The Feynman-Kac Equation For Levy Walks With The Waiting Time Being Exponentially Truncated Power Law Distribution

As a kind of abnormalous diffusion with limited speed,Levy walks have a wide range of applications in practice.Levy walking is determined by two parameters,one is the particle velocity v,and the other is the waiting time distribution?(t).When the tail of the waiting time obeys the power-law distribution,different powers determine its different diffusion behavior.However,the power-tail distribution leads to the long-term continuity of the movement.Here,we introduce the exponential truncated power-law distribution,in the Laplace space?(s)=1-?s-B?(s+?)~?+B??~?,and control Levy walks'behavior through the exponential decay parameter?.We obtain the prop-agator of Levy walks through modeling,and derive the Feynman-Kac equation of Levy walks when the waiting time obeys the exponential truncated power law distribution,and analyze the critical parameters.The change in?establishes a link between normal and abnormal proliferation.Finally,we calculate the common statistics in Levy walks,such as variance and second moment.