I ntn. J. Math. & Math. Sci.
Vol. 3 No. 3 (1980) 549-558
549
GENERALIZED RANDOM PROCESSES AND CAUCHY’S PROBLEM FOR
SOME PARTIAL DIFFERENTIAL SYSTEMS
MAHMOUD M. EL-BORAI
Faculty of Science
King Abdulaziz University
P.O. Box 1540 Jeddah
SAUDI ARABIA
(Received February i, 1979 and in revised form October 22, 1979)
ABSTRACT.
the form D
In this paper we consider a parabolic partial differential system of
t
H
t
L(t,x,D) H t.
The generalized stochastic solutions
responding to the generalized stochastic initial conditions H
Ht,
cor-
are given.
Some properties concerning these generalized stochastic solutions are also
obtained.
KEY WORDS AND PHRASES.
Generalized Stochastic Solutions, Strongly Parabolic
Systems.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES: 60H15, 35R60.
1.
INTRODUCTION.
Consider the system
D u
t
Lu
(i.i)
M. M. EL-BORAI
550
.
where
--,
Dt
D
kl
k
I
+
k
+
Euclidean space E
(-i)
k
k
D
I
L--
Ikl
k
n
n
...D
1
D
Lk(t,x)
< 2b
)X
r
r
D
k
1
n
r
kn,
t
(0,T), T>0, x
is an element of the n-dimensional
n’
and
(Lk(t,x) Ikl
<- 2b)
is a family of square matrices
of order N.
We assume that (i.i) is a strongly parabolic system on
0,T],
t
every
x En
7 En’
{(t,x):
Gn+I
and every (t,x)
6Gn+I
Lk(t,x) o
Re
Ikl
aN),
(a I
in the sense that for every complex vector a
2b
k
where
k
’lal 2
aS +
olk
k
n
n
I
+ a,
2
2b
n
and
is a positive constant
2)b
(see [I]).
In the
above inequality and in-the following, we denote the scalar product of two
N-vector functions u and v by the bracket notation (u,v).
As usual, we denote by Cm
functions defined on E
n
0 < m < =, the set of all real-valued
(En)
which have continuous partial derivatives of order
up to and including m (of order <
set of all
matrices
(a)
(h
ector functions h
with compact support, r
Lk(t,x), Ikl
<
if m
I
N.
i,
=).
N)
h
By
Cmo (En’ N)
we denote the
such that every h
r
is in
We assume that the elements of the
2b, satisfy the following conditions:
They are bounded on
Gn+I
and satisfy a Holder condition of order
with respect to x, (0 < a < i).
(b)
For every
x En
Cm(En
they
are continuous functions in t
[0, T].
551
GENERALIZED RANDOM PROCESSES AND CAUCHY’S PROBLEM
(c)
(u
functions.
0, T], they are C (E)
n
For every t in
u
I
N)
satisfy the initial condition
u(x, o)
where
UoN),
(Uol
u
o
[u
r
u
or
We say that u is of the class S(E
UrC2b(En)
Let u
n)
o
6 C(E
(1.2)
(x),
n)
are bounded on E
if for each t
(0,T),
n
N].
r--1
Dtur C(En)
and
N.
i
It has been proved [ 2] that, under conditions (a) and (b), there exists a
fundamental matrix Z(t,0,x,y) of the system (i.i) such that
I
u(t x)
Z(t 0 x y) u (y) dy, dy
o
dYl
dy
(1.3)
r
E
n
represents the unique solution of the Cauchy problem (i.i), (1.2) in the class
S(En).
Let (V
r
r
N) be a family of Gaussian random measures in the
i
2].
sense of Gelfand and Vilenkin
defined on E
Ig
r
(s)12
1
We say that
gr
gr
Let
be a complex-valued function
is of the class
Kr
if the integral
dF (s) exists, where F is a positive measure such that
r
r
E
for any two Borel sets B
Fr(B I B 2)
[Vr(BI) Vr(B2)]
I
and B
2
o the real line [r
i,
N and E (.)
denotes the expectation of (.)].
Let H be an N-vector of generalized stochastic processes, which associates
with every h
Co
(En, N) an N-vector of random variables defined by
Hr(h)
(h))
(HI(h),
H(h)
E
gro(S)
1
d
Vr(S),
(1.4)
552
M. M. EL-BORAI
gro(S)
I
(I r(x,s), h(x)) dx,
E
n
where (I
on E
I,
r
r
N) is a family of N-vectors of continuous functions
n+l"
It is assumed also that all the components of I
dependently of s.
