We prove that a family of linear bounded evolution operators ${\left(\mathbf{G}(t,s)\right)}_{t\ge s\in I}$ can be associated, in the space of vector-valued bounded and continuous functions, to a class of systems of elliptic operators $\mathcal{A}$ with unbounded coefficients defined in $I\times {\mathbb{R}}^{d}$ (where $I$ is a right-halfline or $I=\mathbb{R}$) all having the same principal part. We establish some continuity and representation properties of ${\left(\mathbf{G}(t,s)\right)}_{t\ge s\in I}$ and a sufficient condition for the evolution operator to be compact in ${C}_{b}({\mathbb{R}}^{d};{\mathbb{R}}^{m})$. We prove also a uniform weighted gradient estimate and some of its more relevant consequence.
Keywords: Nonautonomous parabolic systems, unbounded coefficients, evolution operators, compactness, gradient estimates, semilinear systems, stochastic games
@article{COCV_2017__23_3_937_0, author = {Addona, Davide and Angiuli, Luciana and Lorenzi, Luca and Tessitore, Gianmario}, title = {On coupled systems of {Kolmogorov} equations with applications to stochastic differential games}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {937--976}, publisher = {EDP-Sciences}, volume = {23}, number = {3}, year = {2017}, doi = {10.1051/cocv/2016019}, zbl = {1371.35144}, mrnumber = {3660455}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016019/} }
TY - JOUR AU - Addona, Davide AU - Angiuli, Luciana AU - Lorenzi, Luca AU - Tessitore, Gianmario TI - On coupled systems of Kolmogorov equations with applications to stochastic differential games JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 937 EP - 976 VL - 23 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016019/ DO - 10.1051/cocv/2016019 LA - en ID - COCV_2017__23_3_937_0 ER -
%0 Journal Article %A Addona, Davide %A Angiuli, Luciana %A Lorenzi, Luca %A Tessitore, Gianmario %T On coupled systems of Kolmogorov equations with applications to stochastic differential games %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 937-976 %V 23 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016019/ %R 10.1051/cocv/2016019 %G en %F COCV_2017__23_3_937_0
Addona, Davide; Angiuli, Luciana; Lorenzi, Luca; Tessitore, Gianmario. On coupled systems of Kolmogorov equations with applications to stochastic differential games. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 3, pp. 937-976. doi : 10.1051/cocv/2016019. http://archive.numdam.org/articles/10.1051/cocv/2016019/
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