Splitting Probabilities of Symmetric Jump Processes.

Area of Science:

• Mathematical Physics
• Statistical Mechanics
• Stochastic Processes

Background:

• Continuous jump processes are fundamental in modeling various physical phenomena.
• Understanding the probability of a process reaching one boundary before another is critical for predicting system behavior.
• Existing models often simplify microscopic dynamics, potentially overlooking crucial effects.

Purpose of the Study:

• To derive a universal, exact asymptotic form for the splitting probability in symmetric continuous jump processes.
• To explicitly determine the transmission probability, highlighting the limitations of continuous limits.
• To provide a quantitative tool for characterizing light scattering in heterogeneous media.

Main Methods:

• Derivation of an exact asymptotic form for the splitting probability using mathematical analysis.
• Analysis of the transmission probability in the limit of the starting position approaching zero.
• Application and illustration using paradigmatic models of jump processes.

Main Results:

• A universal, exact asymptotic form for the splitting probability is established.
• The transmission probability is explicitly determined, contradicting the trivial prediction from continuous limits.
• The importance of microscopic jump process properties is demonstrated.

Conclusions:

• The derived formula offers a precise prediction for the splitting probability in continuous jump processes.
• Microscopic dynamics play a vital role in transmission phenomena, necessitating their inclusion in models.
• The results provide experimentally measurable predictions for light scattering in 3D heterogeneous media.