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We prove that densities of the measures in a strictly stable semigroup (h t ) of symmetric measures on a homogeneous group, if they exist, have the following asymptotic behaviour: $$\mathop {\lim |}\limits_{|x| \to \infty } x|^{Q + \alpha } \cdot h_1 (x) = k(\bar x),$$ where α is the characteristic exponent, $$\bar x = |x|^{ - 1} x$$ , andk is the density of the Lévy measure associated to the semigroup. Moreover, if $$k(\bar x) = 0$$ a more precise description is given.

We explore some of the connections between the local picture left by the trace of simple random walk on a cylinder $${(\mathbb {Z} / N\mathbb {Z})^d \times \mathbb {Z}}$$ , d ≥ 2, running for times of order N 2d and the model of random interlacements recently introduced in Sznitman ( http://www.math.ethz.ch/u/sznitman/preprints ). In particular, we show that for large N in the neighborhood of a point of the cylinder with vertical component of order N d the complement of the set of points visited by the walk up to times of order N 2d is close in distribution to the law of the vacant set of random interlacements with a level which is determined by an independent Brownian local time. The limit behavior of the joint distribution of the local pictures in the neighborhood of finitely many points is also derived.

The classical theorem by Pitman states that a Brownian motion minus twice its running infimum enjoys the Markov property. On the one hand, Biane understood that Pitman’s theorem is intimately related to the representation theory of the quantum group $${{\mathcal {U}}}_q\left( {\mathfrak sl}_2 \right) $$ , in the so-called crystal regime $$q \rightarrow 0$$ . On the other hand, Bougerol and Jeulin showed the appearance of exactly the same Pitman transform in the infinite curvature limit $$r \rightarrow \infty $$ of a Brownian motion on the hyperbolic space $${{\mathbb {H}}}^3 = SL _2({{\mathbb {C}}})/ SU _2$$ . This paper aims at understanding this phenomenon by giving a unifying point of view. In order to do so, we exhibit a presentation $${{\mathcal {U}}}_q^\hbar \left( {\mathfrak sl}_2 \right) $$ of the Jimbo–Drinfeld quantum group which isolates the role of curvature r and that of the Planck constant $$\hbar $$ . The simple relationship between parameters is $$q=e^{-r}$$ . The semi-classical limits $$\hbar \rightarrow 0$$ are the Poisson–Lie groups dual to $$ SL _2({{\mathbb {C}}})$$ with varying curvatures $$r \in {{\mathbb {R}}}_+$$ . We also construct classical and quantum random walks, drawing a full picture which includes Biane’s quantum walks and the construction of Bougerol–Jeulin. Taking the curvature parameter r to infinity leads indeed to the crystal regime at the level of representation theory ( $$\hbar >0$$ ) and to the Bougerol–Jeulin construction in the classical world ( $$\hbar =0$$ ). All these results are neatly in accordance with the philosophy of Kirillov’s orbit method.

In this paper, we prove, using Malliavin calculus, that under a local Hörmander condition the solution of a stochastic differential equation with time depending coefficients admits aC ∞ density with respect to Lebesgue measure. An application of this result to nonlinear filtering is developed in this paper to prove the existence of aC ∞ density for the filter associated with a correlated system whose observation is one dimensional with unbounded coefficients.

 New multiplicative and statistically self-similar measures μ are defined on ℝ as limits of measure-valued martingales. Those martingales are constructed by multiplying random functions attached to the points of a statistically self-similar Poisson point process defined in a strip of the plane. Several fundamental problems are solved, including the non-degeneracy and the multifractal analysis of μ. On a bounded interval, the positive and negative moments of diverge under broad conditions.

Let {T(t)}t≧0 be a strongly continuous semi-group of Markov operators on C(X) with generator G. If m∃C(X) is strictly positive, mG generates a semigroup. If {T(t)} is a group given by a flow, m may have isolated zeros and, under some regularity conditions, mG will still generate a flow, constructed explicitly. The connection between some ergodic properties of the new and original flow is studied. For the Markov semi-groups, the new one is strongly ergodic if and only if the original one is strongly ergodic.