The set Z is a conve x cone closed under almost sure convergence.
In particular, we prove in Proposition 2.16 that, for any u,v G ZY,
Zu®v Zu + Zv, N
-a.s., for every X G D .
The main distinction with the Dynkin-Kuznetsov definition is the fact that
stochastic boundary values are defined almost surely under the excursion measures
of the Brownian snake.
In addition, Proposition 2.18 gives a crucial result that has no analogue in the
superprocess setting:
Let o~(Z) be the cr-field generated by the elements of Z. For every x,x' G D,
nx ~ Nx on a(Z),
meaning that the two measures N^ and Nx/ are mutually absolutely continuous on
the a-field o~(Z) (see Proposition 2.18).
We provide formulas for the stochastic boundary values of various classes of
If r C dD is a Borel set, we prove in Proposition 2.30 that
Zv := Zur = oclgD
n r
If v is a measure of finite energy on dD (y G A/2), we prove in Proposition 2.26 that
Zv := ZUu Av00.
If v G A/o, there exists a sequence vn G A/2 such that vn \ v and then
^-00 T %v '•= zUv.
By abuse of notation, we put A1^ := Z„ even if v 0 A/2.
Fin e trace.
Singular points on the boundary. We prove in Proposition 2.31 an analogue of
a fundamental formula for stochastic boundary values shown by Dynkin in 1997
(see [Dy97]) in the superprocess framework. Given v G M\ and u G U, we have,
for every x G D,
(8) Nx(Zuexp(-Zu)) = J kD{x,y)E^y(expl-4f u(Bs)ds J J i/(dy),
where P ^ ^ stands for the law of Brownian motion started at x and conditioned
to exit D at ?/, with lifetime denoted by (" (see Section 2.1.2). The superprocess
analogue of this formula was used by Dynkin in [Dy97] to introduce a new definition
of the singular points on the boundary adapted to the supercritical case. A point
y G dD is a singular point for u if and only if
/ u(Bs)ds = 00, P^_, -a.s., for every x G D.
We denote by SG(w) C dD the set of all singular points of u on the boundary.
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