Time Reversions of Markov Processes

Nagasawa, Masao

doi:10.1017/s0027763000011405

Cited by 104 publications

(70 citation statements)

References 12 publications

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“…According to the theory of Nagasawa [18] That is, if Lx is the time of the last exit from (-00,a] before Sx, then F{A(LX) = X(LX -),0 < Lx < Sx) = 0. A localization argument similar to that of Theorem 3.1 effects the replacement of the Sx by the fixed times t, settling the issue under the assumption that A" has no continuous upward passages.…”

mentioning

confidence: 99%

Zero-One Laws and the Minimum of a Markov Process

Millar¹

1977

Transactions of the American Mathematical Society

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Abstract. If {Xrt > 0} is a real strong Markov process whose paths assume a (last) minimum at some time M strictly before the lifetime, then conditional on /, the value of this minimum, the process [X(M + t),t> 0} is shown to be Markov with stationary transitions which depend on /. For a wide class of Markov processes, including those obtained from Levy processes via time change and multiplicative functional, a zen -one law is shown to hold at M in the sense that C\>0o[X(M + s),s < i) -» o{X(M)}, modulo null sets. When such a law holds, the evolution of (X(M + t),t > 0) depends on events before M only through X(M) and /.

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mentioning

confidence: 99%

Zero-One Laws and the Minimum of a Markov Process

Millar¹

1977

Transactions of the American Mathematical Society

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“…Moreover, a time reversal relationship exists between the solutions of (4.18) and those of (2.4) stopped at y akin to the one between the 3-dimensional Bessel process and the killed Brownian motion established by Williams [26]. This relationship will be proved by a time reversal result of Nagasawa [20]. The following version of this result is taken from Sharpe [25].…”

Section: Bessel-type Motions and Stepmentioning

confidence: 80%

“…Indeed, if y = 0 and X is a Brownian motion killed at 0, the SDE that we obtain in Step 2 is the SDE associated to the 3-dimensional Bessel process. In Section 4 we give the SDE characterisations of these Bessel-type motions -a term adapted from McKean in his study of excursions of diffusions [18] -and prove a time reversal result akin to those that can be found in the seminal paper of Williams [26] using a theorem due to Nagasawa [20].…”

Section: Introductionmentioning

confidence: 90%

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Path transformations for local times of one-dimensional diffusions

Çetın

2018

Stochastic Processes and their Applications

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Abstract. Let X be a regular one-dimensional transient diffusion and L y be its local time at y. The stochastic differential equation (SDE) whose solution corresponds to the process X conditioned on [L y ∞ = a] for a given a ≥ 0 is constructed and a new path decomposition result for transient diffusions is given. In the course of the construction Bessel-type motions as well as their SDE representations are studied. Moreover, the Engelbert-Schmidt theory for the weak solutions of one dimensional SDEs is extended to the case when the initial condition is an entrance boundary for the diffusion. This extension was necessary for the construction of the Bessel-type motion which played an essential part in the SDE representation of X conditioned on [L y ∞ = a].

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“…The following lemma is essentially due to H. Tanaka, and is a generalization of the Sinai-Volkoviskii's result on the JT-property of the ideal gas model Proof. First, we will show 6) Such a random time T(W) is called £-time which was introduced by M. Nagasawa [7]. …”

Section: Remark 28 If the Equilibrium Process Associated With \J{t mentioning

confidence: 99%

Ergodic Properties of the Equilibrium Process Associated with Infinitely Many Markovian Particles

Shiga¹,

Takahashi

1973

Publ. Res. Inst. Math. Sci.

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Consider a system of independent identically distributed Markov processes which have an invariant measure A. It is known that if each process starts from each point of a ^-Poisson point process at time zero,, these particles are /l-Poisson distributed at every later time £>0 Q].In the present paper we are concerned with the ergodic properties of the stationary processes obtained from such a system of particles, which is called the equilibrium process. Sinai's ideal gas model is a special example of the equilibrium processes Q4]. In §1 we will give some preliminaries and the definition of the equilibrium process, and §2 is devoted to the study of the ergodic properties (metrical transitivity, mixing properties and pure nondeterminism) of the equilibrium processes. In §3 we will discuss the Bernoulli property in the strong sense of the shift flow {® t}-<*> ^) be a ff-finite measure space, and denote by Jf(JT) a

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Time Reversions of Markov Processes

Cited by 104 publications

References 12 publications

Zero-One Laws and the Minimum of a Markov Process

Zero-One Laws and the Minimum of a Markov Process

Path transformations for local times of one-dimensional diffusions

Ergodic Properties of the Equilibrium Process Associated with Infinitely Many Markovian Particles

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