We construct the fundamental solution of ∂t + ∆x + q(t, x), for functions q with a certain integral space-time relative smallness, in particular for those satisfying a relative negligibility. The… (More)

Given a connected open set U 6= ∅ in Rd, d ≥ 2, a relatively closed set A in U is called unavoidable in U , if Brownian motion, starting in x ∈ U \A and killed when leaving U , hits A almost surely… (More)

Let (X,W) be a balayage space, 1 ∈ W, or – equivalently – let W be the set of excessive functions of a Hunt process on a locally compact space X with countable base such that W separates points,… (More)

The restriction of an isotropic α-stable Lévy process Xα in Rn on the fractal support F of certain finite Borel measures μ is introduced in three different ways: by means of the trace of its… (More)

Let us suppose that we have a right continuous Markov semigroup on Rd, d ≥ 1, such that its potential kernel is given by convolution with a function G0 = g(| · |), where g is decreasing, has a mild… (More)

Given a (local) Kato1 measure μ on Rd \ {0}, d ≥ 2, let H 0 (U) be the convex cone of all continuous real solutions u ≥ 0 to the equation ∆u+ uμ = 0 on the punctured unit ball U satisfying lim|x|→1… (More)

In a setting, where only “exit measures” are given, as they are associated with a right continuous strong Markov process on a separable metric space, we provide simple criteria for scaling invariant… (More)

Let φ be a locally upper bounded Borel measurable function on a Greenian open set Ω in Rd and, for every x ∈ Ω, let vφ(x) denote the infimum of the integrals of φ with respect to Jensen measures for… (More)

Let (X,W) be a balayage space, 1 ∈ W, or – equivalently – let W be the set of excessive functions of a Hunt process on a locally compact space X with countable base such that W separates points,… (More)

Liouville’s theorem states that every bounded holomorphic function on C is constant. Let us recall that holomorphic functions f on open subsets U of the complex plane have the mean value property,… (More)