  Splitting probabilities quantify the likelihood of a given outcome out of
competitive events for general random processes. This key observable of random
walk theory, historically introduced as the Gambler's ruin problem for a player
in a casino, has a broad range of applications beyond mathematical finance in
evolution genetics, physics and chemistry, such as allele fixation, polymer
translocation, protein folding and more generally competitive reactions. The
statistics of competitive events is well understood for memoryless (Markovian)
processes. However, in complex systems such as polymer fluids, the motion of a
particle should typically be described as a process with memory. Appart from
scaling theories and perturbative approaches in one-dimension, the outcome of
competitive events is much less characterized analytically for processes with
memory. Here, we introduce an analytical approach that provides the splitting
probabilities for general $d$-dimensional non-Markovian Gaussian processes.
This analysis shows that splitting probabilities are critically controlled by
the out of equilibrium statistics of reactive trajectories, observed after the
first passage. This hallmark of non-Markovian dynamics and its quantitative
impact on splitting probabilities are directly evidenced in a prototypical
experimental reaction scheme in viscoelastic fluids. Altogether, these results
reveal both experimentally and theoretically the importance of memory effects
on competitive reactions.
