Abstract
We obtain upper Gaussian estimates of transition probabilities of a spatially non-homogeneous random walk confined to a multidimensional orthant. For the proof, we use comparison arguments based on discrete potential theory and variants of the Harnack principle.
Highlights
Random walks conditioned to stay in cones is an important topic in probability theory, as they appear naturally in many contexts: non intersecting paths and multidimensional random walks in Weyl chambers [33, 14, 19, 11], eigenvalues of random matrices [12], queueing theory [8], modeling of biological and physical phenomena [3, 16, 24], finance [9], etc
Let us notice that relaxing the spatial homogeneity hypothesis induces a crucial change in the recurrence/transience behavior of zero-mean random walks compared to the homogeneous case, since it is possible to build two-dimensional, zero-drift random walks with bound increments that are transient, as soon as spatial homogeneity is no longer required [17]
Such a choice for F2 is guaranteed by the following result which ensures the existence and uniqueness of a positive harmonic function in O+d vanishing on the boundary ∂O+d
Summary
Random walks conditioned to stay in cones is an important topic in probability theory, as they appear naturally in many contexts: non intersecting paths and multidimensional random walks in Weyl chambers [33, 14, 19, 11], eigenvalues of random matrices [12], queueing theory [8], modeling of biological and physical phenomena [3, 16, 24], finance [9], etc. Let us notice that relaxing the spatial homogeneity hypothesis induces a crucial change in the recurrence/transience behavior of zero-mean random walks compared to the homogeneous case, since it is possible to build two-dimensional, zero-drift random walks with bound increments that are transient, as soon as spatial homogeneity is no longer required [17]. It is more realistic for applications and modeling. Most often C is reserved to denote large constants and c small ones
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