Exit times in non-Markovian drifting continuous-time random-walk processes.

Area of Science:

• * Stochastic Processes
• * Probability Theory
• * Mathematical Physics

Background:

• * Continuous-time random walks (CTRWs) are fundamental models in various scientific fields.
• * Understanding the mean exit time is crucial for analyzing system dynamics and stability.
• * Non-Markovian processes introduce complexities not captured by standard Markovian models.

Purpose of the Study:

• * To derive and analyze the equations governing the mean exit time of a CTRW with drift.
• * To investigate the impact of non-Markovian properties on the mean exit time.
• * To find closed-form solutions for specific cases of drift and jump behavior.

Main Methods:

• * Application of renewal theory to CTRW models.
• * Mathematical derivation of integral equations for mean exit time.
• * Analysis of non-Markovian corrections.
• * Case study with Erlang distributed holding times.

Main Results:

• * Determined the exact equations for mean exit time, considering both coincident and non-coincident jump instants.
• * Quantified the corrections arising from the process's non-Markovian nature.
• * Achieved closed-form solutions for integral equations when drift and jumps share the same sign.
• * Provided a detailed analysis for CTRWs with Erlang distributed holding times.

Conclusions:

• * Renewal theory provides a powerful framework for analyzing complex random walk dynamics.
• * The non-Markovian nature significantly influences mean exit time calculations.
• * Closed-form solutions are attainable under specific conditions, simplifying analysis.
• * The findings offer insights into systems exhibiting drift and random jumps with Erlangian waiting times.