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The purpose of this article is to give a rather thorough understanding of the compact support property for measure-valued processes corresponding to semi-linear equations of the form ut = Lu + βu − αu p in R u(x, 0) = f (x) in R d ; u(x, t) ≥ 0 in R d × [0, ∞). In particular, we shall investigate how the interplay between the underlying motion (the(More)
a i,j (x) ∂ 2 ∂x i ∂x j + n i=1 b i (x) ∂ ∂x i is a non-degenerate elliptic operator, g ∈ C(R n) and the reaction term f converges to −∞ at a super-linear rate as u → ∞. We give a sharp minimal growth condition on f , independent of L, in order that there exist a universal, a priori upper bound for all solutions to the above Cauchy problem—that is, in order(More)
Let the random variable Z n,k denote the number of increasing subsequences of length k in a random permutation from Sn, the symmetric group of permutations of {1, ..., n}. In a recent paper [4] we showed that the weak law of large numbers holds for Z n,kn if kn = o(n 2 5); that is, lim n→∞ Z n,kn EZ n,kn = 1, in probability. The method of proof employed(More)
Let D R d be an unbounded domain and let B(t) be a Brownian motion in D with normal reflection at the boundary. We study the transcience/recurrence dichotomy, focusing mainly on domains of the form D = {(x, z) ∈ R l+m : |z| < H(|x|)}, where d = l + m and H is a sufficiently regular function. This class of domains includes various horn-shaped domains and(More)
Let D ⊂ Rd be a bounded domain and let P(D) denote the space of probability measures on D. Consider a Brownian motion in D which is killed at the boundary and which, while alive, jumps instantaneously at an exponentially distributed random time with intensity γ > 0 to a new point, according to a distribution μ ∈ P(D). From this new point it repeats the(More)
The diffusion operator HD = − 1 2 d dx a d dx − b d dx = − 1 2 exp(−2B) d dx a exp(2B) d dx , where B(x) = R x 0 b a (y)dy, defined either on R + = (0, ∞) with the Dirichlet boundary condition at x = 0, or on R, can be realized as a self-adjoint operator with respect to the density exp(2Q(x))dx. The operator is unitarily equivalent to the Schrödinger-type(More)
Let D ⊂ R d be a bounded domain and denote by P(D) the space of probability measures on D. Let L = 1 2 ∇ · a∇ + b∇ be a second order elliptic operator. Let µ ∈ P(D) and δ > 0. Consider a Markov process X(t) in D which performs diffusion in D generated by the operator δL and is stopped at the boundary, and which while running, jumps instantaneously,(More)