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Understanding Markov Processes

Markov Processes, named after the Russian mathematician Andrey Markov, are stochastic processes that exhibit a unique property known as the Markov property. This property states that the future state of the system depends solely on its current state and is independent of how the system arrived at its current state. In other words, the future evolution of the system is "memoryless," meaning it does not depend on the history of events leading up to the current state.

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Key Concepts:

• State Space: A fundamental concept in Markov Processes, the state space represents the set of all possible states that the system can occupy. It serves as the foundation for analyzing the behavior and evolution of the system over time. Understanding the state space is crucial for defining the boundaries and characteristics of the Markov chains Process under study.
• Transition Probabilities: Central to the definition of a Markov Process are transition probabilities, which quantify the likelihood of the system transitioning from one state to another in a single time step. These probabilities encode the dynamics of the system and govern its evolution over time. Analyzing transition probabilities allows researchers to understand the probabilistic nature of state transitions and predict future states of the system.
• Markov Property: One of the defining characteristics of Markov Processes is the Markov property, which states that the future state of the system depends only on its current state and is independent of the history of events that preceded it. This property implies that the system's behavior is memoryless, with the transition probabilities determining the future state solely based on the present state. Understanding and leveraging the Markov property is essential for modeling and analyzing dynamic systems with probabilistic transitions.
• Stationary Distribution: In certain cases, Markov Processes converge to a stationary distribution, where the probability of being in each state remains constant over time. The stationary distribution provides valuable insights into the long-term behavior of the system, indicating the relative likelihood of the system occupying different states in equilibrium. Analyzing the stationary distribution allows researchers to characterize the steady-state behavior of the Markov Process and identify stable patterns or trends in the system's dynamics.

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