What is the probability density function over time for a 1-D random walk on a line with boundaries?

To understand the probability density function (PDF) for a one-dimensional random walk with boundaries, we first need to clarify what a random walk is. In simple terms, a random walk involves a particle that moves along a line, taking steps either to the left or right with equal probability. When we introduce boundaries, we need to consider how these affect the particle's movement and the resulting probability distribution over time.

Defining the Random Walk

In a one-dimensional random walk, let's say a particle starts at position zero on a line. At each time step, it can move to the left (position -1) or to the right (position +1) with equal probability of 0.5. This process continues for a number of steps, and we can analyze the position of the particle at any given time.

Introducing Boundaries

When boundaries are introduced, such as a reflecting boundary at position 0 and a reflecting boundary at position N, the behavior of the random walk changes. If the particle reaches one of these boundaries, it will reflect back into the allowed region instead of continuing outside the boundaries. This reflection alters the probability distribution of the particle's position over time.

Probability Density Function Over Time

The probability density function for a random walk with boundaries can be derived using combinatorial methods or through the use of generating functions. For a simple case with reflecting boundaries, the PDF can be expressed in terms of binomial coefficients. Specifically, if we denote the position of the particle after \( n \) steps as \( X_n \), the probability of finding the particle at position \( k \) after \( n \) steps can be given by:

• For \( k = 0 \): \( P(X_n = 0) = \frac{1}{2^n} \sum_{j=0}^{n/2} \binom{n}{2j} \)
• For \( 1 \leq k \leq N-1 \): \( P(X_n = k) = \frac{1}{2^n} \sum_{j=0}^{(n-k)/2} \binom{n}{k+j} \binom{n-k}{j} \)
• For \( k = N \): \( P(X_n = N) = \frac{1}{2^n} \sum_{j=0}^{n/2} \binom{n}{2j} \)

Here, \( \binom{n}{k} \) represents the binomial coefficient, which counts the number of ways to choose \( k \) successes (right moves) in \( n \) trials (total steps).

Behavior Over Time

As time progresses (i.e., as \( n \) increases), the distribution of the particle's position approaches a steady state. This steady-state distribution can be approximated by a uniform distribution across the allowed positions between the boundaries. The exact shape of the PDF will depend on the specific parameters of the walk, such as the number of steps and the positions of the boundaries.

Visualizing the Distribution

To visualize this, imagine a histogram representing the positions of the particle after many steps. Initially, the distribution may be quite spread out, but as time goes on, it becomes more concentrated around the center of the allowed region, reflecting the influence of the boundaries. This behavior is akin to the diffusion process, where particles spread out over time but are constrained by barriers.

Conclusion

In summary, the probability density function for a one-dimensional random walk with boundaries is influenced by the reflection at the boundaries and can be derived using combinatorial methods. Over time, the distribution tends to stabilize, reflecting the constraints imposed by the boundaries. Understanding this concept is crucial for applications in various fields, including physics, finance, and biology, where random processes play a significant role.