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Stochastic flows of diffeomorphisms for one-dimensional SDE with discontinuous drift
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Stefano Attanasio, Scuola Normale Superiore
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Abstract
The existence of a stochastic flow of class C1,α, for α< 1/2, for a 1-dimensional SDE will be proved under mild conditions on the regularity of the drift. The diffusion coefficient is assumed constant for simplicity, while the drift is an autonomous BV function with distributional derivative bounded from above or from below. To reach this result the continuity of the local time with respect to the initial datum will also be proved
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Full text: PDF
Pages: 213-226
Published on: June 9, 2010
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Bibliography
- Ambrosio, Luigi. Transport equation and Cauchy problem for $BV$ vector fields. Invent. Math. 158 (2004), no. 2, 227--260. MR2096794 (2005f:35127)
- Attanasio, Stefano; Flandoli, Franco. Zero-noise solutions of linear transport equations without uniqueness: an example. C. R. Math. Acad. Sci. Paris 347 (2009), no. 13-14, 753--756. MR2543977 (Review)
- Flandoli, F.; Gubinelli, M.; Priola, E. Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180 (2010), no. 1, 1--53. MR2593276
- Gradinaru, Mihai; Herrmann, Samuel; Roynette, Bernard. A singular large deviations phenomenon. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), no. 5, 555--580. MR1851715 (2002j:60042)
- Hu, Yueyun; Warren, Jon. Ray-Knight theorems related to a stochastic flow. Stochastic Process. Appl. 86 (2000), no. 2, 287--305. MR1741809 (2001a:60091)
- Ouknine, Youssef; Rutkowski, Marek. Strong comparison of solutions of one-dimensional stochastic differential equations. Stochastic Process. Appl. 36 (1990), no. 2, 217--230. MR1084976 (92b:60052)
- Rajeev, B. From Tanaka's formula to Itô's formula: the fundamental theorem of stochastic calculus. Proc. Indian Acad. Sci. Math. Sci. 107 (1997), no. 3, 319--327. MR1467435 (98d:60104)
- Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357 (2000h:60050)
- Zvonkin, A. K. A transformation of the phase space of a diffusion process that will remove the drift. (Russian) Mat. Sb. (N.S.) 93(135) (1974), 129--149, 152. MR0336813 (49 #1586)
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Electronic Communications in Probability. ISSN: 1083-589X |
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