  We generalize the concept of coarse hypercyclicity, introduced by Feldman in
\cite{Fe1}, to that of coarse topological transitivity on open cones. We show
that a bounded linear operator acting on an infinite dimensional Banach space
with a coarsely dense orbit on an open cone is hypercyclic and a coarsely
topologically transitive (mixing) operator on an open cone is topologically
transitive (mixing resp.). We also "localize" these concepts by introducing two
new classes of operators called coarsely $J$-class and coarsely $D$-class
operators and we establish some results that may make these classes of
operators potentially interesting for further studying. Namely, we show that if
a backward unilateral weighted shift on $l^2(\mathbb{N})$ is coarsely $J$-class
(or $D$-class) on an open cone then it is hypercyclic. Then we give an example
of a bilateral weighted shift on $l^{\infty}(\mathbb{Z})$ which is coarsely
$J$-class, hence it is coarsely $D$-class, and not $J$-class. Note that,
concerning the previous result, it is well known that the space
$l^{\infty}(\mathbb{Z})$ does not support $J$-class bilateral weighted shifts,
see \cite{CosMa2}. Finally, we show that there exists a non-separable Banach
space which supports no coarsely $D$-class operators on open cones. Some open
problems are added.
