\contentsline {section}{\numberline {1}Introduction}{2}{section.1}% 
\contentsline {section}{\numberline {2}Kinetic samplers}{4}{section.2}% 
\contentsline {subsection}{\numberline {2.1}General setting}{4}{subsection.2.1}% 
\contentsline {subsection}{\numberline {2.2}Velocity jumps}{6}{subsection.2.2}% 
\contentsline {subsection}{\numberline {2.3}Particular known cases}{8}{subsection.2.3}% 
\contentsline {section}{\numberline {3}The Gaussian case}{9}{section.3}% 
\contentsline {subsection}{\numberline {3.1}The process}{9}{subsection.3.1}% 
\contentsline {subsection}{\numberline {3.2}Convergence toward the Hamiltonian dynamics}{12}{subsection.3.2}% 
\contentsline {subsection}{\numberline {3.3}Drift limit and factorization}{14}{subsection.3.3}% 
\contentsline {subsubsection}{\numberline {3.3.1}Gibbs velocity jump processes}{15}{subsubsection.3.3.1}% 
\contentsline {subsubsection}{\numberline {3.3.2}Multi-time-stepping}{15}{subsubsection.3.3.2}% 
\contentsline {subsection}{\numberline {3.4}Non-irreducibility}{15}{subsection.3.4}% 
\contentsline {section}{\numberline {4}Hypocoercivity}{16}{section.4}% 
\contentsline {section}{\numberline {5}Simulation of velocity-jump processes}{22}{section.5}% 
\contentsline {subsection}{\numberline {5.1}General strategy}{22}{subsection.5.1}% 
\contentsline {subsection}{\numberline {5.2}Bounds on the corrected rate}{23}{subsection.5.2}% 
\contentsline {subsection}{\numberline {5.3}Sampling according to the corrected kernel}{24}{subsection.5.3}% 
\contentsline {subsubsection}{\numberline {5.3.1}General strategy}{24}{subsubsection.5.3.1}% 
\contentsline {subsubsection}{\numberline {5.3.2}Proposal distributions}{24}{subsubsection.5.3.2}% 
\contentsline {paragraph}{Gamma proposal}{24}{section*.7}% 
\contentsline {paragraph}{Exponential proposal}{25}{section*.8}% 
\contentsline {paragraph}{Shifted Rayleigh proposal}{25}{section*.9}% 
\contentsline {paragraph}{Mixture between Rayleigh and Gaussian distribution}{25}{section*.10}% 
\contentsline {paragraph}{Gaussian proposal}{26}{section*.11}% 
\contentsline {subsubsection}{\numberline {5.3.3}Choice of the proposal}{26}{subsubsection.5.3.3}% 
\contentsline {section}{\numberline {6}Numerical experiments}{27}{section.6}% 
\contentsline {paragraph}{Motivation}{27}{section*.13}% 
\contentsline {paragraph}{Simulation parameters}{27}{section*.14}% 
\contentsline {paragraph}{Irreducibility issues}{28}{section*.15}% 
\contentsline {paragraph}{Results --- short trajectories}{28}{section*.16}% 
\contentsline {paragraph}{Results --- long non-ergodic trajectories}{29}{section*.19}% 
\contentsline {paragraph}{Results -- mixing}{29}{section*.21}% 
\contentsline {paragraph}{Conclusion}{31}{section*.23}% 
\contentsline {section}{\numberline {7}Supplementary material}{31}{section.7}% 
