We study the differential operator A = d 4 dx 4 acting on a connected network G along with L 2 , the square of the discrete Laplacian acting on a connected discrete graph G. For both operators we discuss well-posedness of the associated linear parabolic problemsIn view of the well-known lack of parabolic maximum principle for all elliptic differential operators of order 2N for N > 1, our most surprising finding is that, after some transient time, the parabolic equations driven by −A may display Markovian features, depending on the imposed transmission conditions in the vertices.Analogous results seem to be unknown in the case of general domains and even bounded intervals. Our analysis is based on a detailed study of bi-harmonic functions complemented by simple combinatorial arguments. We elaborate on analogous issues for the discrete bi-Laplacian; a characterization of complete graphs in terms of the Markovian property of the semigroup generated by −L 2 is also presented.

We give general conditions to state the weighted Hardy inequalitywith respect to a probability measure dµ. Moreover, the optimality of the constant c 0,µ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potentialfor which the existence of positive solutions to the corresponding parabolic problem can be investigated. The hypotheses on dµ allow the drift term to be of type ∇µ µ = −|x| m−2 x with m > 0.

We prove that the realization A_p in Lp(R^N); 1 < p < 1, of the elliptic operator\ud A = (1+|x|^\alpha)\Delta +b|x|^{\alpha -2}x\nabla -c|x|^\beta with domain D(Ap) = fu 2\ud W^{2;p}(R^N) j Au 2 Lp(RN)g generates a strongly continuous analytic semigroup\ud T(t) provided that \alpha > 2; \beta > \alpha - 2 and any constants b in R and c > 0. This\ud generalizes recent results in in the litterature]. Moreover we show that T(t) is\ud consistent, immediately compact and ultracontractive

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