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In probability theory , a Levy process , named after the French mathematician Paul Levy , is a stochastic process with independent , stationary increments  : it represents the motion of a point whose successive displacements are random and independent , and statistically identical over different time intervals of the same length .. probability theory. probability theory. Paul Levy. Paul Levy ( mathematician ). stochastic process. stochastic process. random. random. A Levy process may thus be viewed as the continuous-time analog of a random walk .. random. random. random walk. random walk. The most well known examples of Levy processes are the Wiener process , often called the Brownian motion process , and the Poisson process .. Wiener process. Wiener process. Poisson process. Poisson process. Aside from Brownian motion with drift , all other proper Levy processes have discontinuous paths .. discontinuous. discontinuous