A FUNCTIONAL LIMIT THEOREM FOR LOCALLY PERTURBED
RANDOM WALKS
Alexander Iksanov
Andrey Pilipenko
Abstract: A particle moves randomly over the integer points of the real line. Jumps of the particle
outside the membrane (a fixed “locally perturbating set”) are i.i.d., have zero mean
and finite variance, whereas jumps of the particle from the membrane have other
distributions with finite means which may be different for different points of the
membrane; furthermore, these jumps are mutually independent and independent of
the jumps outside the membrane. Assuming that the particle cannot jump over the
membrane, we prove that the weak scaling limit of the particle position is a skew
Brownian motion with parameter . The path of a skew Brownian motion is
obtained by taking each excursion of a reflected Brownian motion, independently of
the others, positive with probability and negative with probability
. To prove the weak convergence result, we give a new approach which
is based on the martingale characterization of a skew Brownian motion. Among
others, this enables us to provide the explicit formula for the parameter . In the
previous articles, the explicit formulae for the parameter have only been obtained
under the assumption that outside the membrane the particle performs unit jumps.
2010 AMS Mathematics Subject Classification: Primary: 60F17; Secondary:
60G50.
Keywords and phrases: Functional limit theorem, locally perturbed random walk,
martingale characterization, skew Brownian motion.