Paul Lévy was brought up in the world of math. His father and grandfather were both math professors in Paris, where Lévy was born in 1886. Not only did Lévy continue the family tradition by making a career in math, but he passed the math gene on to his daughter, Marie-Hélène, who became a math professor herself.

While four generations of brilliant mathematicians is a fun historic quirk, what is most notable about Lévy is the groundbreaking work he produced on probability and Brownian motion. These concepts laid the foundation for big changes in the way we understand physics and economics.

Inventing probability theory

When Lévy first set out on a career in academia, there was no theory of probability. Lévy was one of the key figures who helped establish probability theory as a major branch of mathematics.

In 1925, Lévy published the first of many books on probability, Calculs des Probabilités, and it was groundbreaking. The book “contained the first systematic treatise of random variables, their probability distributions, and their characteristic functions,” wrote Swedish mathematician Harald Cramer in 1980.

Lévy continued to be a leading light on probability theory. His 1937 book, Théorie de l’addition et des variables aléatoires (Theory of Addition and Random Variables), “was the first modern technical book on probability; it inspired a generation of students,” wrote Joseph L. Doob, the acclaimed American mathematician.

“If there is one person who has influenced the establishment and growth of probability theory more than any other, that person must be Paul Lévy.”

British mathematician S.J. Taylor, writing in the Bulletin of the London Mathematical Society in 1975

Ignoring Brownian motion before embracing it

In the 1920s, Paul Lévy played an unfortunate role in derailing the career and sullying the reputation of fellow French mathematician Louis Bachelier. At the urging of a colleague, Lévy wrote a letter to the University of Dijon stating that a key work by Bachelier published a decade earlier was fundamentally flawed. Bachelier was “blackballed” from the university and relegated to academic obscurity for the rest of his days.

Lévy later admitted that Bachelier wasn’t quite as wrong as he initially believed. And in the 1930s Lévy went on to build upon the theories of stochastic process and diffusion process that Bachelier had put forth decades earlier.

In later decades, the work on these concepts by Bachelier and Lévy had a profound impact on the world of finance. Prominent economists such as Fischer Black, Myron Scholes, Robert Merton, and Eugene Fama incorporated their theories on the random movement of particles and prices to shape their view of the stock market.

Triumphing over persecution

Lévy’s career was interrupted and nearly derailed twice by war. First, he had to take a break from academia to fight in World War I. Two-and-a-half decades later, Lévy, who was Jewish, was forced into hiding to evade capture by the Nazis and their allies in the French Vichy government.

After France’s liberation, Lévy got right back to work. Only a few years later, in 1948, he published an influential analysis of the stochastic process and Brownian motion. Writing in 2012, the French mathematician Jean-François Le Gall described the paper as “paving the way for work done by a number of other well-known probabilists of the 20th century.”

Lévy continued to refine his mathematical concepts until his retirement in 1959 at the age of 73.
Besides his daughter, his legacy includes a variety of mathematical concepts that bear his name, including Lévy process, Lévy flight, Lévy’s constant, Lévy distribution, Lévy C curve, and Lévy’s arcsine law.