Given a second-order elliptic operator on R d , with bounded diffusion coefficients and unbounded drift, which is the generator of a strongly continuous semigroup on L 2 (R d ) represented by an integral, we study the time behavior of the integral kernel and prove estimates on its decay at infinity. If the diffusion coefficients are symmetric, a local lower estimate is also proved.

Let H be a separable Hilbert space and let A A ⊂ H → H be a self-adjoint operator with A ≤ I, > 0 and Tr −A −1 < . We endow H with the centered Gaussian measure with covariance operator Q = − 2 1 A −1 and consider a function U ∈ C 3 H with bounded second and third order derivatives, the SDE dX = AX − DU X dt + dW t , X 0 = x and the associated transition semigroup P t . We define the class BV H of bounded variation functions with respect to the probability measure dx = Z −1 e −2U x dx , where Z is the normalization constant, through an integration by parts formula and prove that P t u ∈ W 1 1 H for t > 0, u ∈ BV H , and that u ∈ BV H if and only if the limit of DP t u L 1 H as t → 0 is finite.

Functions of bounded variation in an abstract Wiener space, i.e., an infinite-dimensional Banach space endowed with a Gaussian measure and a related differential structure, have been introduced by M. Fukushima and M. Hino using Dirichlet forms, and their properties have been studied with tools from analysis and stochastics. In this paper we reformulate, in an integral-geometric vein and with purely analytical tools, the definition and the main properties of BV functions, and investigate further properties.

We study elliptic operators L with Dirichlet boundary conditions on a bounded domain Ω whose diffusion coefficients degenerate linearly at ∂Ω in tangential directions. We compute the domain of L and establish existence, uniqueness and (maximal) regularity of the elliptic and parabolic problems for L in L p -spaces and in spaces of continuous functions. Moreover, the analytic semigroups generated by L are consistent, positive, compact and exponentially stable.

In this paper we provide two different characterizations of sets with finite perimeter and functions of bounded variation in Carnot groups, analogous to those that hold in Euclidean spaces, in terms of the short-time behavior of the heat semigroup. The second one holds under the hypothesis that the reduced boundary of a set of finite perimeter is rectifiable, a result that presently is known in Step 2 Carnot groups.