Paul Lévy

Paul Lévy (1886–1971) was a French mathematician whose work shaped probability theory, especially the study of stable distributions, Brownian motion, and stochastic processes.

Probability theory: The mathematical study of random events, outcomes, and uncertainty. It is the field in which Lévy recast randomness as a rigorous mathematical subject.
Lévy process: A stochastic process with stationary independent increments and paths that are right-continuous with left limits. It gathers the independent-increment processes that became central to Lévy’s theory.
Andrey Kolmogorov: A Russian mathematician who formulated probability theory using measure theory and developed foundational results on stochastic processes. Kolmogorov’s axiomatic framework complemented Lévy’s work on random processes.
Lévy–Itō calculus: A stochastic calculus for processes with jumps, extending techniques developed for Brownian motion. It brings Lévy processes into differential equations and models with discontinuous paths.
Markov process: A stochastic process whose future, conditional on its present, is independent of its past. Lévy processes form a special class of Markov processes with stationary independent increments.
Stochastic process: A collection of random variables indexed by time or another parameter. Lévy helped establish the mathematical study of these evolving random systems.
Lévy–Khintchine formula: A representation of the characteristic exponent of an infinitely divisible probability distribution. It gives the general analytic form behind the infinitely divisible laws Lévy studied.
Norbert Wiener: An American mathematician who established mathematical foundations for Brownian motion and developed Wiener measure. Wiener’s construction gave rigorous foundations to the Brownian motion Lévy studied.
Fractional diffusion: A diffusion model governed by fractional-order spatial or temporal derivatives. Stable distributions and Lévy motion provide probabilistic models for its nonlocal spreading.
Martingale: A stochastic process whose conditional expected future value equals its current value. Martingales describe a different structural property than Lévy’s independent-increment condition.

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