Spatial extent of branching Brownian motion

Area of Science:

• Stochastic Processes
• Statistical Physics
• Mathematical Biology

Background:

• Branching Brownian motion models systems with diffusion, death, and branching.
• Understanding the extreme values (rightmost and leftmost sites) is crucial for characterizing system spread.
• Focus is on the regime where branching rate (b) is less than or equal to death rate (a), leading to strong correlations.

Purpose of the Study:

• Investigate the correlation between the rightmost (X(max)) and leftmost (X(min)) visited sites in one-dimensional branching Brownian motion.
• Determine the stationary joint probability distribution function (PDF) of these extreme values.
• Analyze the distribution of the span (ζ), the distance between X(max) and X(min).

Main Methods:

• Derivation of exact analytical results for the joint PDF of extreme values.
• Calculation of the stationary PDF of the dimensionless span (ζ = (X(max) - X(min))/√(D/b)).
• Monte Carlo simulations to verify theoretical findings.

Main Results:

• Established that the joint PDF of X(max) and X(min) becomes stationary at large times.
• Demonstrated a non-zero correlation between X(max) and X(min) in the stationary state.
• Characterized the span distribution: linear for small spans, with distinct power-law or exponential tails for large spans depending on the ratio a/b.

Conclusions:

• The correlation between extreme spatial points in branching Brownian motion persists even in the stationary state.
• The derived span distribution provides precise insights into the system's spatial extent.
• Asymptotic behaviors of the span distribution confirm the influence of X(max)-X(min) correlations.