Hunt process

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 Hunt processes were important in the study of probabilistic potential theory until they were superseded by right processes in the 1970s.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 significantly 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." More recently, Marc Yor 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 together called "hypothesis (A)". A stochastic process $X$ satisfies hypothesis (A) if the following three assumptions hold:

First assumption: $X$ is a Markov process on a Polish space with càdlàg paths.

Second assumption: $X$ satisfies the strong Markov property.

Third assumption: $X$ is quasi-left continuous on $[0,\infty )$.

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 $X$, then $X$ is called a "standard processes", a term that was introduced by Eugene Dynkin.

Rise and fall

The textbook "Markov Processes and Potential Theory" (1968) by Blumenthal and Getoor codified standard and Hunt processes as the archetypal Markov processes. Over the next decade 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, and in particular completely removed 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 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 $X$ is a strong Markov process on a Polish space that is càdlàg and quasi-left continuous; that is, if $(T_{n})$ is an increasing sequence of stopping time with limit $T$, then

$$\mathbb {P} {\big (}\lim _{n\to \infty }X_{T_{n}}=X_{T}{\big |}T<\infty {\big )}=1.$$

Verbose definition

Let $E$ be a Radon space and ${\mathcal {E}}$ the $\sigma$-algebra of universally measurable subsets of $E$, and let $(P_{t})$ be a Markov semigroup on $(E,{\mathcal {E}})$ that preserves ${\mathcal {E}}$. A Hunt process is a collection $X=(\Omega ,{\mathcal {G}},{\mathcal {G}}_{t},X_{t},\theta _{t},\mathbb {P} ^{x})$ satisfying the following conditions:

(i) $(\Omega ,{\mathcal {G}},{\mathcal {G}}_{t})$ is a filtered measurable space, and each $\mathbb {P} ^{x}$ is a probability measure on $(\Omega ,{\mathcal {G}})$.

(ii) For every $x\in E$, $X_{t}$ is an $E$-valued stochastic process on $(\Omega ,{\mathcal {G}},\mathbb {P} ^{x})$, and is adapted to $({\mathcal {G}}_{t})$.

(iii) (normality) For every $x\in E$, $\mathbb {P} ^{x}(X_{0}=x)=1$.

(iv) (Markov property) For each $x\in E$, $\mathbb {P} ^{x}(f(X_{t+s})|{\mathcal {G}}_{t})=P_{s}f(X_{t})$ for all $t,s\geq 0,f\in b{\mathcal {E}}$.

(v) $(\theta _{t})_{t\geq 0}$ is a collection of maps $:\Omega \to \Omega$ such that for each $t,s\geq 0$, $\theta _{t}\circ \theta _{s}=\theta _{t+s}$ and $X_{t}\circ \theta _{s}=X_{t+s}.$

(vi) $({\mathcal {G}}_{t})$ is augmented and right continuous.

(vii) (right-continuity) For every $x\in E$, every $\alpha >0$, and every $\alpha$-excessive (with respect to $(P_{t})$) function $f$, the map $t\mapsto f(X_{t})$ is almost surely right continuous under $\mathbb {P} ^{x}$.

(viii) (quasi-left continuity) For every $x\in E$, if $(T_{n})$ is an increasing sequence of stopping times with limit $T$, then $\mathbb {P} ^{x}(\lim _{n\to \infty }X_{T_{n}}=X_{T}|T<\infty )=1$.

Sharpe shows in Lemma 2.6 that conditions (i)-(v) imply measurability of the map $x\mapsto \mathbb {P} ^{x}(X_{t}\in B)$ for all $B\in {\mathcal {E}}$, and in Theorem 7.4 that (vi)-(vii) imply the strong Markov property with respect to $({\mathcal {G}}_{t})$.

Connection to other Markov processes

The following inclusions hold among various classes of Markov process:

{Lévy} $\subset$ {Itô} $\subset$ {Feller} $\subset$ {Hunt} $\subset$ {special standard} $\subset$ {standard} $\subset$ {right} $\subset$ {strong Markov}

See also

• Markov process
• Borel right process
• Gilbert Hunt

Sources

• Blumenthal, Robert M. and Getoor, Ronald K. "Markov Processes and Potential Theory". Academic Press, New York, 1968.
• Chung, Kai Lai and Walsh, John B. "Markov Processes, Brownian Motion, and Time Symmetry". Springer. 2005. ISBN 0-387-22026-7.
• Çinlar, Erhan. "Probability and Stochastics ". Springer, New York. 2011. ISBN 978-0-387-87858-4
• Getoor, Ronald K. "Markov Processes: Ray Processes and Knight Processes". Springer Berlin, Heidelberg. 1975. ISBN 978-3-540-07140-2.
• Hunt, G. A. "Markoff Processes and Potentials. I, II, III." , Illinois J. Math. 1 (1957) 44–93; 1 (1957), 313–369; 2 (1958), 151–213.