Suppose that d ≥ 1 and α ∈ (1, 2). Let Y be a rotationally symmetric α-stable process on R d and b a R d -valued measurable function on R d belonging to a certain Kato class of Y . WeThroughout this paper, unless otherwise stated, d ≥ 1 and α ∈ (1, 2). A rotationally symmetric α-stable process Y in R d is a Lévy process with characteristic function given by(1.1) The infinitesimal generator of Y is the fractional Laplacian ∆ α/2 := −(−∆) α/2 . Here we use ":=" to denote a definition. Denote by B(x, r) the open ball in R d centered at x ∈ R d with radius r > 0 and dx the Lebesgue measure on R d . Definition 1.1. For a real-valued function f on R d and r > 0, define M α f (r) := sup x∈R d B(x,r) |f (y)| |x − y| d+1−α dy. (1.2) A function f on R d is said to belong to the Kato class K d,α−1 if lim r↓0 M α f (r) = 0.Using Hölder's inequality, it is easy to see that for every p > d/(α−1), L ∞ (R d ; dx)+L p (R d ; dx) ⊂ K d,α−1 . Throughout this paper we will assume b = (b 1 , · · · , b d ) is a R d -valued function on R d such that |b| ∈ K d,α−1 .

We prove the existence of a smooth density for a convolution semigroup on a symmetric space and obtain its spherical representation.

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Let X(t), t ∈ R, be a d-dimensional vector-valued Brownian motion,that is the asymptotical behavior of tail distribution of vector-valued analog of Shepp-statistics for X; we cover not only the case of a fixed time-horizon T > 0 but also cases where T → 0 or T → ∞. Results for high level excursion probabilities of vector-valued processes are rare in the literature, with currently no available approach suitable for our problem. Our proof exploits some distributional properties of vectorvalued Brownian motion, and results from quadratic programming problems. As a by-product we derive a new inequality for the 'supremum' of vector-valued Brownian motions.

We give the asymptotics of the tail of the distribution of the first exit time of the isotropic αstable Lévy process from the Lipschitz cone in R d . We obtain the Yaglom limit for the killed stable process in the cone. We construct and estimate entrance laws for the process from the vertex into the cone. For the symmetric Cauchy process and the positive half-line we give a spectral representation of the Yaglom limit.Our approach relies on the scalings of the stable process and the cone, which allow us to express the temporal asymptotics of the distribution of the process at infinity by means of the spatial asymptotics of harmonic functions of the process at the vertex; on the representation of the probability of survival of the process in the cone as a Green potential; and on the approximate factorization of the heat kernel of the cone, which secures compactness and yields a limiting (Yaglom) measure by means of Prokhorov's theorem.Keywords. Yaglom limit ⋆ stable process ⋆ Lipschitz cone ⋆ quasi-stationary measure ⋆ Green function ⋆ Martin kernel ⋆ excursions.