Clearly,
gro
r
are bounded on E
n
in-
Kr.
is of the class
The theoretical development in section 2 exhibits the use of formula (1.3)
in order to integrate (i.i) when the initial condition is an N-vector of general-
ized stochastic processes, which is defined by
(1.4).
Also, some essential
properties are derived in section 3.
2.
GENERALIZED STOCHASTIC SOLUTIONS.
An N-vector w(t,x,s) of functions is said to be of the class
C(En+I,
N) if,
for each t in (0,T), the components of w(t,x,s) represent continuous functions
En+I and
of (x,s) on
they are bounded on E
n
the generalized stochastic vector H
[Sr(t,x,s)
Co (En, N), Ht(h)
family
H
t(h)
S C’ C(En+l
(gl
We say that
is of the class V if there exists a
N), r
N-1 such that, for each h in
1,
can be represented in the form
J
E
g
t
independently of s.
g(t,s) dV(s),
1
gN )’ gr(t’s)
I
h(x))dx,
(Sr (t ,x, s)
E
n
H (h)
t
(Hit(h)-
N--H’-t(h))
(h)
H
rt
)| gr
E
It is clear that, for each t in (0,T),
given by
grl E
Kr.
(t,s)dV (s).
r
1
The expectation of
.IHrt’12
is
GENERALIZED RANDOM PROCESS AND CAUCHY’S PROBLEM
I
IHrt 12
E
E
If
d
d--{
Dtgr(t,s)__
Hrt (h)
.
Ikl
for each t in (0,T), then we define
E
d V
r
(s)
I Dt
E
1
gr(t,s)
At,s)
Agr (t’s)
gr(t
s) d
Vr(S)
1
and l.i.m, denotes limit in the mean,
Dtgr(t,s)
At
2
dFr(S)
0.
1
IklDk Lk,,
(-1)
< 2b
adjoint matrices to
At
I
t->o
.
E
gr(t +
lim
L are C
Kr
Agr(t’ s)
I
l.i.m
t/0
Agr(t,s)
e
Fr(S).
d
by
d- Hrt(h)
Let
2
1
exists and belongs to
d
where
Igr(t,s)
553
(Lk, Ikl
-
<
where
2b).
(Be, Ikl
<
2b) is the family of
Since the coefficients of the operator
(En) functions, it follows that, for every h in C (E
o
n
is also in C
o
_(En,N).
We call H
dH
system (1.1) if H
t
and
t
d--
dH
Co (En,N)
h
are of the class V and
h)
t
H
t(h t)
and t in (0,T).
H (h)
o
(2.1)
We assume that
H(h)
(2.2)
where H is defined by (1.4).
THEOREM i:
The Cauchy problem (2.1), (2.2) has a unique generalized
stochastic solution H
PROOF:
t
in the class V.
Let (S (t,x,s)
r
t
a generalized stochastic solution of the
t
d--{--
for every h in
N), L h
r
i,
system (I.i) with the initial conditions"
N) be a family of solutions of the
554
M. M. EL-BORAI
S (O,x,s)
r
I (x,s), r
N.
i
r
(1.3), one gets
Using formula
Sr(t,x,s)
Ir(Y,S)
| Z(t,O,x,y)
E
(2.3)
dy.
n
According to the properties of the fundamental matrix Z, we find
r
N.
1,
Sr E C(En+I’
N),
Set,
J
Ht(h)--
E
g(t,s) dV (s)
1
and
gr(t’s)
I
E
where
Sl(t,x,s),
D
t
[
E
are defined by (2.3).
N),
(Sr(t,x,s)
Sr C(En+I,
Since
I
h(x))dx
(DtSr(t,x,s)
h(x))dx
E
n
n
(Sr(t,x,s), h *(x)
t
dx.
E
The last formula proves that D
grl
t
6
Kr.
Now we already have
d
dt
Ht(h)
(Sr(t
E
where
d
H
t
1
E
We also have
H (h)
o
x,s), h (x)) dx dV(s)
t
n
is of the class V.
.
E
g(O,s) dV(s)
1
N),
Using again the properties of Z, we get
is of the class V.
t
o (En’
n
SN(t,x,s)
it follows that H
h(x))dx with hl E C
(Sr(t,x,s),
H
t
(h)
GENERAL RANDOM PROCESSES AND CAUCHY’S PROBLEM
555
where
I
gr(O’s)
E
(Ir(X,S),
h(x)) dx.
n
Thus the existence of the generalized stochastic solution H
condition H
H is proved.
o
To prove the uniqueness of H
H(h)
then E
[Hrol 2
t(h)
[gr(S) 12
it is sufficient
t
o
Co (En,N)
0 for every h in
I
E
with the initial
(2.1) with the initial condition H (h)
to show that the only solution of
0 is H
t
dF(s)
and t in (O,T).
