By J. Fabera
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29 30 L. BOGACHEV, G. DERFEL, S. MOLCHANOV, AND J. OCKENDON in a mathematical model for the dynamics of an overhead current collection system on an electric locomotive (with the physically relevant value q < 1) . At about the same time, a systematic analysis of solutions to the pantograph equation was started by Fox et at. , where various analytical, perturbation, and numerical techniques were discussed at length (for both q < 1 and q > 1). 1) (with q > 1) was derived more than 25 years earlier by Ambartsumian  to describe the absorption of light by the interstellar matter.
As a consequence, the terms in the random seri es will infinit ely ofte n exceed the value 1, all other t erms being nonnegati ve. s. 3) it follows that lim inf X U n ~ n - ex> In t urn, this implies that To < as claimed. s. , and so fo(x ) = 1 for all x > 0, 5 . Jump diffusions. We now pur sue a more general (and more systematic) approach. 2) ar e linked in a natural way wit h certain continuous-time Markov pr ocesses (more speci fically, diffusions with multiplication j umps) . To describe this class of pr ocesses , let us consider a Brownian mot ion B~ ' v , starting at the origin, with diffusion coefficient", 2': and nonpositi ve (constant) drift - v ~ 0, ° or equivalently t 2': 0, where B t = B; 'o is a st andard Br owni an mot ion (wit h continuo us sample paths).
Probability Theory and Related Fields, 96: 177-204. , Multivariate point processes: predictable projection, Radon-Nikodyrn derivatives, representation martingales. Z. Wahrsch. Verw. Gebiete (1975) , 31 : 235-53.  JACOD J. , Limit Theorems for Stochastic Processes . SpringerVerlag , New York, Heidelberg, Berlin (1987 ). S. N. (1989) , Theory of Martingales. Kluwer , Dordrecht (Russian edition 1986) .  PUHALSKlI A. (2001), Large Deviations and Idempotent Probability, Chapman & Hall /CRC Press .
Equadiff IV by J. Fabera