Continuous-time random walk

<h2 id="motivation">Motivation</h2>
<p>CTRW was introduced by <a href="/facts/Elliott_Waters_Montroll/U7gG1Gua">Montroll</a> and <a href="/facts/George_Herbert_Weiss/1nGYmHVe">Weiss</a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> as a generalization of physical diffusion processes to effectively describe <a href="/facts/Anomalous_diffusion/vSVM53kY">anomalous diffusion</a>, i.e., the super- and sub-diffusive cases. An equivalent formulation of the CTRW is given by generalized <a href="/facts/Master_equation/4QbTFyak">master equations</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> A connection between CTRWs and diffusion equations with <a href="/facts/Fractional_derivative/PxN1pMcj">fractional time derivatives</a> has been established.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> Similarly, <a href="/facts/Fractional_calculus/PxN1pMcj">time-space fractional diffusion equations</a> can be considered as CTRWs with continuously distributed jumps or continuum approximations of CTRWs on lattices.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p>
<h2 id="formulation">Formulation</h2>
<p>A simple formulation of a CTRW is to consider the stochastic process 
  
    
      
        X
        (
        t
        )
      
    
    {\displaystyle X(t)}
  
 defined by
</p>

X
        (
        t
        )
        =
        
          X
          
            0
          
        
        +
        
          ∑
          
            i
            =
            1
          
          
            N
            (
            t
            )
          
        
        Δ
        
          X
          
            i
          
        
        ,
      
    
    {\displaystyle X(t)=X_{0}+\sum _{i=1}^{N(t)}\Delta X_{i},}

<p>whose increments 
  
    
      
        Δ
        
          X
          
            i
          
        
      
    
    {\displaystyle \Delta X_{i}}
  
 are <a href="/facts/Iid/othIRaWt">iid</a> random variables taking values in a domain 
  
    
      
        Ω
      
    
    {\displaystyle \Omega }
  
 and 
  
    
      
        N
        (
        t
        )
      
    
    {\displaystyle N(t)}
  
 is the number of jumps in the interval 
  
    
      
        (
        0
        ,
        t
        )
      
    
    {\displaystyle (0,t)}
  
. The probability for the process taking the value 
  
    
      
        X
      
    
    {\displaystyle X}
  
 at time 
  
    
      
        t
      
    
    {\displaystyle t}
  
 is then given by
</p>

P
        (
        X
        ,
        t
        )
        =
        
          ∑
          
            n
            =
            0
          
          
            ∞
          
        
        P
        (
        n
        ,
        t
        )
        
          P
          
            n
          
        
        (
        X
        )
        .
      
    
    {\displaystyle P(X,t)=\sum _{n=0}^{\infty }P(n,t)P_{n}(X).}

<p>Here 
  
    
      
        
          P
          
            n
          
        
        (
        X
        )
      
    
    {\displaystyle P_{n}(X)}
  
 is the probability for the process taking the value 
  
    
      
        X
      
    
    {\displaystyle X}
  
 after 
  
    
      
        n
      
    
    {\displaystyle n}
  
 jumps, and 
  
    
      
        P
        (
        n
        ,
        t
        )
      
    
    {\displaystyle P(n,t)}
  
 is the probability of having 
  
    
      
        n
      
    
    {\displaystyle n}
  
 jumps after time 
  
    
      
        t
      
    
    {\displaystyle t}
  
.
</p>
<h2 id="montrollweiss-formula">Montroll–Weiss formula</h2>
<p>We denote by 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 the waiting time in between two jumps of 
  
    
      
        N
        (
        t
        )
      
    
    {\displaystyle N(t)}
  
 and by 
  
    
      
        ψ
        (
        τ
        )
      
    
    {\displaystyle \psi (\tau )}
  
 its distribution. The <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transform</a> of 
  
    
      
        ψ
        (
        τ
        )
      
    
    {\displaystyle \psi (\tau )}
  
 is defined by
</p>

ψ
              ~
            
          
        
        (
        s
        )
        =
        
          ∫
          
            0
          
          
            ∞
          
        
        d
        τ
        
        
          e
          
            −
            τ
            s
          
        
        ψ
        (
        τ
        )
        .
      
