The Annals of Probability

Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation

Salah-Eldin A. Mohammed, Torstein K. Nilssen, and Frank N. Proske

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In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms

\[\mathbb{R}^{d}\ni x\quad\longmapsto\quad\phi_{s,t}(x)\in\mathbb{R}^{d},\qquad s,t\in\mathbb{R}\] for a stochastic differential equation (SDE) of the form

\[dX_{t}=b(t,X_{t})\,dt+dB_{t},\qquad s,t\in\mathbb{R},X_{s}=x\in\mathbb{R}^{d}.\] The above SDE is driven by a bounded measurable drift coefficient $b:\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ and a $d$-dimensional Brownian motion $B$. More specifically, we show that the stochastic flow $\phi_{s,t}(\cdot)$ of the SDE lives in the space $L^{2}(\Omega;W^{1,p}(\mathbb{R}^{d},w))$ for all $s,t$ and all $p\in(1,\infty)$, where $W^{1,p}(\mathbb{R}^{d},w)$ denotes a weighted Sobolev space with weight $w$ possessing a $p$th moment with respect to Lebesgue measure on $\mathbb{R}^{d}$. From the viewpoint of stochastic (and deterministic) dynamical systems, this is a striking result, since the dominant “culture” in these dynamical systems is that the flow “inherits” its spatial regularity from that of the driving vector fields.

The spatial regularity of the stochastic flow yields existence and uniqueness of a Sobolev differentiable weak solution of the (Stratonovich) stochastic transport equation

\[\cases{d_{t}u(t,x)+(b(t,x)\cdot Du(t,x))\,dt+\sum_{i=1}^{d}e_{i}\cdot Du(t,x)\circ dB_{t}^{i}=0,\cr u(0,x)=u_{0}(x),}\] where $b$ is bounded and measurable, $u_{0}$ is $C_{b}^{1}$ and $\{e_{i}\}_{i=1}^{d}$ a basis for $\mathbb{R}^{d}$. It is well known that the deterministic counterpart of the above equation does not in general have a solution.

Article information

Ann. Probab., Volume 43, Number 3 (2015), 1535-1576.

First available in Project Euclid: 5 May 2015

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Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H15: Stochastic partial differential equations [See also 35R60] 37H05: Foundations, general theory of cocycles, algebraic ergodic theory [See also 37Axx] 37H10: Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15] 34A36: Discontinuous equations

SDEs with measurable coefficients stochastic flows Malliavin derivatives Sobolev spaces stochastic transport equation


Mohammed, Salah-Eldin A.; Nilssen, Torstein K.; Proske, Frank N. Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation. Ann. Probab. 43 (2015), no. 3, 1535--1576. doi:10.1214/14-AOP909.

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  • [1] Ambrosio, L. (2004). Transport equation and Cauchy problem for $BV$ vector fields. Invent. Math. 158 227–260.
  • [2] Attanasio, S. (2010). Stochastic flows of diffeomorphisms for one-dimensional SDE with discontinuous drift. Electron. Commun. Probab. 15 213–226.
  • [3] Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd ed. Birkhäuser, Basel.
  • [4] Davie, A. M. (2007). Uniqueness of solutions of stochastic differential equations. Int. Math. Res. Not. IMRN 24 Art. ID rnm124, 26.
  • [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.
  • [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.
  • [7] DiPerna, R. J. and Lions, P.-L. (1989). Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 511–547.
  • [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.
  • [11] Fedrizzi, E. and Flandoli, F. (2013). Noise prevents singularities in linear transport equations. J. Funct. Anal. 264 1329–1354.
  • [12] Fedrizzi, E. and Flandoli, F. (2013). Hölder flow and differentiability for SDEs with nonregular drift. Stoch. Anal. Appl. 31 708–736.
  • [13] Flandoli, F. (2011). Random Perturbation of PDEs and Fluid Dynamic Models. Lecture Notes in Math. 2015. Springer, Heidelberg.
  • [14] Flandoli, F., Gubinelli, M. and Priola, E. (2010). Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180 1–53.
  • [15] Hajłasz, P. (1993). Change of variables formula under minimal assumptions. Colloq. Math. 64 93–101.
  • [16] Heinonen, J., Kilpeläinen, T. and Martio, O. (1993). Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Univ. Press, New York.
  • [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.
  • [18] Kufner, A. (1985). Weighted Sobolev Spaces. Wiley, New York.
  • [19] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press, Cambridge.
  • [20] Malliavin, P. (1997). Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften 313. Springer, Berlin.
  • [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.
  • [22] Meyer-Brandis, T. and Proske, F. (2010). Construction of strong solutions of SDE’s via Malliavin calculus. J. Funct. Anal. 258 3922–3953.
  • [23] Mohammed, S.-E. A. and Scheutzow, M. K. R. (1998). Spatial estimates for stochastic flows in Euclidean space. Ann. Probab. 26 56–77.
  • [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.
  • [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.
  • [26] Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.
  • [27] Portenko, N. I. (1990). Generalized Diffusion Processes. Translations of Mathematical Monographs 83. Amer. Math. Soc., Providence, RI.
  • [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.
  • [30] Veretennikov, A. Y. (1979). On the strong solutions of stochastic differential equations. Theory Probab. Appl. 24 354–366.
  • [31] Ziemer, W. P. (1989). Weakly Differentiable Functions. Springer, New York.
  • [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.