Residence times of branching diffusion processes

The residence time of a branching Brownian process is the amount of time that the mother particle and all its descendants spend inside a domain. Using the Feynman-Kac formalism, we derive the residence-time equation as well as the equations for its moments for a branching diffusion process with an arbitrary number of descendants. This general approach is illustrated with simple examples in free space and in confined geometries where explicit formulas for the moments are obtained within the long time limit. In particular, we study in detail the influence of the branching mechanism on those moments. The present approach can also be applied to investigate other additive functionals of branching Brownian process.

Figure 1

A schematic representation of two branching diffusion processes up to some time $t$, one starting outside the domain $V$ (green) and one starting inside the domain (red). By diffusion, the particles can re-enter in $V$ an arbitrary number of times and eventually escape to infinity. Furthermore, due to the branching mechanism, they can be absorbed or created anywhere. Only paths drawn with a continuous line contribute to the residence time in $V$.

Figure 2

Three dimensions: Mean residence time and variance inside a sphere of radius unity for different parameters ${\nu }_2$ with $D=1/2$ and $\beta =1/2$. Each case has the same mean number of descendants ${\nu }_1=0.75$. Black line, ${\tau }_1\left(r\right)$; blue line, $\sigma \left(r\right)$ with ${\nu }_2=0$; red line, $\sigma \left(r\right)$ with ${\nu }_2=0.25$. Whatever the starting point ${\mathbf{x}}_{\mathbf{0}}$, the branching mechanism has the effect of increasing the variance (as expected), but its effect remains, however, relatively moderate.

Figure 3

An example of realization of the one-dimensional branching diffusion process in the segment $[-R_2;R_2]$ with reflecting boundary conditions on the borders. Only paths between $[-R_1;R_1]$ contribute to the RT of Region I.

Figure 4

Residence time of reflected branching diffusions: mean and variance in the region I for different set of parameters ${\nu }_2$ with $D=1/2$ and $\beta =5$. Each case has the same mean number of descendants ${\nu }_1=0.75$. Black line: ${\tau }_1\left(x\right)$; Blue line: $\sigma \left(x\right)$ with ${\nu }_2=0$; Red line: $\sigma \left(x\right)$ with ${\nu }_2=0.25$. Due to the branching mechanism, when ${\nu }_2\ne 0$, the variance of the process increases drastically.

Figure 5

Residence time of a particle inside a subvolume $V\subset \mathrm{\Omega }$. The whole domain $\mathrm{\Omega }$ has reflecting boundary conditions on its surface.