  We searched for an isotropic stochastic gravitational wave background in the
second data release of the International Pulsar Timing Array, a global
collaboration synthesizing decadal-length pulsar-timing campaigns in North
America, Europe, and Australia. In our reference search for a power law strain
spectrum of the form $h_c = A(f/1\,\mathrm{yr}^{-1})^{\alpha}$, we found strong
evidence for a spectrally-similar low-frequency stochastic process of amplitude
$A = 3.8^{+6.3}_{-2.5}\times10^{-15}$ and spectral index $\alpha = -0.5 \pm
0.5$, where the uncertainties represent 95\% credible regions, using
information from the auto- and cross-correlation terms between the pulsars in
the array. For a spectral index of $\alpha = -2/3$, as expected from a
population of inspiralling supermassive black hole binaries, the recovered
amplitude is $A = 2.8^{+1.2}_{-0.8}\times10^{-15}$. Nonetheless, no significant
evidence of the Hellings-Downs correlations that would indicate a
gravitational-wave origin was found. We also analyzed the constituent data from
the individual pulsar timing arrays in a consistent way, and clearly
demonstrate that the combined international data set is more sensitive.
Furthermore, we demonstrate that this combined data set produces comparable
constraints to recent single-array data sets which have more data than the
constituent parts of the combination. Future international data releases will
deliver increased sensitivity to gravitational wave radiation, and
significantly increase the detection probability.
