t) < 0,

(3.5)

E n z o Orsingher et hi.

178

The distribution of Y,~(t) salisfies the hyperbolic equation:

Theorem 3

02 0 Ot 2u(t,x) + 2A

u(t,x)

C2 04n 4~~ c3x4.

--

u(t,x),

(3.6)

where cn = 2c~_1 + 1, co = O. Proof The case n = 1 has been proved in Hochberg and Orsingher [2]. Here is a simpler proof which is suitable for the generalized case we are analyzing. Let us recall that the density p(t, x) of the distribution of P(t, x) = P { T ( t ) < xIT(0) = 0} satisfies Eq. (3.1), with initial value p(0, x) = 5(x). Therefore, H(t, a) = satisfies

F

t

eiO~dP(t, x)

(3.7)

ct

d2 d dt 2 H(t, a) + 2A~--~H(t, a) = c 2 ( - i a ) 2 H ( t , a).

(3.8)

Notice that Y=(t) defined in (3.5) has

EeiaY~(t) : E { E (ei~Yn(t)IT(t)) } =

=

ct

exp

(-1) ha2" -~d-jY dP(t, y)

~

-aexp~z (

= H

y

}

(3.9)

dP(t,y)

(-~)~+lio ~ ) t,

2~n

,

and, therefore, satisfies Eq. (3.8) when the argument is replaced by tetK(t,a)=H

2c.

, that is, if we

t, (-1)~+1i~2"~ ~ ], then d2K 2A d K c2a 2"+~ ---5dt + dt - ~ 4c K,

(3.10)

which is the Fourier transform of Eq. (2.6). This completes the proof. Remark 1 have

Furthermore, if we let L ( a , t ) = H (L~_ + tta,t) for some constant # > 0, we

d2L 2~ dL -c 2 ---W dt + dt = Therefore, q(x, t ) :

+ #a

L = -c 2 -

+ #2az + i#a3

)

L.

2~ f~-~ e - m ~ L ( a , t ) da satisfies b-/~q = - 2 A

q+ ~

~

+ 2.

q,

(3.11)

and q(t, x) is the density of the iterated process

W ( t ) = f B ( T ( t ) ) + #T(t), [ i ( B ( - T ( t ) ) - #T(t)),

T(t) > 0, T(t) ( O.

(3.12)

179


That is, the iterated process obtained by combining the telegraph process and Brownian motion with a drift term represents a solution to a different differential equation. Note that when # = 0 we obtain again Eq. (3.4).

u(t, x) satisfies Eq. (3.6) v(t, x) ~=u(r x). We have

Remark 2

Let us consider a time change in Eq. (3.6). Assume that

and that r

is a twice differentiable function on R+ and set

fftv(t,z) = ff--~su(s,z)ls=r

02

02

~ v ( t , x) = ~ u ( s , x)ls=r Therefore, from (3.6) notice that if r

02 ( Ot2v(t,x) + 2Ar

r

r

In particular, if we take r

02 022 v(t, z) Ot

)

2+

ffsu(s,x)[~=r

r 0,

Ov(t, x) - - (cr

t>0,

Ox4" v ( t , x ) ,

4 c"

xER.

(3.13)

= lnt, t _> O, (3.13) reads O

+ -(2A- t+ 1)

v(t, x) - -

C2

04 n

t24c" Ox4~ v ( t ,

x),

t > O,

x E R.

(3.14)

It is interesting to remark that here we have given processes whose density is a solution to an equation with coefficients depending on time t.

4

M a x i m u m of M u l t i p l y I t e r a t e d B r o w n i a n M o t i o n s

Let B1, B 2 be two independent Brownian motions starting at 0. Consider the process defined by (2.3). We are here interested in the explicit law of the maximum of the first passage time of (2.3). Clearly, P~max

[.0