xii INTRODUCTION
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
x
a.s., for every X G D .
The main distinction with the DynkinKuznetsov 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 crfield 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 afield o~(Z) (see Proposition 2.18).
We provide formulas for the stochastic boundary values of various classes of
solutions:
If r C dD is a Borel set, we prove in Proposition 2.30 that
Zv := Zur = oclgD
n r
/
0

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(expl4f 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.
Jo
We denote by SG(w) C dD the set of all singular points of u on the boundary.