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The transport equation and zero quadratic variation processes
1. | Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Valparaíso, Chile |
2. | Departamento de Matemática, Universidade Estadual de Campinas, 13.081-970-Campinas-SP, Brazil |
3. | Laboratoire Paul Painlevé, Université de Lille 1, F-59655 Villeneuve d'Ascq, France |
References:
[1] |
P. Catuogno and C. Oliveira, $L ^p$ solutions of the stochastic transport equation, Random Operators and Stochastic Equations, 21 (2013), 125-134.
doi: 10.1515/rose-2013-0007. |
[2] |
P. L. Chow, Stochastic Partial Differential Equations, $2^{nd}$ edition. Advances in Applied Mathematics. CRC Press, Boca Raton, FL, 2015. |
[3] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, $3^{rd}$ edition, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 325. Springer-Verlag, 2010.
doi: 10.1007/978-3-642-04048-1. |
[4] |
J. Duan, H. Gao and B. Schmalfuss, Stochastic dynamics of a coupled atmosphere-ocean model, Stochastics and Dynamics, 2 (2002), 357-380.
doi: 10.1142/S0219493702000467. |
[5] |
F. Fedrizzi and F. Flandoli., Noise prevents singularities in linear transport equations, Journal of Functional Analysis, 264 (2013), 1329-1354.
doi: 10.1016/j.jfa.2013.01.003. |
[6] |
M. Ferrante and C. Rovira, Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter $H>\frac{1}{2}$, Bernoulli, 12 (2006), 85-100. |
[7] |
F. Flandoli, M. Gubinelli and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (2010), 1-53.
doi: 10.1007/s00222-009-0224-4. |
[8] |
F. Flandoli and F. Russo, Generalized integration and stochastic ODEs, Annals of Probability, 30 (2002), 270-292.
doi: 10.1214/aop/1020107768. |
[9] |
H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, in Ecole d'été de probabilités de Saint-Flour XII - 1982, Lecture Notes in Mathematics, 1097 (1984), 143-303, Springer, Berlin.
doi: 10.1007/BFb0099433. |
[10] |
H. Kunita, First order stochastic partial differential equations, in Proceedings of the Taniguchi International Symposium on Stochastic Analysis, North-Holland Mathematical Library, 32 (1984), 249-269.
doi: 10.1016/S0924-6509(08)70396-9. |
[11] |
H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990. |
[12] |
P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. I: Incompressible Models, Oxford Lecture Series in Mathematics and its applications, 3, (1996), Oxford University Press. |
[13] |
P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. II: Compressible Models, Oxford Lecture Series in Mathematics and its applications, 10 (1998), Oxford University Press. |
[14] |
M. Maurelli, Wiener chaos and uniqueness for stochastic transport equation, Comptes Rendus Mathematique, 349 (2011), 669-672.
doi: 10.1016/j.crma.2011.05.006. |
[15] |
I. Nourdin, Selected Aspects of Fractional Brownian Motion, Springer, Milan; Bocconi University Press, Milan, 2012.
doi: 10.1007/978-88-470-2823-4. |
[16] |
I. Nourdin and F. Viens, Density formula and concentration inequalities with Malliavin calculus, Electronic Journal of Probability, 14 (2009), 2287-2309.
doi: 10.1214/EJP.v14-707. |
[17] |
D. Nualart, Malliavin Calculus and Related Topics, $2^{nd}$ edition, Springer New York, 2006. |
[18] |
D. Nualart and L. Quer-Sardanyons, Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations, Stoch. Process. Appl., 119 (2009), 3914-3938.
doi: 10.1016/j.spa.2009.09.001. |
[19] |
B. Oksendal, Stochastic Differential Equations, Springer-Verlag, 2003.
doi: 10.1007/978-3-642-14394-6. |
[20] |
C. Olivera, Well-posedness of first order semilinear PDE's by stochastic perturbation, Nonlinear Anal., 96 (2014), 211-215.
doi: 10.1016/j.na.2013.10.022. |
[21] |
C. Olivera and C. A. Tudor, The density of the solution to the transport equation with fractional noise, Journal of Mathematical Analysis and Applications, 431 (2015), 57-72.
doi: 10.1016/j.jmaa.2015.05.030. |
[22] |
B. Perthame, Transport Equations in Biology, Series Frontiers in Mathematics, Birkhauser, 2007. |
[23] |
V. Pipiras and M. Taqqu, Integration questions related to the fractional Brownian motion, Probability Theory and Related Fields, 118 (2001), 251-291.
doi: 10.1007/s440-000-8016-7. |
[24] |
F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration, Probab. Theory Rel. Fileds, 97 (1993), 403-421.
doi: 10.1007/BF01195073. |
[25] |
F. Russo and P. Vallois, Elements of stochastic calculus via regularization, in Séminaire de Probabilités XL, Lecture Notes in Mathematics, 1899 (2007), 147-186.
doi: 10.1007/978-3-540-71189-6_7. |
[26] |
M. Sanz-Solé, Malliavin Calculus. With Applications to Stochastic Partial Differential Equations, Fundamental Sciences, EPFL Press, Lausanne, 2005. |
[27] |
I. Shigekawa, Derivatives of Wiener functionals and absolute continuity of induced measures, J. Math. Kyoto Univ., 20 (1980), 263-289. |
[28] |
C. A. Tudor, Analysis of Variations for Self-similar Processes, Springer 2013.
doi: 10.1007/978-3-319-00936-0. |
show all references
References:
[1] |
P. Catuogno and C. Oliveira, $L ^p$ solutions of the stochastic transport equation, Random Operators and Stochastic Equations, 21 (2013), 125-134.
