## The Annals of Probability

### Stochastic flows for nonlinear second-order parabolic SPDE

Franco Flandoli

#### Abstract

The existence of stochastic flows in $L^2$ -spaces is proved for a stochastic reaction ] diffusion equation of second order in a bounded domain.

#### Article information

Source
Ann. Probab., Volume 24, Number 2 (1996), 547-558.

Dates
First available in Project Euclid: 11 December 2002

https://projecteuclid.org/euclid.aop/1039639354

Digital Object Identifier
doi:10.1214/aop/1039639354

Mathematical Reviews number (MathSciNet)
MR1404520

Zentralblatt MATH identifier
0870.60056

#### Citation

Flandoli, Franco. Stochastic flows for nonlinear second-order parabolic SPDE. Ann. Probab. 24 (1996), no. 2, 547--558. doi:10.1214/aop/1039639354. https://projecteuclid.org/euclid.aop/1039639354

#### References

• 1 BRZEZNIAK, Z., CAPINSKI, M. and FLANDOLI, F. 1992. Stochastic Navier]Stokes equations with multiplicative noise. Stochastic Anal. Appl. 10 523]532. w x.
• 2 BRZEZNIAK, Z. and FLANDOLI, F. 1992. Regularity of solutions and random evolution operator for stochastic parabolic equations. In Stochastic Partial Differential Equa. tions and Applications. G. Da Prato and L. Tubaro, eds. 54]71. Pitman, Harlow. w x. 3 DA PRATO, G. and ZABCZy K, J. 1992. Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press. w x.
• 4 FLANDOLI, F. 1991. Stochastic flows and Ly apunov exponents for abstract stochastic PDEs of parabolic ty pe. Ly apunov Exponents. Lecture Notes in Math. 1486 196]205. Springer, New York. w x.
• 5 FLANDOLI, F. 1995. Regularity Theory and Stochastic Flows for Parabolic SPDE's. Gordon and Breach, Amsterdam. w x.
• 6 FLANDOLI, F. and SCHAUMLFFEL, K.-U. 1990. Stochastic parabolic equations in bounded domains: random evolution operators and Ly apunov exponents. Stochastics Stochastic Rep. 29 461]485. w x.
• 7 IKEDA, N. and WATANABE, S. 1981. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam. w x.
• 8 KRy LOV, N. V. and ROZOVSKII, B. L. 1981. Stochastic evolution equations. J. Soviet Math. 16 1233]1277. w x.
• 9 KUNITA, H. 1990. Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press. w x.
• 10 PARDOUX, E. 1975. Equations aux derivee partielles stochastiques non lineaires mono´ ´ tones. Thesis, Dept. Mathematics, Univ. Paris XI. w x.
• 11 ROZOVSKI, B. L. and SHIMIZU, A. 1981. Smoothness of solutions of stochastic evolution equations and the existence of a filtering transition density. Nagoy a Math. J. 84 195]208. w x.
• 12 SCHAUMLFFEL, K.-U. and FLANDOLI, F. 1991. A multiplicative ergodic theorem with applications to a first order stochastic hy perbolic equation in a bounded domain. Stochastics Stochastic Rep. 34 241]255. w x.
• 13 SKOROHOD, A. V. 1984. Random Linear Operators. Reidel, Dordrecht. w x.
• 14 TUBARO, L. 1988. Some results on stochastic partial differential equations by the stochastic characteristic method. Stochastic Anal. Appl. 6 217]230.