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Предлагается нелинейная модель эволюции случайных полей, основанная на аппроксимации кусочно-линейными дифференциальными уравнениями в частных производных со стохастическими граничными условиями. Ключевые слова и фразы: обобщенные случайные поля, нелинейное эволю ционное уравнение, стохастические граничные условия, Соболевские суперпро странства.We have in mind a generalized random field № = {&, «*)). vec 0°°( G)} in a region G С B. d such that, with time t £ R 1 going, its evolution goes according to non-linear stochastic differential equation dt(t) = A(t,e i )dt+B[t,e i )dri(t)where drift and diffusion operators A(t, £*), £')) depend on a whole «past» = {£(*). s < so (f, «*)) = iV, №o)) + J* (v>, A(a,V)) ds+J* B(s,e)dr,(s)), t > *o, (2) for all test functions ip £ CQ°(G). A problem is on stochastic boundary conditions for the corresponding random field £ = |^(*), * £ over a space-time cylinder G x I, I = (£ 0 ,t*)i which determine £ £ W(G x /) as a unique solution from an appropriate functional class W(G x /). Suppose, the evolution goes in a such way that approximately we have d£{t) = A£(t) dt + B dn(t) (3) on any sufficiently small time interval /д": s < t < s + As, where А, В are appropriate linear operators, depending on the past f * up to a «current» time moment s. Then, we can approach the boundary problem by means of a piecewise linear approximation for f, taking a partition to < , 4) d = У (в*ч>{-Л Mt)), v e CS°(G x /д< ; .)} (4) as a generalized random field which is meansquare continuous over test functions

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A similar result holds, of course, when we replace x(T) + y(T) = 0 by x(T) -y(T) = 0. We want to prove the claim by using that observation. To do that we need to show that this situation happens a.s. with respect to the process {B(t) y Y(t)): 0 ^ t ^ 1}. Indeed, we haveThe zero-set of namely the set {0 ^ t ^ 1: X+(t)X-(t) = 0}, behaves like the zero-set of a Brownian motion (see [2, p. 392], for the latter). From this we conclude that if a zero is isolated on the right then it is a limit of increasing zeros from the left. Also, every neighborhood of T(B, Y) contains (on the left of T) both negative and positive values of B(t) -Y(t) or of B(t) + Y(t). This proves the claim.To finish the proof of Theorem 1 we combine the claim that was just proved and Lemma 2 and we get by standard weak convergence arguments that:where the convergence is in distribution on the space Co It follows now from Lemma 1 that Z n -+ B* and by (27) we finally get В = В* in distribution. This completes the proof of Theorem 1.Acknowledgments. Thanks to S. Chobanyan who introduced me to this subject. Also thanks to A. V. Skorohod for a private communication and to M. Ryznar for some discussions on the subject. REFERENCES