Editing Openai/693343d7-a38c-8012-a67c-11cbed4c0fd9 (section)

===== - Stochastic state transitions: Each agent's state can evolve stochastically based on its environment and interactions. Instead of deterministic updates (e.g., S_{t+1} = C(S_t, ...)), we introduce a stochastic process: ``<code> S_{t+1} = StochasticTransition(S_t, U_t), where U_t ∼ P(U) (random variable drawn from a distribution) </code>`<code> Here, U_t represents random influences (e.g., environmental noise, social interactions), and P(U) is the probability distribution governing them. =====
* Stochastic processes: We can model the evolution of an agent’s state using Markov processes or random walks, depending on the memory and dependence on past states: - Markovian Agent: If the state depends only on the current state, we model using a Markov Chain: </code>`<code> P(S_{t+1} | S_t) = f(S_t)  (probabilistic state transition matrix) </code>`<code> - Random Walks: When agents take random steps within their environment, governed by a probability density function.