O, and hence
gro (s)
If
O on E
Ho
O,
1
1
Therefore,
gro(S)
I
(Ir(X,S)
h(x)) dx
O,
E
which is true for any arbit=mryh in
Since
d
H
t(h)
H
t(h t),
Co (En,
N), and hence
Ir(X,S)
0 on
En+I.
it follows that
therefore,
I
(DtS r(t,x,s)
L S (t,x,s), h(x)) dx
r
O,
E
which implies
DtSr(t,x,s)
L
Sr(t,x,s).
(2.4)
We also have
s r (O,x,s)
The uniqueness of the problem
o.
(2.5)
(2.4), (2.5) gives
s r (t,x,s)
0,
(2.6)
556
M. M. EL-BORAI
(O,T), (x,s) E
t
Using (2.6), one gets H
t(h)
En+I,
I
(r
O, for every h in C
N).
=o (En’N)
and t in (O,T).
This completes the proof.
A CONVERGENCE THEOREM.
3.
(h
Let h
m
m
h ), m-- i, 2,
m
N
I’
G is a bounded open domain of E
Wr E L2(G ),
r
w (x))
r
N and L
i,
xG
If H
THEOREM 2:
dx
(3.1)
0
It is assumed that w (x)
r
0
N.
1
where r
(G, N), where
(G) denotes the set of all Lebesgue
2
measurable square integrable functions on G.
for
o
Suppose that
n
lim | (h (x)
m
J
r
where
be a sequence in C
t(hm)
I gm(t’s)
dV(s),
then
l.i.m.
m+
where
gm(t’s)
(gml(t,s)
r(t, s)
is defined by
PROOF:
t
m)
J
q(t,s) dV(s),
gmN(t,s)),
I (Sr(t’x’s)
gmr (t,s)
J
H (h
(Sr(t,x,s),
h (x)) dx, q
m
(ql
qN
w(x))dx, and the family
(Sr,
r
I,
N)
(2.3).
A straight forward application of the Cauchy
Schwarz inequality
establishes that
lira
I (Sr(t,x,s), hm(X) I (Sr(t,x,s),
dx
G
G
w(x))dx
(3.2)
GENERALIZED RANDOM PROCESSES AND CAUCHY’ S PROBLEM
According to the conditions imposed on the family
(Ir(X,S)
r
i
557
N)
and according to the properties of the fundamental matrix Z, we can find a
constant A such that
Igm
<
(t,s)
(3.3)
A,
r
for all m, s, t
(O,T) and
r
For any positive integers
N.
i
and m,
we have
E
I
Hrt(h )12
Hrt(hm)
Igmr (t,s)- gr (t,s) 2 dFr(S).
By a standard argument based on (3.2) and (3.3), the righthand side of (3.4) can
m)
Thus, H (h
be shown to go to zero.
t
is a Cauchy sequence.
We deduce also
that
lim
I
Igmr (t,s) r(t,s) 2 dFr(S)
O.
The last argument leads to the fact that there exists a stochastic process
R (t) such that E
r
IR
(t)l 2
E
lim
Following Doob
and that
<
IHrt
(hm)
r(t) 12
R
0
3], we find
J r (t s)
R (t)
r
r(t,s)
dV (s)
r
J| (Sr (t ,x, s)
w(x))dx.
This completes the proof.
COROLLARY:
w (x)
r
0 for x
For vector functions (w
w
N)
where
in C
G), there exists a sequence (h)
m
(En,
l.i.m.
m/
H (hm)
o
w
I
H
o(w)
w , L2(Q)
and
N) such that
M. M. EL-BORAI
558
i.i.m.
H
m/=
t
(hm)=H t (w).
The proof can be deduced directly by using theorem 2.
(Compare []).
REFERENCES
[i]
C.D. Idelman, Parabolic Systems. P. 87, Moscow 1964, (Mir Publishers).
[2]
N.Ya. Vilenkin, I.M. Gelfand, Generalized Functions, Vol. 4, 1964
(Translated).
[3]
J.L. Doob, Stochastic Processes.
[]
M.M. Ei-Borai, On Stochastic Differential Equation in a Hilbert Space,
Egyptian Statistical Journal Vol. 1__7, No. 2, 1973.
New York 1953.