thanksmathman said:Usual meaning: Let us assume the walk starts at the origin at time 0, then the term refers to the first time, after it starts, that it gets back to the origin.
A random walk is a mathematical concept used to describe the path of a randomly moving object. It is a sequence of steps taken in a random direction and distance, resulting in a path that appears to be random.
A first return visit in random walks refers to the first time a walker returns to a specific point on their path. This point can be the starting point or any other point on the path.
Studying first return visits in random walks can provide insights into the properties and behavior of random processes. It can also have applications in various fields such as physics, biology, and finance.
A first return time in random walks is the number of steps taken by a walker before returning to a specific point. It can be calculated by counting the steps taken or by using mathematical formulas based on the properties of random walks.
Random walks and first return visits can be observed in various natural phenomena such as the movement of molecules in a gas, the behavior of animals searching for food, and the fluctuations in stock prices. They can also be used to model the spread of diseases, the movement of particles in a liquid, and the behavior of random algorithms.