Titlebook: Probability and Stochastic Processes for Physicists; Nicola Cufaro Petroni Textbook 2020 The Editor(s) (if applicable) and The Author(s),

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Probability and Stochastic Processes for Physicists978-3-030-48408-8Series ISSN 2198-7882 Series E-ISSN 2198-7890

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DistributionsWe will explore now the protocols used to define on . a probability . also called either . or ., and we will start with finite or countable spaces so that . will be defined in an elementary way.

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Limit TheoremsThe Limit Theorems are statements about limits of sequences of sums of .’s when the number of addenda grows to infinity.

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General NotionsThe notion of stochastic process (.). on a probability space . with . has already been introduced in the Section. where we pointed out that it can be considered from two complementary standpoints.

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Dynamical Theory of Brownian MotionIn 1930 L.S. Ornstein and G.F. Uhlenbeck [.] addressed again the problem of elaborating a suitable model for the Brownian motion, and they refined in more detail the Langevin dynamical equation to investigate the phenomenon at time scales shorter than those considered by Einstein [.] and Smoluchowski [.] in 1905-6.

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Nicola Cufaro PetroniBuilds a bridge between physics and probability.Explains the role of Brownian motion in physics.Includes appendices devoted to particular details, calculations and concise treatments of thought-provok

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Heuristic Definitionsabilistic properties. This illuminating and informative procedure, however, can not be easily replicated for other typical, non trivial processes whose trajectories, as we will see later, can only be defined either as limits of suitable approximations or by adopting a more general standpoint