INTRODUCTIO N xi
the path with initial point x and with lifetime zero (see Section 2.2.1), we have for
instance
-logE* (exp(-
ZDJ
)) = N* (1 - exp(-
ZD
J ))
for every x G D and every nonnegative Borel function / on dD.
Solutions defined by hitting probabilities. In Section 2.2.1, we define the exit
points from D of the Brownian snake as a random (possibly empty) subset of dD
denoted by SD. The exit measure ZD is almost surely supported on £D and,
provided D is smooth,
{ZD^O}
=
{£D^0},Nx-&.s..
If T C dD is closed, then the solution ur defined above is given by
(5) ur(x) = N , ( ^
D
n r / 0 ) , xeD.
If r C dD is only supposed to be a Borel set, then the function ur defined by
(5) remains a solution of (3) in D (see Section 2.2A). We say that Y C dD is
boundary polar if and only if Y is almost surely not hit by the Brownian snake,
i.e. if and only if
Nx(£DnT^0)
=ur{x) = 01
for every x G D. Boundary polar sets coincide with removable boundary singulari-
ties for (3) and with sets of boundary capacity zero (see Theorem 2.9).
Additive functional of the Brownian snake. To every measure v G A/2, one
can associate an additive functional of the Brownian snake, which is a continuous
adapted increasing process denoted by
Av
= (A^s ^ 0) (see [DhLG97] and
Section 2.3.5). We have for instance
Au+tu'
=
Av
+
tAv'
if v, v' G A/"2, t ^ 0, and
for every v G A/2, the moderate solutions uv defined above are given by
uv(x) = Nx (1 - exp(-A^)) , x G D.
In addition, if v is supported on a compact set K C dD, we have almost surely
{A»ooo}c{£DnK^0}.
(See (2.31)).
Stochastic boundary values. The concept of stochastic boundary value of a so-
lution was introduced by Dynkin in the context of superprocesses in [Dy98]. We
adapt his definitions in Section 2.3 to the Brownian snake framework. Let w b e a
solution of (3) and let (Dn)n^o be an increasing sequence of smooth subdomains
of D such that
Wn C
Dn+1
and
[JDn =
D.
There exists a random variable Zu with values in [0, 00] called the stochastic bound-
ary value of u defined N^-almost surely for every x G D by
(6) Zu = lim ZDn,u .
n—*oc
(See Section 2.3). For every x G D, we have
(7) u(x) = N* (1 - exp(-Z
n
)).
We denote by Z the set of all stochastic boundary values of solutions. Then Formu-
las (6) and (7) establish a one-to-one correspondence between Z and hi (see Section
2.3.3). Besides, we have the following fundamental property adapted from [Dy98]
and proved in the specific context of the Brownian snake in Proposition 2.16:
Previous Page Next Page