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On constructing a multidimensional diffusion process with a membrane located on a given hyperplane and acting in an oblique direction

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Using the methods of the theory of classical potentials, we have constructed a Feller semigroup of linear operators that generates a multidimensional diffusion process whose diffusion matrix is given by a sufficiently regular function and whose drift vector is given by a generalized function of the type of a δ - function concentrated on a given hyperplane. Such process can serve as a mathematical model for describing the motion of a diffusing particle in a medium where a membrane is located on the hyperplane. The particle is receiving "a pulse of infinite intensity" at those instants of time when it is hitting the hyperplane. The direction of those pulses are determined by a vector field given on that hyperplane. It is important to emphasize that the trajectories of the process constructed are continuous.
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Bibliografia
  • [1] Portenko N.I., Generalized Diffusion Processes, Kyiv, 1982; English transl., Amer. Math. Soc., Providence, RI, 1990.
  • [2] Portenko M.I., Diffusion Processes in Media with Membranes, Institute of Mathematics, of Ukrainian National Acad. of Sci., Kyiv 1995 (in Ukrainian).
  • [3] Baderko E.A., Solution of a problem with oblique derivative for parabolic equation by method of boundary integral equations, Differential Equations 1971, 25, 1, 14-20 (in Russian).
  • [4] Kopytko B.I., Portenko M.I., On a Multidimensional Brownian Motion with a Membrane Located on a Hyperplane and Acting in an Oblique Direction, Probability Theory and Mathematical Statistics, Proceedings, Institute of Mathematics of Ukrainian National Acad. of Sci., Kyiv 2002, 73-84.
  • [5] Pogorzelski W., Étude de la solution fondamental de l’equation parabolique, Richerche di Mat. 1956, 5, 25-27.
  • [6] Pogorzelski W., Study of integrals of parabolic equation and boundary problems in an unbounded domain, Math. Coll. 1959, 47, 4(89), 397-430 (in Russian).
  • [7] Ladyzhenskaya O.A., Solonnikov V.A., Ural’tseva N.N., Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow 1967 (in Russian).
  • [8] Friedman A., Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Clifs, N.J. 1964.
  • [9] Kopytko B.I., Construction of the diffusion process with generalized drift vector by means of solution some conjugation problem for the second-order parabolic type equation, Random Oper. Stoch. Eqs., 1984, 2, 1, 33-38.
  • [10] Pagni M., Su un problema al contorno tipico per l’equazione del calore, Ann. Sc. Norm. Super. Di Pisa 1957, (3), 2, 1-2, 73-115.
  • [11] Anulova S.V., Diffusion processes: discontinuous coefficients, degenerate diffusion, Randomized Drift, DAN USSR 1981, 260, 5, 1036-1040 (in Russian).
  • [12] Wentzel A.D., On boundary conditions for multidimensional diffusion processes, Theor. Prob. and Appl. 1959, 2, 5, 172-185 (in Russian).
  • [13] Kamynin L.I., Maslennikova V.N., On maximum principle for parabolic equation with discontinuous coefficients, Siberian Math. Journal 1961, 2, 3, 384-399 (in Russian).
  • [14] Zaitseva L.L., On a probabilistic approach to the construction of the generalized diffusion processes, Theory of Stochastic Processes 2000, 6(22), 1-2, 141-146.
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bwmeta1.element.baztech-article-BPC6-0024-0005
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