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We consider two-dimensional L\'evy processes reflected to stay in thepositive quadrant. Our focus is on the non-standard regime when the mean of thefree process is negative but the reflection vectors point away from the origin,so that the reflected process escapes to infinity along one of the axes. Underrather general conditions, it is shown that such behaviour is certain and eachcomponent can dominate the other with positive probability for any givenstarting position. Additionally, we establish the corresponding invarianceprinciple providing justification for the use of the reflected Brownian motionas an approximate model. Focusing on the probability that the first componentdominates, we derive a kernel equation for the respective Laplace transform inthe starting position. This is done for the compound Poisson model withnegative exponential jumps and, by means of approximation, for the Brownianmodel. Both equations are solved via boundary value problem analysis, whichalso yields the domination probability when starting at the origin. Finally,certain asymptotic analysis and numerical results are presented.

Probability of total domination for transient reflecting processes in a quadrant