First-passage functionals of Brownian motion in logarithmic potentials and heterogeneous diffusion

Area of Science:

• Statistical Mechanics
• Stochastic Processes
• Mathematical Physics

Background:

• Brownian motion is a fundamental model for random processes.
• Logarithmic potentials introduce unique complexities in stochastic dynamics.
• Random functionals capture integrated properties of stochastic trajectories.

Purpose of the Study:

• To compute statistics of random functionals for Brownian motion in a logarithmic potential.
• To derive exact probability distributions and Laplace transforms.
• To extend findings to heterogeneous diffusion models.

Main Methods:

• Explicit computation of probability density functions (PDFs) and Laplace transforms.
• Analysis of first-exit and first-passage time dynamics.
• Mapping Brownian motion in logarithmic potentials to heterogeneous diffusion.

Main Results:

• Exact PDF for Z computed for γ=0.
• Laplace transform of Z derived for γ≠0, invertible for specific parameters.
• Exact distribution obtained for first-passage time to origin for γ>0 and V0>-D.
• Extension of results to heterogeneous diffusion models.

Conclusions:

• The study provides a comprehensive statistical analysis of random functionals in a specific potential.
• The connection between logarithmic potentials and heterogeneous diffusion is established.
• Theoretical results are validated through numerical simulations.