A stochastic process is a mathematical concept that describes how a system changes randomly over time. It's a collection of random variables, each representing the state of the system at different points in time. Think of it as a sequence of snapshots showing how something evolves unpredictably.
The Markov property is the key characteristic that defines a Markov chain. It states that the future state of the system depends only on the current state, not on how we arrived at the current state. This is often called the memoryless property. In mathematical terms, the conditional probability of the next state given all previous states equals the conditional probability given only the current state.
Now we can define a Markov chain. A Markov chain is simply a stochastic process that satisfies the Markov property. It consists of a set of discrete states and transition probabilities between these states. The system moves from one state to another according to these probabilities, but the choice of the next state depends only on the current state, not on the sequence of states that led to it. This creates a memoryless random walk through the state space.
A Markov chain can be represented using a transition matrix. Each element P[i,j] represents the probability of transitioning from state i to state j. Each row must sum to 1 since the system must transition to some state. Let's look at a simple weather model with two states: sunny and rainy. The transition matrix shows that if it's sunny today, there's a 70% chance it stays sunny tomorrow and 30% chance it becomes rainy. If it's rainy, there's a 60% chance it becomes sunny and 40% chance it stays rainy.
Markov chains have numerous real-world applications across many fields. In computer science, Google's PageRank algorithm uses Markov chains to rank web pages based on link structure. In finance, they model stock prices and market behavior. In biology, they help understand population dynamics and disease spread. The memoryless property makes them particularly useful for modeling systems where the current state captures all relevant information for predicting future behavior.