American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[2]

P. L. Chow, Stochastic Partial Differential Equations, $2^{nd}$ edition. Advances in Applied Mathematics. CRC Press, Boca Raton, FL, 2015.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

F. Flandoli and F. Russo, Generalized integration and stochastic ODEs, Annals of Probability, 30 (2002), 270-292. doi: 10.1214/aop/1020107768.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

I. Nourdin, Selected Aspects of Fractional Brownian Motion, Springer, Milan; Bocconi University Press, Milan, 2012. doi: 10.1007/978-88-470-2823-4.  Google Scholar

[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.  Google Scholar

[17]

D. Nualart, Malliavin Calculus and Related Topics, $2^{nd}$ edition, Springer New York, 2006.  Google Scholar

[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.  Google Scholar

[19]

B. Oksendal, Stochastic Differential Equations, Springer-Verlag, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

B. Perthame, Transport Equations in Biology, Series Frontiers in Mathematics, Birkhauser, 2007.  Google Scholar

[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.  Google Scholar

[24]

F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration, Probab. Theory Rel. Fileds, 97 (1993), 403-421. doi: 10.1007/BF01195073.  Google Scholar

[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.  Google Scholar

[26]

M. Sanz-Solé, Malliavin Calculus. With Applications to Stochastic Partial Differential Equations, Fundamental Sciences, EPFL Press, Lausanne, 2005.  Google Scholar

[27]

I. Shigekawa, Derivatives of Wiener functionals and absolute continuity of induced measures, J. Math. Kyoto Univ., 20 (1980), 263-289.  Google Scholar

[28]

C. A. Tudor, Analysis of Variations for Self-similar Processes, Springer 2013. doi: 10.1007/978-3-319-00936-0.  Google Scholar

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.  Google Scholar

[2]

P. L. Chow, Stochastic Partial Differential Equations, $2^{nd}$ edition. Advances in Applied Mathematics. CRC Press, Boca Raton, FL, 2015.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

F. Flandoli and F. Russo, Generalized integration and stochastic ODEs, Annals of Probability, 30 (2002), 270-292. doi: 10.1214/aop/1020107768.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

I. Nourdin, Selected Aspects of Fractional Brownian Motion, Springer, Milan; Bocconi University Press, Milan, 2012. doi: 10.1007/978-88-470-2823-4.  Google Scholar

[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.  Google Scholar

[17]

D. Nualart, Malliavin Calculus and Related Topics, $2^{nd}$ edition, Springer New York, 2006.  Google Scholar

[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.  Google Scholar

[19]

B. Oksendal, Stochastic Differential Equations, Springer-Verlag, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

B. Perthame, Transport Equations in Biology, Series Frontiers in Mathematics, Birkhauser, 2007.  Google Scholar

[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.  Google Scholar

[24]

F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration, Probab. Theory Rel. Fileds, 97 (1993), 403-421. doi: 10.1007/BF01195073.  Google Scholar

[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.  Google Scholar

[26]

M. Sanz-Solé, Malliavin Calculus. With Applications to Stochastic Partial Differential Equations, Fundamental Sciences, EPFL Press, Lausanne, 2005.  Google Scholar

[27]

I. Shigekawa, Derivatives of Wiener functionals and absolute continuity of induced measures, J. Math. Kyoto Univ., 20 (1980), 263-289.  Google Scholar

[28]

C. A. Tudor, Analysis of Variations for Self-similar Processes, Springer 2013. doi: 10.1007/978-3-319-00936-0.  Google Scholar

[1]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[2]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255
[3]

Caroline Hillairet, Ying Jiao, Anthony Réveillac. Pricing formulae for derivatives in insurance using Malliavin calculus. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 7-. doi: 10.1186/s41546-018-0028-9
[4]

S. Kanagawa, K. Inoue, A. Arimoto, Y. Saisho. Mean square approximation of multi dimensional reflecting fractional Brownian motion via penalty method. Conference Publications, 2005, 2005 (Special) : 463-475. doi: 10.3934/proc.2005.2005.463
[5]

Elisa Gorla, Maike Massierer. Index calculus in the trace zero variety. Advances in Mathematics of Communications, 2015, 9 (4) : 515-539. doi: 10.3934/amc.2015.9.515
[6]

Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199
[7]

Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197
[8]

Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257
[9]

Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281
[10]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005
[11]

Xiaojie Wang. Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 481-497. doi: 10.3934/dcds.2016.36.481
[12]

Hongjun Gao, Fei Liang. On the stochastic beam equation driven by a Non-Gaussian Lévy process. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1027-1045. doi: 10.3934/dcdsb.2014.19.1027
[13]

Paulina Grzegorek, Michal Kupsa. Exponential return times in a zero-entropy process. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1339-1361. doi: 10.3934/cpaa.2012.11.1339
[14]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003
[15]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473
[16]

Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483
[17]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[18]

Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084
[19]

Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015
[20]

Thierry Paul, David Sauzin. Normalization in Banach scale Lie algebras via mould calculus and applications. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4461-4487. doi: 10.3934/dcds.2017191

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (86)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences