  We derive "quenched" subdiffusive lower bounds for the exit time tau(n) from
a box of size n for the simple random walk on the planar invasion percolation
cluster. The first part of the paper is devoted to proving an almost sure
analog of H. Kesten's subdiffusivity theorem for the random walk on the
incipient infinite cluster and the invasion percolation cluster using ideas of
M. Aizenman, A. Burchard and A. Pisztora. The proof combines lower bounds on
the instrinsic distance in these graphs and general inequalities for reversible
Markov chains. In the second part of the paper, we present a sharpening of
Kesten's original argument, leading to an explicit almost sure lower bound for
tau(n) in terms of percolation arm exponents. The methods give tau(n) \geq
n^{2+epsilon_0 + kappa}, where epsilon_0>0 depends on the instrinsic distance
and (assuming the exact value of the backbone exponent) kappa can be taken to
be 17/384 on the hexagonal lattice.
