  The stochastic noise of splitting, defined initially on the (basic) algebra
of unions of intervals, is extended to a maximal class of domains. The
$\sigma$-fields of this largest extension constitute the completion, in the
sense of noise-type Boolean algebras, of the range of the unextended (basic)
splitting noise. A Lebesgue measurable domain to which the extension is
possible is characterized by a certain stability property: under independent
perturbation of Brownian increments off the domain, of the times of Brownian
local maxima belonging to the domain; together with the same property for the
complement of the domain. A set that is equal to an open set modulo a Lebesgue
negligible one, with the same being true of its complement, has this stability
property, but others have it too: the extension is non-trivial. Some domains
are totally unstable with respect to the indicated perturbation, and to them
the extension cannot be made.
