this way we follow a certain general approach to stochastic boundary problems, see for example, [3]-[5].) To conclude our introduction, we have to emphasize ...
B o u n d a r y V a l u e P r o b l e m s for H a r m o n i c Random Fields Y. Rozanov and F. Sansb
1
Introduction
We consider Laplace equation
A~t = 0
(1.1)
and boundary value problems for harmonic functions ( E W(G) of a certain class W(G) in a bounded region G C_ /i~d, whose behaviour near the boundary F = OG is as much chaotic as we see in a case of generalized random fields ~# &aa'acterized by the stochastic equation
A{ = r1
(1.2)
with a stochastic source r] of white noise type in C. The choice of W_W_(G)is motivated by a few important reasons as follows. We Would like to have the corresponding sample ~ E W(C) of chaotic boundacy behaviour in the case of (1.2) with any random Schwartz distribution
O0
*~
which is meansquare continuous over Schwartz test functions cfl with respect to L2-norm:
EI(~,~)I 2 ~ clI~It~ , ~ ~ c~(c)
(1.3)
68
We would like also to get a variety of stochastic boundary conditions such that the corresponding boundary data, given arbitrarily, determine a unique solution ~ E W___(G) of equation (1.1)/(1.2). And apart from of that, we would like not to fail in case, that the region G considered has inside a lot of singularities (totally of zero Lebesgue measure in Rd). These motivations lead us to stochastic Sobolev space.~
w(a)
= w
(a) , p = 2
of random Schwartz distributions
which are meansquare continuous over Schwartz test functions ~2 with respect to the corresponding norm
flail_p= (/'Ic)(.A)I2(I + IAI2)-~'dA) I/2
1.4)
given by means of the corresponding Fourier transform 1 Our approach to stochastic boundary problems for harmonic functions E W~(G), p = 2, is based on the application of an appropriate te~t
function space
x((;)
= [co
which appears as a closure of all Schwartz test functions p E Co~°(G) with respect to norm (1.4). 1A m a t e r i a l on classical ( d e t e r m i n i s t i c ) Sobolev spaces can be f o u n d in nearly any b o o k on P u n c t i o n a l A n a l i s e a n d PDE-see, for example, 1 e 2.
69 It occurs that in this way we get
x(G)
= w;-~(c;)
, p = 2
~s a space of all Schwartz distributions
= (~, ~)
, ~ ~ C J ° ( R d) ,
with supports suppz c_ [(;:] in the closure [G] of G, having a finite Ilorm
l{~{{x × II~{{-~ =
I~(A)I ~
i + IAI ~ -~ dA
, p = 2
(1.5)
Dealing with any ( E W(G), by meansquare continuity we can determine
