In probability theory, a Hunt process is a type of Markov process, named for mathematician Gilbert A. Hunt who first defined them in 1957. Hunt processes were important in the study of probabilistic potential theory until they were superseded by right processes in the 1970s.
History
Background In the 1930-50s the work of mathematicians such as Joseph Doob, William Feller, Mark Kac, and Shizuo Kakutani developed connections between Markov processes and potential theory. In 1957-8 Gilbert A. Hunt published a triplet of papers which deepened that connection. The impact of these papers on the probabilist community of the time was significant. Joseph Doob said that "Hunt’s great papers on the potential theory generated by Markov transition functions revolutionized potential theory." Ronald Getoor described them as "a monumental work of nearly 170 pages that contained an enormous amount of truly original mathematics." Gustave Choquet wrote that Hunt's papers were "fundamental memoirs which were renewing at the same time potential theory and the theory of Markov processes by establishing a precise link, in a very general framework, between an important class of Markov processes and the class of kernels in potential theory which French probabilists had just been studying." One of Hunt's contributions was to group together several properties that a Markov process should have in order to be studied via potential theory, which he called "hypothesis (A)". A stochastic process X {\displaystyle X} satisfies hypothesis (A) if the following three assumptions hold:
First assumption: X {\displaystyle X} is a Markov process on a Polish space with càdlàg paths. Second assumption: X {\displaystyle X} satisfies the strong Markov property. Third assumption: X {\displaystyle X} is quasi-left continuous on [ 0 , ∞ ) {\displaystyle [0,\infty )} . Processes satisfying hypothesis (A) soon became known as Hunt processes. If the third assumption is slightly weakened so that quasi-left continuity holds only on the lifetime of X {\displaystyle X} , then X {\displaystyle X} is called a "standard process", a term that was introduced by Eugene Dynkin.
Rise and fall The book Markov Processes and Potential Theory (1968) by Blumenthal and Getoor codified standard and Hunt processes as the archetypal Markov processes. Over the next few years probabilistic potential theory was concerned almost exclusively with these processes. Of the three assumptions contained in Hunt's hypothesis (A), the most restrictive is quasi-left continuity. Getoor and Glover write: "In proving many of his results, Hunt assumed certain additional regularity hypotheses about his processes. ... It slowly became clear that it was necessary to remove many of these regularity hypotheses in order to advance the theory." Already in the 1960s attempts were being made to assume quasi-left continuity only when necessary. In 1970, Chung-Tuo Shih extended two of Hunt's fundamental results, completely removing the need for left limits (and thus also quasi-left continuity). This led to the definition of right processes as the new class of Markov processes for which potential theory could work. Already in 1975, Getoor wrote that Hunt processes were "mainly of historical interest". By the time that Michael Sharpe published his book "General Theory of Markov Processes" in 1988, Hunt and standard processes were considered obsolete in probabilistic potential theory. Hunt processes are still studied by mathematicians, most often in relation to Dirichlet forms.
Definition
Brief definition A Hunt process X {\displaystyle X} is a strong Markov process on a Polish space that is càdlàg and quasi-left continuous; that is, if ( T n ) {\displaystyle (T_{n})} is an increasing sequence of stopping times with limit T {\displaystyle T} , then
P ( lim n → ∞ X T n = X T | T < ∞ ) = 1. {\displaystyle \mathbb {P} {\big (}\lim _{n\to \infty }X_{T_{n}}=X_{T}{\big |}T<\infty {\big )}=1.}
… excerpt ends here. Continue reading the full article.
