## Sobolev differentiable stochastic flows for SDEs with singular coefficients: applications to the transport equation.(English)Zbl 1333.60127

The authors developed a new approach for constructing a Sobolev differentiable stochastic flow for a stochastic differential equation (SDE) of the form $dX_t=b(t, X_t) dt+dB_t, \quad s,t\in \mathbb R, \,\, X_s=x\in \mathbb R^d,$ where $$b:\mathbb R\times\mathbb R^d\to\mathbb R^d$$ is a bounded measurable coefficient and $$B$$ is a $$d$$-dimensional Brownian motion.
The approach is based on Malliavin calculus ideas coupled with new probabilistic estimates on the spatial weak derivatives of solutions of SDEs. A unique feature of these estimates is that they do not depend on the spatial regularity of the drift coefficient $$b$$. The existence of a Sobolev differentiable stochastic flow for an SDE is exploited to obtain a unique weak solution of the Stratonovich stochastic transport equation.

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60H07 Stochastic calculus of variations and the Malliavin calculus 37H10 Generation, random and stochastic difference and differential equations 37H05 General theory of random and stochastic dynamical systems 34A36 Discontinuous ordinary differential equations
Full Text:

### References:

 [1] Ambrosio, L. (2004). Transport equation and Cauchy problem for $$BV$$ vector fields. Invent. Math. 158 227-260. · Zbl 1075.35087 [2] Attanasio, S. (2010). Stochastic flows of diffeomorphisms for one-dimensional SDE with discontinuous drift. Electron. Commun. Probab. 15 213-226. · Zbl 1226.60086 [3] Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion-Facts and Formulae , 2nd ed. Birkhäuser, Basel. · Zbl 1012.60003 [4] Davie, A. M. (2007). Uniqueness of solutions of stochastic differential equations. Int. Math. Res. Not. IMRN 24 Art. ID rnm124, 26. · Zbl 1139.60028 [5] Da Prato, G., Flandoli, F., Priola, E. and Röckner, M. (2013). Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift. Ann. Probab. 41 3306-3344. · Zbl 1291.35455 [6] Da Prato, G., Malliavin, P. and Nualart, D. (1992). Compact families of Wiener functionals. C. R. Acad. Sci. Paris Sér. I Math. 315 1287-1291. · Zbl 0782.60002 [7] DiPerna, R. J. and Lions, P.-L. (1989). Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 511-547. · Zbl 0696.34049 [8] Di Nunno, G., Øksendal, B. and Proske, F. (2009). Malliavin Calculus for Lévy Processes with Applications to Finance . Springer, Berlin. [9] Fedrizzi, E. (2009). Uniqueness and flow theorems for solutions of SDE’s with low regularity of the drift. Tesi di Laurea in Matematica, Univ. di Pisa. [10] Fedrizzi, E. and Flandoli, F. (2011). Pathwise uniqueness and continuous dependence of SDEs with non-regular drift. Stochastics 83 241-257. · Zbl 1221.60081 [11] Fedrizzi, E. and Flandoli, F. (2013). Noise prevents singularities in linear transport equations. J. Funct. Anal. 264 1329-1354. · Zbl 1273.60076 [12] Fedrizzi, E. and Flandoli, F. (2013). Hölder flow and differentiability for SDEs with nonregular drift. Stoch. Anal. Appl. 31 708-736. · Zbl 1281.60055 [13] Flandoli, F. (2011). Random Perturbation of PDEs and Fluid Dynamic Models. Lecture Notes in Math. 2015 . Springer, Heidelberg. · Zbl 1221.35004 [14] Flandoli, F., Gubinelli, M. and Priola, E. (2010). Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180 1-53. · Zbl 1200.35226 [15] Hajłasz, P. (1993). Change of variables formula under minimal assumptions. Colloq. Math. 64 93-101. · Zbl 0840.26009 [16] Heinonen, J., Kilpeläinen, T. and Martio, O. (1993). Nonlinear Potential Theory of Degenerate Elliptic Equations . Oxford Univ. Press, New York. · Zbl 0780.31001 [17] Krylov, N. V. and Röckner, M. (2005). Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields 131 154-196. · Zbl 1072.60050 [18] Kufner, A. (1985). Weighted Sobolev Spaces . Wiley, New York. · Zbl 0567.46009 [19] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations . Cambridge Univ. Press, Cambridge. · Zbl 0743.60052 [20] Malliavin, P. (1997). Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften 313 . Springer, Berlin. · Zbl 0878.60001 [21] Menoukeu-Pamen, O., Meyer-Brandis, T., Nilssen, T., Proske, F. and Zhang, T. (2013). A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s. Math. Ann. 357 761-799. · Zbl 1282.60057 [22] Meyer-Brandis, T. and Proske, F. (2010). Construction of strong solutions of SDE’s via Malliavin calculus. J. Funct. Anal. 258 3922-3953. · Zbl 1195.60082 [23] Mohammed, S.-E. A. and Scheutzow, M. K. R. (1998). Spatial estimates for stochastic flows in Euclidean space. Ann. Probab. 26 56-77. · Zbl 0937.60056 [24] Mohammed, S.-E. A. and Scheutzow, M. K. R. (2003). The stable manifold theorem for non-linear stochastic systems with memory. I. Existence of the semiflow. J. Funct. Anal. 205 271-305. · Zbl 1039.60060 [25] Mohammed, S.-E. A. and Scheutzow, M. K. R. (2004). The stable manifold theorem for non-linear stochastic systems with memory. II. The local stable manifold theorem. J. Funct. Anal. 206 253-306. · Zbl 1053.60061 [26] Nualart, D. (1995). The Malliavin Calculus and Related Topics . Springer, New York. · Zbl 0837.60050 [27] Portenko, N. I. (1990). Generalized Diffusion Processes. Translations of Mathematical Monographs 83 . Amer. Math. Soc., Providence, RI. · Zbl 0727.60088 [28] Reshetnyak, Y. G. (1966). Some geometrical properties of functions and mappings with generalized derivatives. Sibirsk. Mat. Zh. 7 886-919. [29] Reshetnyak, Y. G. (1987). On the condition $$N$$ for mappings of class $$W_{n,\mathrm{loc}}^{1}$$. Sibirsk. Mat. Zh. 28 149-153. · Zbl 0634.30023 [30] Veretennikov, A. Y. (1979). On the strong solutions of stochastic differential equations. Theory Probab. Appl. 24 354-366. · Zbl 0434.60064 [31] Ziemer, W. P. (1989). Weakly Differentiable Functions . Springer, New York. · Zbl 0692.46022 [32] Zvonkin, A. K. (1974). A transformation of the phase space of a diffusion process that will remove the drift. Mat. Sb. 93 129-149, 152. · Zbl 0306.60049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.