POLYLINEAR ADDITIVE-FUNCTIONALS OF SUPERPROCESSES

Let X be a superdiffusion in a domain E of R-d. A polylinear additive Functional of X corresponding to a positive Borel Function rho is give n by the formulaA(B) = integral(b) dt(l),...,dt(k) integral(Ek) rho(t( l), z(l); ...; t(k), z(k)) X-t1 (dz(l)) ... X-tk (dz(k)). By a passage to the limit, we extend this definition to a certain class of general ized functions rho. More precisely, we associate an additive functiona l A(v) of(X-t, P-mu) with every measure v subject to the condition int egral v(dt(l), dz(l); ...; dt(k), dz(k)) v(dt'(l), dz'(l); ...; dt'(k) , dz'(k)) x q(mu)(t(l), z(l); ...; t(k), dz(k); t'(l), z'(l); ...; t'( k), dz'k) < infinity For a certain kernel q(mu). We prove that, if v{t : t(i) = s} = 0 for i = l,..., k and for all s, then measure A(v) has, a.s., the same property. This result is applicable, in particular, to self-intersection local times of X. (C) 1997 Academic Press.