By Nicholas T. Varopoulos, L. Saloff-Coste, T. Coulhon

The geometry and research that's mentioned during this publication extends to classical effects for normal discrete or Lie teams, and the equipment used are analytical, yet aren't desirous about what's defined nowadays as genuine research. lots of the effects defined during this publication have a twin formula: they've got a "discrete model" with regards to a finitely generated discrete team and a continual model on the topic of a Lie team. The authors selected to middle this publication round Lie teams, yet may possibly simply have driven it in different different instructions because it interacts with the idea of moment order partial differential operators, and chance conception, in addition to with staff idea.

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E. 5 Symmetric submarkovian semigroups 21 Ht = e-tA; indeed (A f, g) = (f, Ag) and (Af,f) = i=1 f IXfl2dm > 0, df,g E Co (V) This last formula also shows that (A f, g) is a Dirichlet form, hence that Ht is symmetric submarkovian. This situation occurs in particular if V = G is a Lie group endowed with its right invariant Haar measure dx and if the vector fields Xi are left invariant. Indeed, let cp E Co (G), and X a left invariant vector field on G. 3 below). Taking cp = fg, one concludes that X is formally skew-adjoint.

Xk} is a Hormander system, one can always connect two points of the manifold V by such a path. One defines in this way a distance which is suitable for the study of 0, and is called the Cannot-Caratheodory distance. , Xk}, a system of C°° vector fields on V, let Cx be the set of all absolutely continuous paths -y: [0, 11 - V, satisfying ry(t) = jk 1 ai(t)Xi(7(t)), for almost every t E [0, 1]. Put 1 1r1= (ta2(t)) dt, f(=1 and for x, y E V, px(x, y) = p(x, y) = inf{Iyi 17 E Cx, 7(0) = x, 7(1) = y} if there exists at least one such path connecting x and y, and +oo otherwise.

This yields h(1-E)t(x) < CEV(Vt-)-1 exp(-p2(x)/4t + p(x)/V), and by changing (1 - e)t into t, ht(x) < exp(-p2(x)/4(1 +e)t) which is the expected conclusion. Our next result is the companion lower bound for the heat kernel ht. 3 Theorem Let G be a nilpotent Lie group, X a Hormander system on G. /)-l exp(-C'p2(x)lt), for all x E G and t > 0. 3 shows that ht (x) > Cht12(e) exp(-C'p2(x)lt), for all x E G and t > 0. 0o is a non-increasing function of t, and ht/2(e) > ht(e). This reduces the proof of the theorem to the proof of the central estimate ht(e) > C'V(f)-1, Vt > 0.