Hunt process

Inprobability theory,aHunt processis a type ofMarkov process,named for mathematicianGilbert A. Huntwho first defined them in 1957. Hunt processes were important in the study of probabilisticpotential theoryuntil they were superseded byright processesin the 1970s.

History

Background

In the 1930-50s the work of mathematicians such asJoseph Doob,William Feller,Mark Kac,andShizuo Kakutanideveloped connections between Markov processes andpotential theory.

In 1957-8Gilbert A. Huntpublished a triplet of paperswhich 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 Getoordescribed them as "a monumental work of nearly 170 pages that contained an enormous amount of truly original mathematics." Gustave Choquetwrote 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$X$satisfies hypothesis (A) if the following three assumptions hold:

First assumption:$X$is a Markov process on aPolish spacewithcàdlàgpaths.

Second assumption:$X$satisfies thestrong 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 process", a term that was introduced byEugene Dynkin.

Rise and fall

The book "Markov Processes and Potential Theory"(1968) byBlumenthaland 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 andGloverwrite: "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 ofright processesas 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 toDirichlet forms.

Definition

Brief definition

A Hunt process$X$is a strong Markov process on aPolish spacethat iscàdlàgand quasi-left continuous; that is, if$(T_{n})$is an increasing sequence ofstopping timeswith 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 aRadon spaceand${\mathcal {E}}$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 afilteredmeasurable space,and each$\mathbb {P} ^{x}$is aprobability measureon$(\Omega,{\mathcal {G}})$.

(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 every$x\in E$,and for all$t,s\geq 0,f\in b{\mathcal {E}}$,$\mathbb {E} ^{x}(f(X_{t+s})|{\mathcal {G}}_{t})=P_{s}f(X_{t})$.

(v)$(\theta _{t})_{t\geq 0}$is a collection of maps${\displaystyle:\Omega \to \Omega }$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)$({\mathcal {G}}_{t})$isaugmentedandright 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$\mathbb {P} ^{x}(\lim _{n\to \infty }X_{T_{n}}=X_{T}|T<\infty )=1$.

Sharpeshows 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}

Time-changed Itô processes

In 1980 Çinlar et al. proved that anyreal-valuedHunt process issemimartingaleif and only if it is a random time-change of an Itô process. More precisely, a Hunt process$X$on$\mathbb {R} ^{m}$(equipped with theBorel$\sigma$-algebra) is a semimartingale if and only if there is an Itô process$Y$and ameasurable function$f$with$0\leq f\leq 1$such that$X_{t}=Y_{A_{t}},t\geq 0$,where $$A_{t}=\int _{0}^{t}f(Y_{s})\mathrm {d} s.$$ Itô processes were first named due to their role in this theorem, thoughItôhad previously studied them.

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.
• 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.