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fixed typo in notation
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papad
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Consider a simple random walk in one dimension with reflective boundaries at $n=1$ and $n=N$. We can express it via the master equation: \begin{equation} P(n,t) = \frac{1}{2}P(n-1,t-1) + \frac{1}{2}P(n+1,t-1) \quad \text{if } n\neq 1 \;{\rm and} \;n\neq N, \end{equation} \begin{equation} P(n,t) = P(n-1,t-1) \quad \text{if } n = 1, \end{equation}With the following boundary conditions: \begin{equation} P(n,t) = P(n+1,t-1) \quad \text{if } n = N. \end{equation}\begin{align} P(0,t) =P(N+1,t) = 0. \end{align}

I would like to know the exact expression of $P(n,t)$. I thought this would be simple to find but I can't find a reference with a simple expression for it... Am I missing something?

Consider a simple random walk in one dimension with reflective boundaries at $n=1$ and $n=N$. We can express it via the master equation: \begin{equation} P(n,t) = \frac{1}{2}P(n-1,t-1) + \frac{1}{2}P(n+1,t-1) \quad \text{if } n\neq 1 \;{\rm and} \;n\neq N, \end{equation} \begin{equation} P(n,t) = P(n-1,t-1) \quad \text{if } n = 1, \end{equation} \begin{equation} P(n,t) = P(n+1,t-1) \quad \text{if } n = N. \end{equation}

I would like to know the exact expression of $P(n,t)$. I thought this would be simple to find but I can't find a reference with a simple expression for it... Am I missing something?

Consider a simple random walk in one dimension with reflective boundaries at $n=1$ and $n=N$. We can express it via the master equation: \begin{equation} P(n,t) = \frac{1}{2}P(n-1,t-1) + \frac{1}{2}P(n+1,t-1) \quad \text{if } n\neq 1 \;{\rm and} \;n\neq N, \end{equation} With the following boundary conditions: \begin{align} P(0,t) =P(N+1,t) = 0. \end{align}

I would like to know the exact expression of $P(n,t)$. I thought this would be simple to find but I can't find a reference with a simple expression for it... Am I missing something?

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papad
papad
  • 274
  • 1
  • 11

Probability of random walk on confined lattice with reflective boundaries

Consider a simple random walk in one dimension with reflective boundaries at $n=1$ and $n=N$. We can express it via the master equation: \begin{equation} P(n,t) = \frac{1}{2}P(n-1,t-1) + \frac{1}{2}P(n+1,t-1) \quad \text{if } n\neq 1 \;{\rm and} \;n\neq N, \end{equation} \begin{equation} P(n,t) = P(n-1,t-1) \quad \text{if } n = 1, \end{equation} \begin{equation} P(n,t) = P(n+1,t-1) \quad \text{if } n = N. \end{equation}

I would like to know the exact expression of $P(n,t)$. I thought this would be simple to find but I can't find a reference with a simple expression for it... Am I missing something?