Unique continuation for stochastic heat equations∗
Revised: 24 April 2014
We establish a unique continuation property for stochastic heat equations evolving in a domain G ⊂ Rn(n ∈N). Our result shows that the value of the solution can be determined uniquely by means of its value on an arbitrary open subdomain of G at any given positive time constant. Further, when G is convex and bounded, we also give a quantitative version of the unique continuation property. As applications, we get an observability estimate for stochastic heat equations, an approximate result and a null controllability result for a backward stochastic heat equation.
Mathematics Subject Classification: 60H15 / 93B05
Key words: Stochastic heat equations / unique continuation property / backward stochastic heat equations / approximate controllability / null controllability
This work is partially ERC advanced grant 266907 (CPDENL) of the 7th Research Framework Programme (FP7), the NSF of China under grant 11101070, the project MTM2011-29306 of the Spanish Science and Innovation Ministry, the Fundamental Research Funds for the Central Universities in China under grants ZYGX2012J115 and the General Fund Project of Sichuan Provincial Department of Education in China under the grant 124632.
© EDP Sciences, SMAI 2014