In probability theory, a superprocess is a measure-valued stochastic process that is usually constructed as a special limit of near-critical branching diffusions.

Informally, a superprocess can be seen as a branching process where each particle splits and dies at infinite rates, and evolves in a state space E according to a diffusion equation. We follow the rescaled population of particles, seen as a measure on E.

For any integer ${\displaystyle N\geq 1}$, consider a branching Brownian process ${\displaystyle Y^{N}(t,dx)}$ defined as follows:

The notation ${\displaystyle Y^{N}(t,dx)}$ means should be interpreted as: at each time ${\displaystyle t}$, the number of particles in a set ${\displaystyle A\subset \mathbb {R} }$ is ${\displaystyle Y^{N}(t,A)}$. In other words, ${\displaystyle Y}$ is a measure-valued random process.

Now, define a renormalized process:

${\displaystyle X^{N}(t,dx):={\frac {1}{N}}Y^{N}(t,dx)}$

Then the finite-dimensional distributions of ${\displaystyle X^{N}}$ converge as ${\displaystyle N\to +\infty }$ to those of a measure-valued random process ${\displaystyle X(t,dx)}$, which is called a ${\displaystyle (\xi ,\phi )}$-superprocess, with initial value ${\displaystyle X(0)=\mu }$, where ${\displaystyle \phi (z):={\frac {z^{2}}{2}}}$ and where ${\displaystyle \xi }$ is a Brownian motion (specifically, ${\displaystyle \xi =(\Omega ,{\mathcal {F}},{\mathcal {F}}_{t},\xi _{t},{\textbf {P}}_{x})}$ where ${\displaystyle (\Omega ,{\mathcal {F}})}$ is a measurable space, ${\displaystyle ({\mathcal {F}}_{t})_{t\geq 0}}$ is a filtration, and ${\displaystyle \xi _{t}}$ under ${\displaystyle {\textbf {P}}_{x}}$ has the law of a Brownian motion started at ${\displaystyle x}$).

As will be clarified in the next section, ${\displaystyle \phi }$ encodes an underlying branching mechanism, and ${\displaystyle \xi }$ encodes the motion of the particles. Here, since ${\displaystyle \xi }$ is a Brownian motion, the resulting object is known as a Super-brownian motion.