Geodesic
On a certain stochastic quasilinear hyperbolic equation
Sbornik. Mathematics, Tome 44 (1983) no. 3, pp. 363-388 Cet article a éte moissonné depuis la source Math-Net.Ru

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The author considers the first boundary value problem for the equation $$ \frac{\partial^2u(t,x)}{\partial t^2}+k\,\frac{\partial u}{\partial t}-\Delta u+|u|^\rho u=\frac{\partial w(t,x)}{\partial t},\qquad t>0, \quad x\in\mathscr O\Subset\mathbf R^n, $$ where $k\geqslant0$, $\rho>0$, and $w(t)$ is a Wiener process in the space $L^2(\mathscr O)$. The initial values are assumed random and independent of the process $w(t)$. The existence of a space-time statistical solution is proved and (under a certain restriction on $\rho$) the existence of a strong solution. A steady state space-time statistical solution is constructed for $k>0$. Bibliography: 12 titles. 
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     author = {D. A. Khrychev},
     title = {On a certain stochastic quasilinear hyperbolic equation},
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D. A. Khrychev. On a certain stochastic quasilinear hyperbolic equation. Sbornik. Mathematics, Tome 44 (1983) no. 3, pp. 363-388. http://geodesic.mathdoc.fr/item/SM_1983_44_3_a7/

[1] Vishik M. I., Komech A. I., Fursikov L. B., “Nekotorye matematicheskie zadachi statisticheskoi gidromekhaniki”, UMN, 34:5 (1979), 135–210 | MR | Zbl

[2] Viot M., Solutions faibles d'équations aux dérivées partielles stochastiques non linéaires, Thése, Paris, 1976

[3] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[4] Girya T. V., “Razreshimost pryamogo uravneniya Kolmogorova dlya nelineinogo giperbolicheskogo uravneniya s belym shumom”, Vestnik MGU, ser. matem., 1980, no. 1, 65–70 | MR

[5] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971

[6] Khasminskii R. 3., Ustoichivost sistem differentsialnykh uravnenii pri sluchainykh vozmuscheniyakh ikh parametrov, Nauka, M., 1969 | MR

[7] Getoor R. K., Sharpe M. J., “Conformal martingales”, Invent. Math., 16:4 (1972), 271–308 | DOI | MR | Zbl

[8] Vishik M. I., Komech A. I., “O razreshimosti zadachi Koshi dlya uravneniya Khopfa, sootvetstvuyuschego nelineinomu giperbolicheskomu uravneniyu”, Tr. seminara im. I. G. Petrovskogo, vyp. 3, MGU, M., 1978, 19–42 | MR

[9] Vishik M. I., Fursikov A. V., Matematicheskie zadachi statisticheskoi gidromekhaniki, Nauka, M., 1980 | MR

[10] Billingsli P., Skhodimost veroyatnostnykh mer, Nauka, M., 1977 | MR

[11] Burbaki N., Integrirovanie. Mery na lokalno kompaktnykh prostranstvakh. Prodolzhenie mery. Integrirovanie mer. Mery na otdelimykh prostranstvakh, Nauka, M., 1977

[12] Fleming W. H., Viot M., Some measure-valued Markov processes in population genetics theory, Preprint | MR