# Hunt Process

Chapter

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## Abstract

Let

*{X*_{ t }*, š*_{ t }*t ā*T} be a (homogeneous) Markov process with state space*(E*_{ ā }*,Īµ*_{ Ć“ }*)*and transition function*(P*_{ š }*)*, as specified in Ā§1.1 and Ā§1.2. Here*š*_{ t }is the*š*_{ t }defined in Ā§2.3. Such a process is called a*Hunt process*iff- (i)
it is right continuous;

Ā - (ii)
it has the strong Markov property (embodied in Theorem 1 of Ā§2.3);

Ā - (iii)
it is quasi left continuous (as described in Theorem 4 of Ā§2.4).

Ā

## Keywords

Brownian MotionĀ Compact SubsetĀ Excessive FunctionĀ Optional TimeĀ Resolvent EquationĀ
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## Notes On Chapter 3

- Ā§3.1.Hunt process is named after G. A. Hunt, who was the authorās classmate in Princeton. The basic assumptions stated here correspond roughly with Hypothesis
*(A)*, see Hunt [2]. The Borelian character of the semigroup is not always assumed. A more general process, called āstandard processā, is treated in Dynkin [1] and Blumenthal and Getoor [1]. Another technical variety called āRay processā is treated in Getoor [1].Google Scholar - Ā§3.2.The definition of an excessive function and its basic properties are due to Hunt [2]. Another definition via the resolvents given in (14) is used in Blumenthal and Getoor [1]. Though the former definition is more restrictive than the latter, it yields more transparent results. On the other hand, there are results valid only with the resolvents.Google Scholar
- Ā§3.3.We follow Huntās exposition in Hunt [3]. Although the theory of analytical sets is essential here, we follow Hunt in citing Choquetās theorem without details. A proof of the theorem may be found in Helms [1]. But for the probabilistic applications (projection and section) we need an algebraic version of Choquetās theory without endowing the sample space
*Ī©*with a topology. This is given in Meyer [1] and Dellacherie and Meyer [1].Google Scholar - Ā§3.4 and 3.5.These sections form the core of Huntās theory. Theorem 6 is due to Doob [3], who gave the proof for a āsubparabolicā function. Its extension to an excessive function is carried over by Hunt using his Theorem 5. The transfinite induction used there may be concealed by some kind of maximal argument, see Hunt [3] or Blumenthal and Getoor [1]. But the quickest proof of Theorem 6 is by means of a projection onto the optional field, see Meyer [2]. This is a good place to learn the new techniques alluded to in the Notes on Ā§1.4.Google Scholar
- Dellacherieās deep result (Theorem 8) is one of the cornerstones of what may be called ārandom set theoryā. Its proof is based on a random version of the Cantor-Bendixson theorem in classical set theory, see Meyer [2]. Oddly enough, the theorem is not included in Dellacherie [2], but must be read in Dellacherie [1], see also Dellacherie [3].Google Scholar
- Ā§3.6. Theorem 1 originated with H. Cartan for the Newtonian potential (Brownian motion in
*R*^{3}); see e.g., Helms [1]. This and several other important results for Hunt processes have companions which are valid for left continuous moderate Markov processes, see Chung and Glover [1]. Since a general Hunt process reversed in time has these properties as mentioned in theGoogle Scholar - Notes on Ā§2.4, such developments may be interesting in the future. āLast exit timeā is hereby rechristened āquitting timeā to rhyme with āhitting timeā. Although it is the obvious concept to be used in defining recurrence and transience, it made a belated entry in the current vocabulary of the theory. No doubt this is partly due to the former prejudice against a random time that is not optional, despite its demonstrated power in older theories such as random walks and Markov chains. See Chung [2] where a whole section is devoted to last exits. The name has now been generalized to āco-optionalā and āco-terminalā; see Meyer-Smythe-Walsh [1]. Intuitively the quitting time of a set becomes its hitting time when the sense of time is reversed (āfrom infinityā, unfortunately). But this facile interpretation is only a heuristic guide and not easy to make rigorous. See Ā§5.1 for a vital application.Google Scholar
- Ā§3.7.For a somewhat more general discussion of the concepts of transience and recurrence, see Getoor [2]. The Hunt theory is mainly concerned with the transient case, having its origin in the Newtonian potential. Technically, transience can always be engineered by considering (Math) with Ī± > 0, instead of
*(P*_{t}This amounts to killing the original process at an exponential rate*e-*^{at}, independently of its evolution. It can be shown that the resulting killed process is also a Hunt process.Google Scholar - Ā§3.8.Hunt introduced his Hypothesis
*(B)*in order to characterize the balayage (hitting) operator*P*_{A}in a way recognizable in modern potential theory, see Hunt [1 ; Ā§3.6]. He stated that he had not found āsimple and general conditionsā to ensure its truth. Meyer [3] showed that it is implied by the duality assumptions and noted its importance in the dual theory. It may be regarded as a subtle generalization of the continuity of the paths. Indeed AzĆ©ma [1] and Smythe and Walsh [1] showed that it is equivalent to the quasi left continuity of a suitably reversed process. We need this hypothesis in Ā§5.1 to extend a fundamental result in potential theory from the continuous case to the general case. A new condition for its truth will also be stated there.Google Scholar

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Ā©Ā Springer Science+Business Media New YorkĀ 1982