Home Lipschitz stability for a semi-linear inverse stochastic transport problem
Article
Licensed
Unlicensed Requires Authentication

Lipschitz stability for a semi-linear inverse stochastic transport problem

  • Zhongqi Yin EMAIL logo
Published/Copyright: October 16, 2019

Abstract

This paper is addressed to a semi-linear stochastic transport equation with three unknown parameters. It is proved that the initial displacement, the terminal state and the random term in diffusion are uniquely determined by the state on partial boundary and a Lipschitz stability of the inverse problem is established. The main tool we employ is a global Carleman estimate for stochastic transport equations.

Award Identifier / Grant number: 11401404

Funding statement: This work is partially supported by the NSF of China under Grant 11401404.

References

[1] G. Bal, Inverse problems for homogeneous transport equations. I. The one-dimensional case, Inverse Problems 16 (2000), no. 4, 997–1011. 10.1088/0266-5611/16/4/308Search in Google Scholar

[2] G. Bal, Inverse problems for homogeneous transport equations. II. The multidimensional case, Inverse Problems 16 (2000), no. 4, 1013–1028. 10.1088/0266-5611/16/4/309Search in Google Scholar

[3] G. Bal and A. Tamasan, Inverse source problems in transport equations, SIAM J. Math. Anal. 39 (2007), no. 1, 57–76. 10.1137/050647177Search in Google Scholar

[4] G. Bao, S.-N. Chow, P. Li and H. Zhou, Numerical solution of an inverse medium scattering problem with a stochastic source, Inverse Problems 26 (2010), no. 7, Article ID 074014. 10.1088/0266-5611/26/7/074014Search in Google Scholar

[5] L. Cavalier and A. Tsybakov, Sharp adaptation for inverse problems with random noise, Probab. Theory Related Fields 123 (2002), no. 3, 323–354. 10.1007/s004400100169Search in Google Scholar

[6] P. L. Chow and L. Maestrello, Stochastic inverse problem in the radiation of noise, SIAM J. Appl. Math. 35 (1978), no. 4, 665–677. 10.1137/0135055Search in Google Scholar

[7] D. Crisan, Y. Otobe and S. Peszat, Inverse problems for stochastic transport equations, Inverse Problems 31 (2015), no. 1, Article ID 015005. 10.1088/0266-5611/31/1/015005Search in Google Scholar

[8] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl. 44, Cambridge University, Cambridge, 1992. 10.1017/CBO9780511666223Search in Google Scholar

[9] T. Deck and J. Potthoff, On a class of stochastic partial differential equations related to turbulent transport, Probab. Theory Related Fields 111 (1998), no. 1, 101–122. 10.1007/s004400050163Search in Google Scholar

[10] H. Gjessing, Wick calculus with applications to anticipating stochastic differential equations, Manuscript, University of Bergen, 1994. Search in Google Scholar

[11] M. V. Klibanov and S. E. Pamyatnykh, Global uniqueness for a coefficient inverse problem for the non-stationary transport equation via Carleman estimate, J. Math. Anal. Appl. 343 (2008), no. 1, 352–365. 10.21236/ADA448486Search in Google Scholar

[12] L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics. Vol. 10: Physical Kinetics, Pergamon Press, New York, 1981. Search in Google Scholar

[13] X. Liu, Global Carleman estimate for stochastic parabolic equations, and its application, ESAIM Control Optim. Calc. Var. 20 (2014), no. 3, 823–839. 10.1051/cocv/2013085Search in Google Scholar

[14] Q. Lü, Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems, Inverse Problems 28 (2012), no. 4, Article ID 045008. 10.1088/0266-5611/28/4/045008Search in Google Scholar

[15] Q. Lü, Observability estimate and state observation problems for stochastic hyperbolic equations, Inverse Problems 29 (2013), no. 9, Article ID 095011. 10.1088/0266-5611/29/9/095011Search in Google Scholar

[16] Q. Lü, Observability estimate for stochastic Schrödinger equations and its applications, SIAM J. Control Optim. 51 (2013), no. 1, 121–144. 10.1137/110830964Search in Google Scholar

[17] Q. Lü, Exact controllability for stochastic transport equations, SIAM J. Control Optim. 52 (2014), no. 1, 397–419. 10.1007/978-3-030-82331-3_8Search in Google Scholar

[18] Q. Lü, Stochastic well-posed systems and well-posedness of some stochastic partial differential equations with boundary control and observation, SIAM J. Control Optim. 53 (2015), no. 6, 3457–3482. 10.1137/151002605Search in Google Scholar

[19] Q. Lü and X. Zhang, Global uniqueness for an inverse stochastic hyperbolic problem with three unknowns, Comm. Pure Appl. Math. 68 (2015), no. 6, 948–963. 10.1002/cpa.21503Search in Google Scholar

[20] G. Yuan, Determination of two kinds of sources simultaneously for a stochastic wave equation, Inverse Problems 31 (2015), no. 8, Article ID 085003. 10.1088/0266-5611/31/8/085003Search in Google Scholar

[21] X. Zhang, Carleman and observability estimates for stochastic wave equations, SIAM J. Math. Anal. 40 (2008), no. 2, 851–868. 10.1137/070685786Search in Google Scholar

Received: 2018-12-16
Revised: 2019-05-10
Accepted: 2019-09-29
Published Online: 2019-10-16
Published in Print: 2020-04-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 18.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jiip-2018-0115/html
Scroll to top button