An
(
α
,
d
,
β
)
-superprocess,
X
(
t
,
d
x
)
, is a stochastic process on
R
×
R
d
that is usually constructed as a special limit of branching diffusion where the branching mechanism is given by its factorial moment generating function:
Φ
(
s
)
=
1
1
+
β
(
1
−
s
)
1
+
β
+
s
and the spatial motion of individual particles is given by the
α
-symmetric stable process with infinitesimal generator
Δ
α
.
The
α
=
2
case corresponds to standard Brownian motion and the
(
2
,
d
,
1
)
-superprocess is called the Dawson-Watanabe superprocess or super-Brownian motion.
One of the most important properties of superprocesses is that they are intimately connected with certain nonlinear partial differential equations. The simplest such equation is
Δ
u
−
u
2
=
0
o
n
R
d
.