doi: 10.1515/rose-2013-0007. |
[2] |
P. L. Chow, Stochastic Partial Differential Equations, $2^{nd}$ edition. Advances in Applied Mathematics. CRC Press, Boca Raton, FL, 2015. |
[3] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, $3^{rd}$ edition, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 325. Springer-Verlag, 2010.
doi: 10.1007/978-3-642-04048-1. |
[4] |
J. Duan, H. Gao and B. Schmalfuss, Stochastic dynamics of a coupled atmosphere-ocean model, Stochastics and Dynamics, 2 (2002), 357-380.
doi: 10.1142/S0219493702000467. |
[5] |
F. Fedrizzi and F. Flandoli., Noise prevents singularities in linear transport equations, Journal of Functional Analysis, 264 (2013), 1329-1354.
doi: 10.1016/j.jfa.2013.01.003. |
[6] |
M. Ferrante and C. Rovira, Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter $H>\frac{1}{2}$, Bernoulli, 12 (2006), 85-100. |
[7] |
F. Flandoli, M. Gubinelli and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (2010), 1-53.
doi: 10.1007/s00222-009-0224-4. |
[8] |
F. Flandoli and F. Russo, Generalized integration and stochastic ODEs, Annals of Probability, 30 (2002), 270-292.
doi: 10.1214/aop/1020107768. |
[9] |
H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, in Ecole d'été de probabilités de Saint-Flour XII - 1982, Lecture Notes in Mathematics, 1097 (1984), 143-303, Springer, Berlin.
doi: 10.1007/BFb0099433. |
[10] |
H. Kunita, First order stochastic partial differential equations, in Proceedings of the Taniguchi International Symposium on Stochastic Analysis, North-Holland Mathematical Library, 32 (1984), 249-269.
doi: 10.1016/S0924-6509(08)70396-9. |
[11] |
H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990. |
[12] |
P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. I: Incompressible Models, Oxford Lecture Series in Mathematics and its applications, 3, (1996), Oxford University Press. |
[13] |
P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. II: Compressible Models, Oxford Lecture Series in Mathematics and its applications, 10 (1998), Oxford University Press. |
[14] |
M. Maurelli, Wiener chaos and uniqueness for stochastic transport equation, Comptes Rendus Mathematique, 349 (2011), 669-672.
doi: 10.1016/j.crma.2011.05.006. |
[15] |
I. Nourdin, Selected Aspects of Fractional Brownian Motion, Springer, Milan; Bocconi University Press, Milan, 2012.
doi: 10.1007/978-88-470-2823-4. |
[16] |
I. Nourdin and F. Viens, Density formula and concentration inequalities with Malliavin calculus, Electronic Journal of Probability, 14 (2009), 2287-2309.
doi: 10.1214/EJP.v14-707. |
[17] |
D. Nualart, Malliavin Calculus and Related Topics, $2^{nd}$ edition, Springer New York, 2006. |
[18] |
D. Nualart and L. Quer-Sardanyons, Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations, Stoch. Process. Appl., 119 (2009), 3914-3938.
doi: 10.1016/j.spa.2009.09.001. |
[19] |
B. Oksendal, Stochastic Differential Equations, Springer-Verlag, 2003.
doi: 10.1007/978-3-642-14394-6. |
[20] |
C. Olivera, Well-posedness of first order semilinear PDE's by stochastic perturbation, Nonlinear Anal., 96 (2014), 211-215.
doi: 10.1016/j.na.2013.10.022. |
[21] |
C. Olivera and C. A. Tudor, The density of the solution to the transport equation with fractional noise, Journal of Mathematical Analysis and Applications, 431 (2015), 57-72.
doi: 10.1016/j.jmaa.2015.05.030. |
[22] |
B. Perthame, Transport Equations in Biology, Series Frontiers in Mathematics, Birkhauser, 2007. |
[23] |
V. Pipiras and M. Taqqu, Integration questions related to the fractional Brownian motion, Probability Theory and Related Fields, 118 (2001), 251-291.
doi: 10.1007/s440-000-8016-7. |
[24] |
F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration, Probab. Theory Rel. Fileds, 97 (1993), 403-421.
doi: 10.1007/BF01195073. |
[25] |
F. Russo and P. Vallois, Elements of stochastic calculus via regularization, in Séminaire de Probabilités XL, Lecture Notes in Mathematics, 1899 (2007), 147-186.
doi: 10.1007/978-3-540-71189-6_7. |
[26] |
M. Sanz-Solé, Malliavin Calculus. With Applications to Stochastic Partial Differential Equations, Fundamental Sciences, EPFL Press, Lausanne, 2005. |
[27] |
I. Shigekawa, Derivatives of Wiener functionals and absolute continuity of induced measures, J. Math. Kyoto Univ., 20 (1980), 263-289. |
[28] |
C. A. Tudor, Analysis of Variations for Self-similar Processes, Springer 2013.
doi: 10.1007/978-3-319-00936-0. |
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