We establish a unique continuation property for stochastic heat equations evolving in a domain $G\subset {R}^{n}$($n\in N$). 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.

DOI: 10.1051/cocv/2014027

Keywords: Stochastic heat equations, unique continuation property, backward stochastic heat equations, approximate controllability, null controllability

^{1}; Yin, Zhongqi

^{2}

@article{COCV_2015__21_2_378_0, author = {L\"u, Qi and Yin, Zhongqi}, title = {Unique continuation for stochastic heat equations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {378--398}, publisher = {EDP-Sciences}, volume = {21}, number = {2}, year = {2015}, doi = {10.1051/cocv/2014027}, mrnumber = {3348404}, zbl = {1316.60108}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2014027/} }

TY - JOUR AU - Lü, Qi AU - Yin, Zhongqi TI - Unique continuation for stochastic heat equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 378 EP - 398 VL - 21 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014027/ DO - 10.1051/cocv/2014027 LA - en ID - COCV_2015__21_2_378_0 ER -

%0 Journal Article %A Lü, Qi %A Yin, Zhongqi %T Unique continuation for stochastic heat equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 378-398 %V 21 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2014027/ %R 10.1051/cocv/2014027 %G en %F COCV_2015__21_2_378_0

Lü, Qi; Yin, Zhongqi. Unique continuation for stochastic heat equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 2, pp. 378-398. doi : 10.1051/cocv/2014027. http://archive.numdam.org/articles/10.1051/cocv/2014027/

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