Probabilistic models of parabolic conservation and balance laws and systems with switching regimes

Ya. I. Belopolskaya

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 24, Tome 454 (2016), pp. 5-42

Résumé

We construct a probabilistic representation of the Cauchy problem classical solution for systems of semilinear parabolic equations such that their second order terms enter in a diagonal way with different coefficients while lower order term enter in nondiagonal way. Systems of this kind arise as mathematical models of parabolic conservation and balance laws and as mathematical models of dynamical systems with switching regimes.

Bibliographie

[1] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, 1967 | MR

[2] L. C. Evans, “A strong maximum principle for parabolic systems in a convex set”, Proc. AMS, 138:9 (2010), 3179–3185 | DOI | MR | Zbl

[3] M. Freidlin, Functional Integration and Partial Differential Equations, Princeton Univ. Press, 1985 | MR | Zbl

[4] D. Becherer, M. Schweizer, “Classical solutions to reaction-diffusion systems for hedging problems with interacting Ito and point processes”, Ann. Appl. Probab., 15:2 (2005), 1111–1144 | DOI | MR | Zbl

[5] S. I. Boyarchenko, S. Z. Levendorskii, “Pricing American options in regime-switching models”, SIAM J. Control Optim., 48 (2009), 1353–1376 | DOI | MR | Zbl

[6] Ya. I. Belopolskaya, Yu. L. Daletskii, “Issledovanie zadachi Koshi dlya kvazilineinykh parabolicheskikh sistem pri pomoschi markovskikh sluchainykh protsessov”, Izv. vuzov. Matem., 1978, no. 12, 6–17 | MR | Zbl

[7] Ya. Belopolskaya, Yu. Dalecky, Stochastic Equations and Differential Geometry, Kluwer Academic Publishers, 1990 | MR | Zbl

[8] Yu. Tyutyunov, I. Senina, R. Arditi, “Clustering due to acceleration in the response to population gradient: a simple self-organization model”, Amer. Naturalist, 164:6 (2004), 722–735

[9] R. Arditi, J.-M. Callois, Yu. Tyutyunov, C. Jost, “Does mutual interference always stabilize predator-prey dynamics? A comparison of models”, Comptes Rendus Biologies, 327 (2004), 1037–1057 | DOI

[10] E. Pardoux, “Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order”, Stochast. Anal. Relat. Topics, The Geilo Workshop, Birkhäuser, 1996, 79–127 | MR

[11] E. Pardoux, F. Pradeilles, Z. Rao, “Probabilistic interpretation of a system of semi-linear parabolic partial differential equations.”, Ann. Inst. Henri Poincaré (B) Probab. Statist., 33:4 (1997), 467–490 | DOI | MR | Zbl

[12] Ya. I. Belopolskaya, “Pryamye-obratnye stokhasticheskie uravneniya, svyazannye s sistemami kvazilineinykh parabolicheskikh uravnenii i teoremy sravneniya”, Zap. nauchn. semin. POMI, 412, 2013, 15–46 | MR

[13] Ya. Belopolskaya, “Probabilistic counterparts of nonlinear parabolic PDE systems Modern Stochastics and Applications”, Springer Optimization and Its Applications, 90 (2014), 71–94 | DOI | MR | Zbl

[14] Ya. Belopolskaya, “Probabilistic approaches to nonlinear parabolic equations in jet-bundles”, Global Stochast. Anal., 1 (2011), 3–40 | Zbl

[15] Ya. Belopolskaya, W. Woyczynski, “Generalized solution of the Cauchy problem for systems of nonlinear parabolic equations and diffusion processes”, Stochast. Dynamics, 11:1 (2012), 1–31 | MR

[16] J. Jacod, A. N. Shiryayev, Limit Theorems for Stochastic Processes, Springer-Verlag, New York, 1980 | MR

[17] A. V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Amer. Math. Soc., Providence, RI, 1989 | MR

[18] G. Yin, C. Zhu, “Properties of solutions of stochastic differential equations with continuous-state-dependent switching”, J. Diff. Eq., 249 (2010), 2409–2439 | DOI | MR | Zbl

[19] M.-F. Chen, From Markov Chains to Non-equilibrium Particle Systems, World Scientific, Singapore, 2004 | MR | Zbl