Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event.
In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a sequence of random variables in a probability space, where the index of the sequence often has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.
Stochastic is the property of being well-described by a random probability distribution. Stochasticity and randomness are distinct, in that the former refers to a modeling approach and the latter refers to phenomena; these terms are often used synonymously. In probability theory, the formal concept of a stochastic process is also referred to as a random process.
In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by Maurice Fréchet who commented that the “development of probability theory and expansion of area of its applications have led to necessity to pass from schemes where (random) outcomes of experiments can be described by number or a finite set of numbers, to schemes where outcomes of experiments represent, for example, vectors, functions, processes, fields, series, transformations, and also sets or collections of sets.”
Anders Martin-Löf is a Swedish physicist and mathematician. He has been a professor in insurance mathematics and mathematical statistics since 1987 at the Department of Mathematics of Stockholm University.
In probability theory, the Schramm–Loewner evolution with parameter κ, also known as stochastic Loewner evolution (SLEκ), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensional lattice models in statistical mechanics. Given a parameter κ and a domain in the complex plane U, it gives a family of random curves in U, with κ controlling how much the curve turns. There are two main variants of SLE, chordal SLE which gives a family of random curves from two fixed boundary points, and radial SLE, which gives a family of random curves from a fixed boundary point to a fixed interior point. These curves are defined to satisfy conformal invariance and a domain Markov property.
In statistics, an exchangeable sequence of random variables is a sequence X1, X2, X3, ... whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. In other words, the joint distribution is invariant to finite permutation. Thus, for example the sequences
Jean-François Le Gall is a French mathematician working in areas of probability theory such as Brownian motion, Lévy processes, superprocesses and their connections with partial differential equations, the Brownian snake, random trees, branching processes, stochastic coalescence and random planar maps. He received his Ph.D. in 1982 from Pierre and Marie Curie University under the supervision of Marc Yor. He is currently professor at the University of Paris-Sud in Orsay and is a senior member of the Institut universitaire de France. He was elected to French academy of sciences, December 2013.
Bálint Tόth is a Hungarian mathematician whose work concerns probability theory, stochastic process and probabilistic aspects of mathematical physics. He obtained PhD in 1988 from the Hungarian Academy of Sciences, worked as senior researcher at the Institute of Mathematics of the HAS and as professor of mathematics at TU Budapest. He holds the Chair of Probability at the University of Bristol and is a research professor at the Alfréd Rényi Institute of Mathematics, Budapest.
Harry Kesten was a Jewish American mathematician best known for his work in probability, most notably on random walks on groups and graphs, random matrices, branching processes, and percolation theory.
Jan Hendrik van Schuppen is a Dutch mathematician and Professor at the Department of Mathematics of the Vrije Universiteit, known for his contributions in the field of systems theory, particularly on control theory and system identification, on probability, and on a number of related practical applications.
Geoffrey Richard GrimmettOLY is an English mathematician known for his work on the mathematics of random systems arising in probability theory and statistical mechanics, especially percolation theory and the contact process. He is the Professor of Mathematical Statistics in the Statistical Laboratory, University of Cambridge, and was the Master of Downing College, Cambridge, from 2013 to 2018.
Rollo Davidson was a probabilist, alpinist, and Fellow-elect of Churchill College, Cambridge, who died aged 25 on Piz Bernina. He is known for his work on semigroups, stochastic geometry, and stochastic analysis, and for the Rollo Davidson Prize, given in his name to early-career probabilists.
Carl Svante Janson is a Swedish mathematician. A member of the Royal Swedish Academy of Sciences since 1994, Janson has been the chaired professor of mathematics at Uppsala University since 1987.
Grégory Miermont is a French mathematician working on probability, random trees and random maps.
Gérard Ben Arous is a French mathematician, specializing in stochastic analysis and its applications to mathematical physics. He served as the director of the Courant Institute of Mathematical Sciences at New York University from 2011 to 2016.
Steven Neil Evans is an Australian-American statistician and mathematician, specializing in stochastic processes.
Bruce Edward Hajek is a Professor in the Coordinated Science Laboratory, the head of the Department of Electrical and Computer Engineering, and the Leonard C. and Mary Lou Hoeft Chair in Engineering at the University of Illinois Urbana–Champaign. He does research in communication networking, auction theory, stochastic analysis, combinatorial optimization, machine learning, information theory, and bioinformatics.
Nike Sun is a probability theorist who works as an associate professor of mathematics at the Massachusetts Institute of Technology, on leave from the department of statistics at the University of California, Berkeley. She won the Rollo Davidson Prize in 2017. Her research concerns phase transitions and the counting complexity of problems ranging from the Ising model in physics to the behavior of random instances of the Boolean satisfiability problem in computer science.
