Let L = 1/2 ∑ᵏi,j=1 aᵢⱼ(x)(∂²/∂ xᵢ ∂ xⱼ) + ∑ᵏᵢ₌₁ bᵢ(x)(∂/∂ xᵢ) be an elliptic operator such that aᵢⱼ(•) are continuous and bᵢ(•) are measurable and bounded on compacts. Criteria for transience, null recurrence, and positive recurrence of diffusions on Rᵏ governed by L are derived in terms of the coefficients of L.
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Ritwik Bhattacharya (1978) studied this question.