    
    {\displaystyle {\tilde {\psi }}(s)=\int _{0}^{\infty }d\tau \,e^{-\tau s}\psi (\tau ).}

<p>Similarly, the <a href="/facts/Characteristic_function_(probability_theory)/rl3uAGKM">characteristic function</a> of the jump distribution 
  
    
      
        f
        (
        Δ
        X
        )
      
    
    {\displaystyle f(\Delta X)}
  
 is given by its <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a>:
</p>

f
              ^
            
          
        
        (
        k
        )
        =
        
          ∫
          
            Ω
          
        
        d
        (
        Δ
        X
        )
        
        
          e
          
            i
            k
            Δ
            X
          
        
        f
        (
        Δ
        X
        )
        .
      
    
    {\displaystyle {\hat {f}}(k)=\int _{\Omega }d(\Delta X)\,e^{ik\Delta X}f(\Delta X).}

<p>One can show that the Laplace–Fourier transform of the probability 
  
    
      
        P
        (
        X
        ,
        t
        )
      
    
    {\displaystyle P(X,t)}
  
 is given by
</p>

P
                  ~
                
              
              ^
            
          
        
        (
        k
        ,
        s
        )
        =
        
          
            
              1
              −
              
                
                  
                    ψ
                    ~
                  
                
              
              (
              s
              )
            
            s
          
        
        
          
            1
            
              1
              −
              
                
                  
                    ψ
                    ~
                  
                
              
              (
              s
              )
              
                
                  
                    f
                    ^
                  
                
              
              (
              k
              )
            
          
        
        .
      
    
    {\displaystyle {\hat {\tilde {P}}}(k,s)={\frac {1-{\tilde {\psi }}(s)}{s}}{\frac {1}{1-{\tilde {\psi }}(s){\hat {f}}(k)}}.}

<p>The above is called the <a href="/facts/Elliott_Waters_Montroll/U7gG1Gua">Montroll</a>–<a href="/facts/George_Herbert_Weiss/1nGYmHVe">Weiss</a> formula.
</p>
<h2 id="examples">Examples</h2>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Klages, Rainer; Radons, Guenther; Sokolov, Igor M. (2008-09-08). Anomalous Transport: Foundations and Applications. ISBN 9783527622986. <a href="9783527622986" target="_blank">9783527622986</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Paul, Wolfgang; Baschnagel, Jörg (2013-07-11). Stochastic Processes: From Physics to Finance. Springer Science & Business Media. pp. 72–. ISBN 9783319003276. Retrieved 25 July 2014. <a href="9783319003276" target="_blank">9783319003276</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Slanina, Frantisek (2013-12-05). Essentials of Econophysics Modelling. OUP Oxford. pp. 89–. ISBN 9780191009075. Retrieved 25 July 2014. <a href="9780191009075" target="_blank">9780191009075</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Elliott W. Montroll; George H. Weiss (1965). "Random Walks on Lattices. II". J. Math. Phys. 6 (2): 167. Bibcode:1965JMP.....6..167M. doi:10.1063/1.1704269. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>. M. Kenkre; E. W. Montroll; M. F. Shlesinger (1973). "Generalized master equations for continuous-time random walks". Journal of Statistical Physics. 9 (1): 45–50. Bibcode:1973JSP.....9...45K. doi:10.1007/BF01016796. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Hilfer, R.; Anton, L. (1995). "Fractional master equations and fractal time random walks". Phys. Rev. E. 51 (2): R848 – R851. Bibcode:1995PhRvE..51..848H. doi:10.1103/PhysRevE.51.R848. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Gorenflo, Rudolf; Mainardi, Francesco; Vivoli, Alessandro (2005). "Continuous-time random walk and parametric subordination in fractional diffusion". Chaos, Solitons & Fractals. 34 (1): 87–103. arXiv:cond-mat/0701126. Bibcode:2007CSF....34...87G. doi:10.1016/j.chaos.2007.01.052. <a href="/wiki/Rudolf_Gorenflo" target="_blank">/wiki/Rudolf_Gorenflo</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
</ol>

Continuous-time random walk open-in-new

Continuous-time random walk