The transport equation and zero quadratic variation processes

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November 2016, 21(9): 2991-3002. doi: 10.3934/dcdsb.2016083

The transport equation and zero quadratic variation processes

Jorge Clarke ^1, , Christian Olivera ^2, and Ciprian Tudor ^3,

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

Received October 2015 Revised January 2016 Published October 2016

Abstract
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We analyze the transport equation driven by a zero quadratic variation process. Using the stochastic calculus via regularization and the Malliavin calculus techniques, we prove the existence, uniqueness and absolute continuity of the law of the solution. As an example, we discuss the case when the noise is a Hermite process.

Keywords: Malliavin calculus, method of characteristics, stochastic calculus via regularization, fractional Brownian motion, existence of density., Hermite process, Transport equation, zero quadratic variation process.

Mathematics Subject Classification: Primary: 60H15; Secondary: 60H05, 60H0.

Citation: Jorge Clarke, Christian Olivera, Ciprian Tudor. The transport equation and zero quadratic variation processes. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 2991-3002. doi: 10.3934/dcdsb.2016083

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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