Abstract:

The random walk (RW) describes a path that consists of a succession of random steps on some mathematical space such as the integer lattice. Its modification has been successfully applied to define the distance in the field of data science. In this short course, we will discuss the following questions:

1. On the average, how far is the walker from the starting point?

2. What is the probability distribution for the position of the walker?

3. Does the random walker keep returning to the origin or does the walker eventually leave forever?

4. For RWs with absorbing barriers, what is the probability distribution of the walker’s final position?

5. How to model the heat flow on a subset of d-dimensional lattice with a boundary condition?

6. For RWs with absorbing barriers, what is the expected time for the walker to reach an absorbing state?

7. What is relationship between the RWs and discrete harmonic functions?

Reference: Random Walk and the Heat Equation written by Gregory F. Lawler

Link of Lectures:

Lecture 1

Lecture 2

Lecture 3

Lecture 4

Lecture 5