Abstract
We construct a multidimensional generalized diffusion process with the drift coefficient that is the (generalized) derivative of a vector-valued measure satisfying an analog of the Hölder condition with respect to volume. We prove the existence and continuity of the density of transition probability of this process and obtain standard estimates for this density. We also prove that the trajectories of the process are solutions of a stochastic differential equation.
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REFERENCES
N. I. Portenko, Generalized Diffusion Processes [in Russian], Naukova Dumka, Kiev (1982).
M. I. Portenko, Diffusion in Media with Semitransparent Membranes [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1994).
M. I. Portenko, Diffusion Processes in Media with Membranes [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1995).
G. M. Shevchenko, “On a generalized diffusion process with a drift that is the generalized derivative of a singular function,” in: Probability Theory and Mathematical Statistics. Proceedings of the Ukrainian Mathematical Congress, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (2002), pp. 139-148.
E. B. Dynkin, Markov Processes [in Russian], Fizmatlit, Moscow (1963).
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Shevchenko, H.M. On Multidimensional Generalized Diffusion Processes. Ukrainian Mathematical Journal 55, 1560–1566 (2003). https://doi.org/10.1023/B:UKMA.0000018017.13829.ef
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DOI: https://doi.org/10.1023/B:UKMA.0000018017.13829.ef