Abstract
Although Markov’s property is generally given with a preferential time orientation—that from the past to the future—its statement is actually symmetric in both directions, and it could be intuitively expressed by saying that the events of the future and those of the past result mutually independent conditionally to the knowledge of the information available at present. To emphasize this symmetry we will start by giving the following definition of the Markov property, briefly postponing a proof of its equivalence with the other, more familiar formulations.
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Notes
- 1.
With \(0\le s\le t\le t+\Delta t\), the r-vec’s
$$\begin{aligned} \varvec{X}(0)\qquad \quad \varvec{X}(s)-\varvec{X}(0)\qquad \quad \varvec{X}(t)-\varvec{X}(s)\qquad \quad \Delta \varvec{X}(t)=\varvec{X}(t+\Delta t)-\varvec{X}(t) \end{aligned}$$are all independent by definition, and hence \(\Delta \varvec{X}(t)\) and \(\varvec{X}(s)=[\varvec{X}(s)-\varvec{X}(0)]+\varvec{X}(0)\) are independent too.
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Cufaro Petroni, N. (2020). Markovianity. In: Probability and Stochastic Processes for Physicists. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-48408-8_7
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