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Conditional Markov chains - construction and properties

Tomasz R. Bielecki, Jacek Jakubowski, Mariusz Niewęgłowski (2015)

Banach Center Publications

In this paper we study finite state conditional Markov chains (CMCs). We give two examples of CMCs, one which admits intensity, and another one, which does not admit an intensity. We also give a sufficient condition under which a doubly stochastic Markov chain is a CMC. In addition we provide a method for construction of conditional Markov chains via change of measure.

Convex hulls, Sticky particle dynamics and Pressure-less gas system

Octave Moutsinga (2008)

Annales mathématiques Blaise Pascal

We introduce a new condition which extends the definition of sticky particle dynamics to the case of discontinuous initial velocities u 0 with negative jumps. We show the existence of a stochastic process and a forward flow φ satisfying X s + t = φ ( X s , t , P s , u s ) and d X t = E [ u 0 ( X 0 ) / X t ] d t , where P s = P X s - 1 is the law of X s and u s ( x ) = E [ u 0 ( X 0 ) / X s = x ] is the velocity of particle x at time s ≥ 0 . Results on the flow characterization and Lipschitz continuity are also given.Moreover, the map ( x , t ) ↦ M ( x , t ) : = P ( X t ≤ x ) is the entropy solution of a scalar conservation law ∂ t M + ∂ x ( A ( M ) ) = 0 where the flux A represents the particles...

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