Stochastic Models for Nonlinear Cross-Diffusion Systems

  • Yana Belopolskaya
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 231)


Under a priori assumptions concerning existence and uniqueness of the Cauchy problem solution for a system of quasilinear parabolic equations with cross-diffusion, we treat the PDE system as an analogue of systems of forward Kolmogorov equations for some unknown stochastic processes and derive expressions for their generators. This allows to construct a stochastic representation of the required solution. We prove that introducing stochastic test function we can check that the stochastic system gives rise to the required generalized solution of the original PDE system. Next, we derive a closed stochastic system which can be treated as a stochastic counterpart of the Cauchy problem for a parabolic system with cross-diffusion.


Stochastic flow Cross-diffusion PDE generalized solution Probabilistic representation 



The financial support of the RFBR Grant 15-01-01453 is gratefully acknowledged.


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Authors and Affiliations

  1. 1.Saint-Petersburg State University of Architecture and Civil EngineeringSt. PetersburgRussian Federation

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