Appl Math Optim 41:141–154 (2000)
2000 Springer-Verlag New York Inc.
Hilbert-Space-Valued Super-Brownian Motion and
Related Evolution Equations
and P. Sundar
Department of Statistics, University of North Carolina,
Chapel Hill, NC 27599-3260, USA
Department of Mathematics, Louisiana State University,
Baton Rouge, LA 70803, USA
Abstract. A stochastic partial differential equation in which the square root of
the solution appears as the diffusion coefﬁcient is studied as a particular case of
stochastic evolution equations. Weak existence of a solution is proved by the Euler
approximation scheme. The super-Brownian motion on [0, 1] is also studied as a
Hilbert-space-valued equation. In this set up, weak existence, pathwise uniqueness,
and positivity of solutions are obtained in any dimension d.
Key Words. Stochastic partial differential equations, Super-Brownian motion,
Stochastic evolution equations, Convergence of measures.
AMS Classiﬁcation. Primary 60H15, Secondary 60H20, 60B10.
Branching particle systems undergoing migration and critical branching lead to measure-
valued branching Markov processes in the limit. Such processes are also known as
superprocesses. The usual construction of superprocesses proceeds as follows:
At time 0, let J
be the number of particles positioned in a locally compact Polish
space E. Let J
be of the order O(n). Each of these particles moves independently of
others according to a Feller process X till time 1/n. At time 1/n, each particle splits into
The research of P. Sundar was supported by U.S. Army Research Ofﬁce Grant DAAH04-94-G-0249.