TL;DRAbstract
Up to now we have treated mainly Markov processes with continuous paths. To apply our stochastic theory to relativistic quantum particles we need Markov processes of pure-jumps. This chapter is devoted to the mathematics we need in analyzing the movement of relativistic quantum particles. We begin with simple examples of pure-jump processes, and discuss the stochastic integrals by point processes. Then the main theorems on Lévy processes will be briefly reviewed. Bochner’s subordination of semi-groups and subordinate Markov processes will be explained. We will then prove the duality (namely, time reversal) of the subordinate Markov processes. The duality will play a crucial role in discussing the stochastic theory of relativistic quantum particles in an electromagnetic field, which will be treated in Chapters VII.
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Up to now we have treated mainly Markov processes with continuous paths. To apply our stochastic theory to relativistic quantum particles we need Markov processes of pure-jumps. This chapter is devoted to the mathematics we need in analyzing the movement of relativistic quantum particles. We begin with simple examples of pure-jump processes, and discuss the stochastic integrals by point processes. Then the main theorems on Lévy processes will be briefly reviewed. Bochner’s subordination of semi-groups and subordinate Markov processes will be explained. We will then prove the duality (namely, time reversal) of the subordinate Markov processes. The duality will play a crucial role in discussing the stochastic theory of relativistic quantum particles in an electromagnetic field, which will be treated in Chapters VII.
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