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Anomalous recurrence properties of many-dimensional zero-drift random walks

  • Nicholas Georgiou (a1), Mikhail V. Menshikov (a2), Aleksandar Mijatović (a3) and Andrew R. Wade (a2)


Famously, a d-dimensional, spatially homogeneous random walk whose increments are nondegenerate, have finite second moments, and have zero mean is recurrent if d∈{1,2}, but transient if d≥3. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially nonhomogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension d≥2, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by random walks whose increments are supported on ellipsoids that are symmetric about the ray from the origin through the walk's current position; these elliptic random walks generalize the classical homogeneous Pearson‒Rayleigh walk (the spherical case). Our proof of the recurrence classification is based on fundamental work of Lamperti.


Corresponding author

Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. Email address:
Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. Email address:
Department of Mathematics, King's College London, Strand, London WC2R 2LS, UK. Email address:
Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. Email address:


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