Density of extremal measures in parabolic potential theory

It is shown that, for the heat equation on , d ≥ 1, any convex combination of harmonic (= caloric) measures , where U ₁, . . . , U _k are relatively compact open neighborhoods of a given point x, can be approximated by a sequence of harmonic measures such that each W _n is an open neighborhood of x in . Moreover, it is proven that, for every open set U in containing x, the extremal representing measures for x with respect to the convex cone of potentials on U (these measures are obtained by balayage, with respect to U, of the Dirac measure at x on Borel subsets of U) are dense in the compact convex set of all representing measures. Since essential ingredients for a proof of corresponding results in the classical case (or more general elliptic situations; see Hansen and Netuka in Adv. Math. 218(4):1181–1223, 2008) are not available for the heat equation, an approach heavily relying on the transit character of the hyperplanes , , is developed. In fact, the new method is suitable to obtain convexity results for limits of harmonic measures and the density of extremal representing measures on for practically every space–time structure which is given by a sub-Markov semigroup (P _t ) _t>0 on a space X′ such that there are strictly positive continuous densities with respect to a (non-atomic) measure on X′. In particular, this includes many diffusions and corresponding symmetric processes given by heat kernels on manifolds and fractals. Moreover, the results may be applied to restrictions of the space–time structure on arbitrary open subsets. I. Netuka’s research was supported in part by the project MSM 0021620839 financed by MSMT, by the grant 201/07/0388 of the Grant Agency of the Czech Republic, and by CRC-701, Bielefeld.