What Is Stochastic Analysis? From Brownian Motion to Tanaka's Formula

The Origins of Probability Theory

Probability theory is said to have begun in 1654 with the correspondence between Pascal and Fermat concerning a nobleman's wager. It subsequently developed as a pure game of thought, and classical probability theory was culminated in Laplace's 1812 work, "Analytical Theory of Probability." During this period, Bernoulli discovered the weak law of large numbers in 1713, and the central limit theorem, a refinement of this law near the mean, was proven by de Moivre and Laplace. The strong law of large numbers, which states that the law of large numbers holds with probability 1, was later first proven by Borel in 1909. Furthermore, the theory of large deviations, which deals with the asymptotic behavior of the probability of events deviating significantly from the mean when the law of large numbers holds, was studied by Khinchin in 1929 and Cramér in 1938. In probability theory, such limit theorems are a central focus of interest. Modern probability theory is said to have begun with Kolmogorov's 1933 work, "Foundations of the Theory of Probability." In this work, Kolmogorov, leaving the question of "what is randomness?" aside, provided a measure-theoretic foundation for probability from an axiomatic standpoint, after which probability theory underwent rapid development.

Brownian Motion

Modern probability theory is said to be a theory based on Brownian motion. The name Brownian motion originates from the botanist Brown's 1828 observation of pollen grains moving in a zigzag pattern under a microscope. In 1905, Einstein conducted a molecular-thermodynamic study and derived the relationship between the diffusion coefficient of microparticles and Avogadro's number. Thus, while it has been shown physically that Brownian motion is caused by the random collisions of water molecules with microparticles, this remains a topic of ongoing mathematical research, especially in multiple dimensions. The mathematical construction of an idealized Brownian motion was carried out by Wiener starting in 1923 as a Fourier series with random coefficients. In the context of the connection between Brownian motion and analysis, it is known that taking the average of a function with Brownian motion as its argument provides a solution to the initial value problem for the heat equation. Furthermore, by taking the average up to the hitting time of a certain domain, a probabilistic representation of the solution to the Dirichlet problem can also be obtained.

Wiener Functionals, Itô's Formula

However, attempting to do the same for a general diffusion process requires the fundamental solution of the diffusion equation to construct the process in the first place. This leads to the question of whether a general diffusion process can be constructed using probabilistic methods. What made this possible was the theory of stochastic differential equations by Professor Kiyosi Itô. In other words, a general diffusion process is given as a functional of the path of a Brownian motion (this is called a Wiener functional), and it is realized by solving a stochastic differential equation. This calculus concerning the paths of stochastic processes is called stochastic analysis. The path of a Brownian motion is continuous but nowhere differentiable, and its total variation diverges with probability 1, so ordinary calculus cannot be applied. However, in 1942, Professor Itô introduced the stochastic integral with respect to Brownian motion in a very natural way, provided a method for pathwise calculus centered on "Itô's formula" (the chain rule for this stochastic differentiation), and justified stochastic differential equations. As equations of motion describing random phenomena, stochastic differential equations are now applied in a wide range of fields, including physics, engineering, biology, and economics. When the solution to a diffusion equation is given as the expectation of a Wiener functional in this way, it can be shown that the solution for a drift term shift is given using the Maruyama-Girsanov functional, and the solution for a case with a potential function is given using the Feynman-Kac functional. Using this Feynman-Kac formula, one can represent the principal eigenvalue of a Laplacian with a potential. The question of whether the equality between this representation and the one derived from the variational principle for the principal eigenvalue could be proven probabilistically without resorting to analysis was posed by Kac. An answer to this was provided by Donsker and Varadhan in the early 1970s. This also became the starting point for the subsequent major development of the theory of large deviations.

Tanaka's Formula and Wiener Hida Distributions

One way to view Brownian motion from a generalized function perspective is through local time. This is an important quantity that corresponds to the density function of the sojourn time of a Brownian motion at a certain position. Regarding this, Professor Hiroshi Tanaka, who was a professor at this Faculty of Science and Technology from 1981 to 1998, discovered in his youth that by extending Itô's formula to the δ-function, a clear existence proof could be given through stochastic analysis. This formula has since been generalized and is now widely used as "Tanaka's formula." Unfortunately, Professor Tanaka passed away last July. As for general Wiener Hida distributions, around 1980, Malliavin first introduced a pathwise derivative on Wiener space using the Ornstein-Uhlenbeck process, showing that many important Wiener functionals, though discontinuous, are smooth. Subsequently, after research that included results from many Japanese researchers, it was justified by Professor Shinzo Watanabe, who formulated Malliavin calculus as a theory of Wiener Hida distributions. The fundamental solution of a diffusion process itself, which could not be handled within the scope of Itô calculus without the aid of analysis, can now be treated as a Wiener Hida distribution using stochastic analysis methods, and the range of applications has expanded dramatically.

My own interests are centered on problems related to stochastic analysis, such as the theory of large deviations and the asymptotic theory of stochastic processes in random media, in which I had the opportunity to participate in Professor Tanaka's research. However, probability theory is now applied to a wide range of fields beyond those related to stochastic analysis, including problems in mathematical physics such as Gibbs measures, phase boundary models, and percolation models.