In this paper, a signed and asymmetric measure connected with equations 8u/St = ⢠83u/Bx 3 is constructed on a suitable space of functions. It is shown that in.
8.
F. F. Ivanauskas, "Method of decomposition for the solution of systems of nonlinear evolution equations," Liet. Mat. Rinkinys, 30, No. i, 31-43 (1990).
PROCESSES GOVERNED BY SIGNED MEASURES CONNECTED WITH THIRD-ORDER "HEAT-TYPE" EQUATIONS UDC 519.21
Enzo Orsingher
In this paper, a signed and asymmetric measure connected with equations 8u/St = • 83u/Bx 3 is constructed on a suitable space of functions. It is shown that in this case the arc-sine law (valid for Brownian motion and all processes governed by even-order heat-type equations)does not hold, and its explicit counterpart is presented. It is shown that the maximum and minimum distributions differ substantially, and their connection with the transition function of process is given. Key words and phrases: Feynman--Kac functional, arc-sine law, Airy functions, signed measures, higher-order heat-type equations.
I. Introduction It is well-known that the solution of the initial-value
Ou
1
--~.\'
O,
(i.i) .
(x, O) = ~ (x)
induces a probability measure (usually referred to as Wiener measure) on the space of continuous functions. The measure connected with the fundamental solution of even-order "heat-type" equations Ou
])n+]
a-T = ( -
. (.x,
r ,-~v~ - ~ *
--7
O,
v (.,) = and f = 1 in (3.3). comes
otherwise
In this case the Laplace transform of the Feynman--Kac functional be~O
t
w(>,, .~)=J e .... &
exp
-~
0
Some calculations
J [1-sgnx(s)lj.~
suffice to prove that
"' ( Z . . V ) =
e-
Y Z.: b*
-" F si.
l
t: c o s 5I
1/73
I
5 l'~ , ). + ',?>x
'
---
k + [~ x
}/'
I
+
-~-. >, + ,~~
is a bounded solution of the Laplace transform of (3.4). Imposing the conditions of continuity at 0 we obtain
226
~/,.
o
when x < 0
3
HO,, 0)= l/l/z=0,+a). which is the Laplace transform of
,)=~
.,(o,
/ ~ e-~' o VTS(t_y)
(4.8)
!
~I f [l-sgnx(s)]~'
Result (4.8) clearly shows that
has a distribution obtained from
0
(4.1)
with
a rotation
R e m a r k 3.
around
For the
y = t/2
process
y(s)
~- ,
P
and thus
attains
governed
by equation
[1 + s g n y ( s ) ] d s < . v
its
=
o
(1.6)
.
3
--
"
Vy"
(/=7i
t
t' {v;
minimum a t
.r
[i-sg,_v~,0],/.~.(x, /) = E,
{I
when x > ~ . whcn.v, }} {I
exp-fVz(x(s))ds
t"
=E~
0
{ s:: }
aa- &~P"~"'=~+l~'*""{ { ' ' cos
-5_ }r3- ]/r~ x
_I _f ,,_~,,# f ~o~(~, +~",)d~=.
)
+ } ~ - sin
{-'2 l ' ~ ]'/ ~ x )}
(5.10)
(5.ii)
.--v--* e-':'~;'-"
This can be seen by observing that
J_ 9 e-r',,dt . 0
cos(ex
" 3, cos c,_v
-r:
f~+~,~
0
]
t l ~
0
3
09 --
1
+ - ~3t)d~.=
.
~za sin ocv - f~+.~.,; -d~ }=
0
3
~13
{ f {O~ ~I'I'[ '/: ~ ( :~' ~i'` l~'': '[ [
3
.
I'fz';
'
14-2"
J"
!r i
The integrals above can be evaluated on the basis of formulae appearing on pages 413 and 414 of Gradshteyn and Ryzhik, which in our case read (
-~sinA.--
9
r: ~
' - - '1- -+- z7 - -
d2=
e-*'m
~":
A 6
-- -~
0
6
"
k'.l
..,', f
3 ~ --~co~A'-dz = ~'-" S C-4~b' I
(I
2k--1
"
=1
SiR
{ -(2k-l)~ - -+ A -
6COS
(2k-l)n } 6 '
k~l
After some computations,
results (5.10) and (5.11) emerge.
In view of (5.9) we have
I" e_~t -o P f min x ( s ) > - ~ } d t = T - 9
o
O~S~I
07.
~ e-~'~=3 (e-~O(" ~at, Vk-' o
(5.i2)
where, in the last step, formula (5.11) is applied. This confirms the strange result announced in the introductory section. Clearly, an analogous, be established. If
simple, relationship for the distribution of the maximum cannot
~(t, x ) = ~I [ cos(~x-~.at)d~ denotes the main solution of (1.6), we have 0
a
e-~;' o---~ ') P { m a x x ( s ) < ~ } d l = 2 O
s
.
e ' p(t. a)dt--- 5
-~-
0
This evokes the relationship existing between the maximum and the transition function discussed in Hochberg's paper. However, in our case, the situation is more awkward since (5.12) involves the density connected with equation (1.6). Remark 6. For the minimum and maximum of process governed by the signed measure connected with (1.6) the situation is similar to that discussed above, with the distribution of maximum coinciding with that of the minimum, namely (5.9). LITERATURE CITED i. 2. 3.
230
C . W . Gardiner, Handbook of Stochatic Methods, Berlin: Springer-Verlag (1985). I, S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, New York: Academic Press (1980). K . J . Hochberg, "A signed measure on path space related to Wiener measure," Annals of Probability," 6, 433-458 (1978).
4. 5. 6.
V. Yu. Krylov, "Some properties of the distribution corresponding to the equation 3u/~t= (--l)n+la2gu/sx2g, '' Soviet. Math. Dokl., ~, 260-263 (1960). E. Orsingher, "Brownian fluctuations in space--time with applications to vibrations of rods," Stoch. Proc. ApDI. 23, 221-234 (1986). V . S . Smirnov, Cours e Mathematiques Superrieures. Tome III. Deuxieme Partie, Moscow: Editions, Mir (1972).
A PROBLEM WITH FREE BOUNDARY FOR THE NAVIER--STOKES EQUATIONS. II UDC 517.956.223
K. Samaitis and L. Stupelis
3. Estimate of the Change of the Solution of the Auxiliary Problem Under Variation of the Domai n Now we consider, along with the problem (2.5), (2.6), the same problem in the domain where ~eS:~.~(A) is a function close to ~ in the norm of S3,~(A) and satisfying (1.18). Let (w', p') be a solution of it, i.e.,
i!'={xeV:x~
