Abstract
Two methods for symmetrizing Markov processes are discussed. Letu a(x, y) be the potential density of a Lévy process on a compact Abelian groupG. A general condition is given that guarantees thatv(x, y)=ua(x, y)+ua(y, x) is the potential density of a symmetric Lévy process onG. The second method arises by considering the linear space of one-potentialsU 1 f, withf inL 2, endowed with the inner product (U 1 f,U 1 g)=fU 1 g+gU 1 f. If the semigroup ofX(t) is normal, then the completionH of this space is the Dirichlet space of a symmetric processY(t). A set that is semipolar forX(t) is polar forY(t).
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Glover, J., Rao, M. Symmetrizations of Markov processes. J Theor Probab 1, 305–325 (1988). https://doi.org/10.1007/BF01246632
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DOI: https://doi.org/10.1007/BF01246632