SOME REMARKS ON THE HEAT FLOW FOR FUNCTIONS AND FORMS

This note is concerned with the di(cid:11)erentiation of heat semigroups on Riemannian manifolds. In particular, the relation dP t f = P t df is investigated for the semigroup generated by the Laplacian with Dirichlet boundary conditions. By means of elementary martingale arguments it is shown that well-known properties which hold on complete Riemannian manifolds fail if the manifold is only BM-complete. In general, even if M is ﬂat and f smooth of compact support, k dP t f k 1 cannot be estimated on compact time intervals in terms of f or df . ,


Introduction
Let (M, g) be a Riemannian manifold and ∆ its Laplacian. Consider the minimal heat semigroup associated to 1 2 ∆ on functions given by It is a well-known consequence of the spectral theorem that on a complete Riemannian manifold M d P t f = P (1) t df (1.4) holds for all f ∈ C ∞ c (M ) (compactly supported C ∞ functions on M ) if, for instance, for any x ∈ M and any compact subset K ⊂ M .
with A ∈ Γ(R r ⊗ T M), A 0 ∈ Γ(T M) and Z an R r -valued Brownian motion on some filtered probability space satisfying the usual completeness conditions. For x ∈ M , let be the filtration generated by X(x) starting at x. Then, by [4], A and A 0 in the SDE (1.6) can be chosen in such a way that Suppose that, instead of (1.5), we have for any x ∈ M and any compact subset K ⊂ M . Then

Differentiation of semigroups
We follow the methods of [7]. In the sequel we write occasionally T x f instead of df x for the differential of a function f to avoid mix-up with stochastic differentials. Finally, we denote by B(M ) the bounded measurable functions on M and by bC 1 (M ) the bounded C 1 -functions on M with bounded derivative.
are local martingales (with respect to the underlying filtration).
Proof To see the first claim, note that (P t−. f) X.(x) is a local martingale depending on x in a differentiable way. Thus, the derivative with respect to x is again a local martingale, see [1]. The second claim is reduced to the first one by conditioning with respect to F.(x) to filter out redundant noise. The second part may also be checked directly using the Weitzenböck formula Then dN s m = 0 (equality modulo differentials of local martingales) follows by means of Itô's formula.
Notation For the Brownian motion X.(x) on M , let is a continuous local martingale. Then is again a continuous local martingale for any adapted T x M -valued process h of locally bounded variation. In particular, Proof Indeed, by Itô's lemma, where m = stands for equality modulo local martingales. The second part can be seen using the formula This proves the Lemma. Lemma 2.2 leads to explicit formulae for dP t f by means of appropriate choices for h which make the local martingales in Lemma 2.2 to uniformly integrable martingales. This can be done as in [7].

Theorem 2.3 [7]
Let f: M → R be bounded measurable, x ∈ M and v ∈ T x M . Then, for any bounded adapted process h with paths in H(R + , T x M ) such that τD∧t 0 |ḣ s | 2 ds 1/2 ∈ L 1 , and the property that h 0 = v, h s = 0 for all s ≥ τ D ∧ t, the following formula holds: where τ D is the first exit time of X(x) from some relatively compact open neighbourhood D of x.
Proof (i) Of course, dP s f = P of Lemma 2.1 to be a martingale for 0 ≤ s ≤ t, which gives by taking expectations On the other hand, by Lemma 2.2, Thus

Remark 2.6
In the abstract framework of the Γ 2 -theory of Bakry and Emery (e.g. [2]) lower bounds on the Ricci curvature Ric ≥ α (i.e. Γ 2 ≥ αΓ) may be expressed equivalently in terms of the semigroup as |dP t f| 2 ≤ e −αt P t |df | 2 , t ≥ 0, for f in a sufficiently large algebra of bounded functions on M . However, in general, the setting does not include the Laplacian on metrically incomplete manifolds. On such spaces, we may have lim sup , as can be seen from the examples below.

An example
Let R 2 \ {0} be the plane with origin removed. For n ≥ 2, let M n be an n-fold covering of R 2 \ {0} equipped with the flat Riemannian metric. See [6] for a detailed analysis of the heat kernel on such BM-complete spaces. In terms of polar coordinates is a bounded eigenfunction of ∆ on M n (with eigenvalue −1); here J 1/n denotes the Bessel function of order 1/n. Note that J 1/n (r) = O(r 1/n ) as r 0, consequently dh is unbounded on M n . The martingale property of implies P t h = e −t/2 h which means that dP t h is unbounded on M n as well. for f ∈ C ∞ (M n ) of compact support.
Proof Otherwise (3.2) holds true for all compactly supported f ∈ C ∞ (M n ). Fix t > 0. Then by Theorem 2.4 (ii) for any compactly supported f ∈ C ∞ (M n ). On the other hand, we may choose a sequence (f ) of nonnegative, compactly supported elements in C ∞ (M n ) such that f h c := h + c with h given by (3.1) and c a constant such that h + c ≥ 0. But then (see Chavel [3] p. 187 Lemma 3; note that this is a local argument which can be applied on any open relatively compact subset of M ) P t f P t h c and dP t f → dP t h c as → ∞.

By (3.3) we would have
in contradiction to dP t h c ∞ = e −t/2 dh ∞ = ∞.
Remark 3.2 In [5] it is shown that if a stochastic dynamical system of the type (1.6) is strongly 1-complete, and if for each compact set K there is a δ > 0 such that then d P t f = P (1) t df holds true for functions f ∈ bC 1 (M ). Example 3.1 shows that the strong 1-completeness is necessary and can not be replaced by completeness.