Who was the first mathematician to study stochastic processes, and why did he do such research?

Stochastic process is a quantitative description of the dynamic relationship of a series of random events. Stochastic process theory is closely related to other branches of mathematics, such as potential theory, differential equation, mechanics and complex variable function theory, and is an important tool for studying stochastic phenomena in natural science, engineering science and social science. Stochastic process theory has been widely used at present, and it is often used to build mathematical models in many fields, such as weather forecast, statistical physics, astrophysics, operational decision-making, economic mathematics, security science, population theory, reliability, computer science and so on. Generally speaking, a set of random variables is defined as a random process. When studying stochastic processes, people describe the inherent laws of necessity through superficial contingency and describe these laws in the form of probability, and realize that necessity is the charm of this subject from contingency. The theoretical basis of the whole stochastic process discipline was laid by Andre Andrey Kolmogorov and Du Budian. This discipline originated from the study of physics, such as the study of statistical mechanics by Gibbs, Boltzmann and Poincare, and the pioneering work of Einstein, Weiner and Levy on Brownian motion. Around 1907, Markov studied a series of random variables with specific dependencies, which were later called Markov chains. Wiener gave the mathematical definition of Brownian motion in 1923, and this process is still an important research topic today. Generally speaking, the research on the general theory of stochastic processes began in the 1930s. In 193 1, the analysis method of probability theory was published, and in 1934, Yin published the related theory of stationary process. These two works laid a theoretical foundation for Markov process and stationary process. 1953, Dube published his famous book "Theory of Stochastic Processes", which systematically and strictly described the basic theory of stochastic processes. There are many methods to study stochastic processes, which are mainly divided into two categories: one is probability method, which uses orbital properties, stopping time and stochastic differential equations; The other is analytical method, which uses measure theory, differential equation, semigroup theory, function heap and Hilbert space. In practical research, the two methods are often used in combination. In addition, combinatorial method and algebraic method also play a certain role in the study of some special stochastic processes. The main contents of the study include: multi-index stochastic processes, infinite particles and Markov processes, probability and potential, and special discussions on various special processes. Chinese scholars have done well in stationary processes, Markov processes, martingale theory, limit theorems, stochastic differential equations and so on. The actual stochastic process is any process dominated by probability. Examples are: ① Population development dominated by Mendelian inheritance; (2) Brownian motion of microscopic particles affected by molecular collision, or star motion in macroscopic space; (3) a series of gambling in casinos; (4) the passage of cars at designated points on the highway. In each case, the stochastic system is evolutionary, which means that its state changes with time, so the state at time t is accidental, it is a random variable x(t), and the set of parameters t is usually an interval (a stochastic process with continuous parameters) or an integer set (a stochastic process with discrete parameters). However, some authors use the term stochastic process only in the case of continuous parameters. If the state of the system is represented by a number, x(t) is a numerical value. In other cases, x(t) can be a vector value or a more complex value. In this paper, the discussion is usually limited to numerical cases. When the state changes, its value determines a time function-sample function, and the probability law governing the process determines the probability of giving the sample function various possible properties. Mathematical stochastic process is a mathematical structure caused by the concept of actual stochastic process. People study this process, either because it is a mathematical model of actual stochastic process or because of its inherent mathematical significance and its application outside the field of probability theory. Mathematical stochastic process can be simply defined as a set of random variables, that is, a parameter set is specified, and a random variable x(t) is specified for each parameter point T. If we recall that the random variable itself is a function, a point in the domain of the random variable x(t) is represented by ω, and the value of the random variable in ω is represented by x(t, ω), then the stochastic process is completely defined by the point pair (t). If t is fixed, this binary function defines a function of ω, that is, a random variable represented by x(t). If ω is fixed, this binary function defines a function of t, which is the sample function of the process. The probability distribution of probabilistic stochastic processes is usually given by the joint distribution of random variables that specify it. These joint distributions and the probabilities derived from them can be interpreted as the probabilities of the properties of sample functions. For example, if to is a parameter value, then the probability that the sample function takes a positive value at to is the probability that the random variable x(to) takes a positive value. The basic theorem at this level: any given self-consistent joint probability distribution corresponds to a random process. The concept of stochastic process is very extensive, so the study of stochastic process includes almost all the contents of probability theory. Although we can't give a useful and narrow definition, when using the term stochastic process, probability theorists usually think of stochastic processes with some meaningful relationship between random variables, such as independence. Before the term stochastic process was put forward, the sequence of independent variables was a stochastic process that had been studied for a long time. For historical reasons, such a sequence is generally not regarded as a random process (although its simulation has continuous parameters-the process with independent increments will be discussed later, it is regarded as a random process). The rest of this paper is a general discussion of some special stochastic process classes. Because these process classes are very important in mathematical and non-mathematical applications, they have attracted great attention. The joint distribution of any finite external random variables in stochastic processes such as stationary processes is not affected by parameter shift, that is, the distribution of x(t 1+h), …, x(tn+h) has nothing to do with H. In today's knowledge system of higher education, the basic knowledge of stochastic processes is mainly introduced in the two courses of applied stochastic processes and stochastic process theory. The former is an undergraduate course, which is generally offered in junior year. It briefly introduces discrete-time Markov chain, continuous-time Markov chain, Brownian motion and so on. The latter is a postgraduate course, which introduces martingale theory, strict and steady process and other knowledge. In addition, the theory is also involved in electronic communication disciplines such as communication principles and systems.