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Suppose d ≥ 2 and α ∈ (1, 2). Let D be a (not necessarily bounded) C1,1 open set in ℝd and μ = (μ1, . . . , μd) where each μj is a signed measure on ℝd belonging to a certain Kato class of the rotationally symmetric α-stable process X. Let Xμ be an α-stable process with drift μ in ℝd and let Xμ,D be the subprocess of Xμ in D. In this paper, we derive sharp two-sided estimates for the transition density of Xμ,D.

In this paper, by using probabilistic methods, we establish sharp two-sided large time estimates for the transition densities of relativistic α-stable processes with mass m ∈ (0, 1] (i.e., for the Dirichlet heat kernels of m − (m 2/α − Δ) α/2 with m ∈ (0, 1]) in half-space-like C 1, 1 open sets. The estimates are uniform in m in the sense that the constants are independent of m ∈ (0, 1]. Combining with the sharp two-sided small time estimates, established in Chen et al. (Ann Probab, 2011), valid for all C 1, 1 open sets, we have now sharp two-sided estimates for the transition densities of relativistic α-stable processes with mass m ∈ (0, 1] in half-space-like C 1, 1 open sets for all times. Integrating the heat kernel estimates with respect to the time variable, one can recover the sharp two-sided Green function estimates for relativistic α-stable processes with mass m ∈ (0, 1] in half-space-like C 1, 1 open sets established recently in Chen et al. (Stoch Process their Appl, 2011).

We give a tight upper bound for Schrodinger-type perturbations of integral kernels. 1 Main results and introduction Consider an arbitrary set X with a σ-algebraM and a (nonnegative) σ-finite measure m defined onM. To simplify the notation we will write dz for m(dz) in what follows. Let p : R×X ×R×X → [0,∞) and q : R×X → [0,∞) be jointly measurable, where R is equipped with the Borel sets. For notational convenience we will also assume that p(s, x, t, y) = 0 if t ≤ s. We define p0(s, x, t, y) = p(s, x, t, y), and pn(s, x, t, y) = ∫ R ∫ X pn−1(s, x, u, z)q(u, z)p(u, z, t, y) dz du , n = 1, 2, . . . . We note that pn(s, x, t, y) = 0 if s ≥ t. We define p(s, x, t, y) = ∞ ∑ n=0 pn(s, x, t, y) . (1) It is of interest to estimate the perturbed kernel p under appropriate assumptions on the kernel p and the perturbation q. The problem is motivated ∗This research was partially supported by grants KBN and YSP. 1 by the case when p is a transition density, i.e. satisfies (16). In this case p is also a transition density, one generated by a time-dependent additive (Schrodinger) perturbation of the generator of p, see Example 3 and [3]. For recent developments in Schrodinger perturbations of time-nonhomogeneous transition probabilities we refer the reader to [3], and [13], [14], [20]. We will assume that there are 0 ≤ η < ∞ and a function Q : R × R → [0,∞) satisfying the following condition of super-additivity: Q(u, r) +Q(r, v) ≤ Q(u, v) , for all u < r < v , (2) and such that for all s < t ∈ R and x, y ∈ X, ∫ t

For a hyperbolic α-stable process in the hyperbolic space $\mathbb {H}^{d}, d\ge 2$ , we prove that the mean exit time from a halfspace $H(a)=\{x_{d}>a\}\subset \mathbb {H}^{d} $ is equal to $\mathbb {E}^{x} \tau _{H(a)} = c(\alpha , d) \delta ^{\alpha /2}_{H(a)} (x),$ where δ D (x) is the (hyperbolic) distance of x to D c . Based on this exact result we provide a sharp estimate of the mean exit time from a hyperbolic ball B(x 0,R) of radius R and center x 0: $\mathbb {E}^{x}\tau _{B(x_{0},R)}\approx (\delta _{B(x_{0},R)}(x) \tanh R)^{\alpha /2}, x\in \mathbb {H}^{d}$ . By usual isomorphism argument the same estimate holds in any other model of real hyperbolic space.

We prove non-explosion results for Schr\"odinger perturbations of symmetric transition densities and Hardy inequalities for their quadratic forms by using explicit supermedian functions of their semigroups., Comment: 21 pages, updated references