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March  2015, 5(1): 165-176. doi: 10.3934/mcrf.2015.5.165

A quantitative internal unique continuation for stochastic parabolic equations

Zhongqi Yin 1,

1. 

School of Mathematics, Sichuan Normal University, Chengdu, 610068, China

Received  January 2014 Revised  August 2014 Published  January 2015

This paper is addressed to a quantitative internal unique continuation property for stochastic parabolic equations, i.e., we show that each of their solutions can be determined by the observation on any nonempty open subset of the whole region in which the equations evolve. The proof is based on a global Carleman estimate.
Keywords: global Carleman estimate., internal unique continuation property, Stochastic parabolic equations.
Mathematics Subject Classification: Primary: 60H15; Secondary: 34A1.
Citation: Zhongqi Yin. A quantitative internal unique continuation for stochastic parabolic equations. Mathematical Control & Related Fields, 2015, 5 (1) : 165-176. doi: 10.3934/mcrf.2015.5.165
References:
[1]

L. Escauriaza, Carleman inequalities and the heat operator, Duke Math. J., 104 (2000), 113-127. doi: 10.1215/S0012-7094-00-10415-2.  Google Scholar

[2]

L. Escauriaza and L. Vega, Carleman inequalities and the heat operator II, Indiana U. Math. J., 50 (2001), 1149-1169. doi: 10.1512/iumj.2001.50.1937.  Google Scholar

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V. Isakov, Carleman type estimates in an anisotropic case and applications, J. Differential Equations, 105 (1993), 217-238. doi: 10.1006/jdeq.1993.1088.  Google Scholar

[4]

H. Li and Q. Lü, A quantitative boundary unique continuation for stochastic parabolic equations, J. Math. Anal. Appl., 402 (2013), 518-526. doi: 10.1016/j.jmaa.2013.01.038.  Google Scholar

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F. H. Lin, A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math., 43 (1990), 127-136. doi: 10.1002/cpa.3160430105.  Google Scholar

[6]

Q. Lü, Observability estimate for stochastic Schrödinger equations and its applications, SIAM J. Control Optim., 51 (2013), 121-144. doi: 10.1137/110830964.  Google Scholar

[7]

Q. Lü, Observability estimate and state observation problems for stochastic hyperbolic equations, Inverse Problems, 29 (2013), 095011, 22 pp. doi: 10.1088/0266-5611/29/9/095011.  Google Scholar

[8]

C. C. Poon, Unique continuation for parabolic equations, Comm. Partial Differential Equations, 21 (1996), 521-539. doi: 10.1080/03605309608821195.  Google Scholar

[9]

J.-C. Saut and B. Scheurer, Unique continuation for soome evolution equations, J. Differential Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X.  Google Scholar

[10]

C. D. Sogge, A unique continuation theorem for second order parabolic differential operators, Ark. Mat., 28 (1990), 159-182. doi: 10.1007/BF02387373.  Google Scholar

[11]

S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations, SIAM J. Control Optim., 48 (2009), 2191-2216. doi: 10.1137/050641508.  Google Scholar

[12]

H. Yamabe, A unique continuation theorem of a diffusion equation, Ann. Math., 69 (1959), 462-466. doi: 10.2307/1970194.  Google Scholar

[13]

X. Zhang, Unique continuation for stochastic parabolic equations, Differential Integral Equations, 21 (2008), 81-93.  Google Scholar

[14]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, in Handbook of Differential Equations: Evolutionary Differential Equations. Vol. III, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007, 527-621. doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

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C. Zuily, Uniqueness and Non-Uniqueness in the Cauchy Problem, Progress in Mathematics, 33, Birkhäuser, Boston, 1983. doi: 10.1007/978-1-4899-6656-8.  Google Scholar

show all references

References:
[1]

L. Escauriaza, Carleman inequalities and the heat operator, Duke Math. J., 104 (2000), 113-127. doi: 10.1215/S0012-7094-00-10415-2.  Google Scholar

[2]

L. Escauriaza and L. Vega, Carleman inequalities and the heat operator II, Indiana U. Math. J., 50 (2001), 1149-1169. doi: 10.1512/iumj.2001.50.1937.  Google Scholar

[3]

V. Isakov, Carleman type estimates in an anisotropic case and applications, J. Differential Equations, 105 (1993), 217-238. doi: 10.1006/jdeq.1993.1088.  Google Scholar

[4]

H. Li and Q. Lü, A quantitative boundary unique continuation for stochastic parabolic equations, J. Math. Anal. Appl., 402 (2013), 518-526. doi: 10.1016/j.jmaa.2013.01.038.  Google Scholar

[5]

F. H. Lin, A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math., 43 (1990), 127-136. doi: 10.1002/cpa.3160430105.  Google Scholar

[6]

Q. Lü, Observability estimate for stochastic Schrödinger equations and its applications, SIAM J. Control Optim., 51 (2013), 121-144. doi: 10.1137/110830964.  Google Scholar

[7]

Q. Lü, Observability estimate and state observation problems for stochastic hyperbolic equations, Inverse Problems, 29 (2013), 095011, 22 pp. doi: 10.1088/0266-5611/29/9/095011.  Google Scholar

[8]

C. C. Poon, Unique continuation for parabolic equations, Comm. Partial Differential Equations, 21 (1996), 521-539. doi: 10.1080/03605309608821195.  Google Scholar

[9]

J.-C. Saut and B. Scheurer, Unique continuation for soome evolution equations, J. Differential Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X.  Google Scholar

[10]

C. D. Sogge, A unique continuation theorem for second order parabolic differential operators, Ark. Mat., 28 (1990), 159-182. doi: 10.1007/BF02387373.  Google Scholar

[11]

S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations, SIAM J. Control Optim., 48 (2009), 2191-2216. doi: 10.1137/050641508.  Google Scholar

[12]

H. Yamabe, A unique continuation theorem of a diffusion equation, Ann. Math., 69 (1959), 462-466. doi: 10.2307/1970194.  Google Scholar

[13]

X. Zhang, Unique continuation for stochastic parabolic equations, Differential Integral Equations, 21 (2008), 81-93.  Google Scholar

[14]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, in Handbook of Differential Equations: Evolutionary Differential Equations. Vol. III, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007, 527-621. doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

[15]

C. Zuily, Uniqueness and Non-Uniqueness in the Cauchy Problem, Progress in Mathematics, 33, Birkhäuser, Boston, 1983. doi: 10.1007/978-1-4899-6656-8.  Google Scholar

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