Lévy

Lévy is a French surname. The name is notably associated with Paul Lévy (1886–1971), a French mathematician whose work laid foundations in probability theory, functional analysis, and the theory of stochastic processes. Several mathematical constructs bear his name.

A Lévy process is a stochastic process with stationary independent increments; it generalizes Brownian motion by allowing jumps and is central to continuous-time stochastic modeling in finance and physics. The characteristic function of a Lévy process is described by the Lévy-Khintchine formula, which expresses it in terms of a triplet consisting of a drift vector, a diffusion component, and a Lévy measure that encodes jump intensity and size. The Lévy distribution is a one-sided stable distribution with heavy tails, often described with a location parameter μ and a scale parameter c; it arises as the distribution of certain first passage times and is part of the family of stable distributions with index 1/2.

A Lévy measure is the measure on the real line excluding zero that appears in the Lévy-Khintchine representation and characterizes the jump component of a Lévy process. Lévy’s continuity theorem provides conditions under which convergence of characteristic functions implies convergence of distributions. Lévy’s lemma appears in the study of concentration of measure on high-dimensional spheres, and Lévy flights describe random walks with heavy-tailed step lengths that yield occasional long jumps.

The surname Lévy is shared by various individuals in different fields; in mathematics, it most commonly evokes Paul Lévy and the related probabilistic and analytic concepts.

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