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Stochastic flows associated with Stratonovich curve-line integrals. (English) Zbl 1351.60079
Summary: In this paper, Stratonovich curve-line integrals are used to describe the evolution of a stochastic flow driven by some noncommuting vector fields and independent double Wiener processes. In fact, we analyze the corresponding stochastic evolution of a stochastic flow driven by noncommuting vector fields \({\{g_{1},\dots,g_{m}\}}\) and independent double Wiener processes \[ \{ W^{i}(t)=(W_{1}^{i}(t_{1}),W_{2}^{i}(t_{2}))\in\mathbb{R}^{2}:t=(t_{1},t_{2})\in D\},\; 1\leq i\leq m. \] It is a significant generalization of the case \({m=1}\), considered in a joint work of V. Damian and C. Vârsan [Math. Rep., Buchar. 14(64), No. 4, 325–332 (2012; Zbl 1289.60114)]. This paper contains two open problems; a good start for a future research.
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
Full Text: DOI
[1] Buckdahn R. and Ma J., Stochastic viscosity solutions for nonlinear stochastic partial differential equations, Stochastic Process. Appl. 93 (2001), 181-204. · Zbl 1053.60065
[2] Damian V. and Vârsan C., Stochastic integral equations associated with Stratonovich curve-line integral, Math. Rep. (Bucur.) 14 (2012), 325-331. · Zbl 1289.60114
[3] Iftimie B., Marinescu M. and Vârsan C., Functional associated with gradient stochastic flows and nonlinear SPDEs, Advanced Mathematical Methods for Finance, Springer, Berlin (2011), 397-416. · Zbl 1233.60038
[4] Iftimie B. and Vârsan C., Evolution system of Cauchy-Kowaleska and parabolic type with stochastic perturbation, Math. Rep. (Bucur.) 10 (2008), 213-238. · Zbl 1174.60035
[5] Morozan T., Periodic solutions of affine stochastic differential equations, Stoch. Anal. Appl. 1 (1986), 87-110. · Zbl 0583.60055
[6] Da Prato G. and Tubaro L., Stochastic Partial Differential Equations and Applications, Marcel Dekker, New York, 2002.
[7] Vârsan C., Applications of Lie Algebras to Hyperbolic and Stochastic Differential Equations, Kluwer Academic Publishers, Amsterdam, 1999. · Zbl 0948.35002
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