Ross G. Pinsky

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where aij , bi, V are nice functions in R (say Lipschitz). Fixing a domain D ⊂ R we denote by CL(D) the class of all positive solutions of the equation Lu = 0 in D. Properties of CL(D) may be studied from an analytic point of view, a probabilistic one using diffusions, or with techniques from both fields combined. The point of the book under review is to(More)
In this paper we investigate uniqueness and nonuniqueness for solutions of the equation ut 1⁄4 Lu þ Vu gu in R ð0;NÞ; uðx; 0Þ 1⁄4 f ðxÞ; xAR; uX0; ðNSÞ where g40; p41; g;VACaðRnÞ; 0pfACðRnÞ and L 1⁄4Pni;j1⁄41 ai;jðxÞ @ @xi@xj þPni1⁄41 biðxÞ @ @xi with ai;j ; biACðRÞ: r 2003 Elsevier Science (USA). All rights reserved.
We compare the spectral gaps and thus the exponential rates of convergence to equilibrium for ergodic one-dimensional diffusions on an interval. One of the results may be thought of as the diffusion analog of a recent result for the spectral gap of one-dimensional Schrödinger operators. We also discuss the similarities and differences between spectral gap(More)
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 × (0,∞), p ∈ (1, 2]; u(x, 0) = f(x) in R; u(x, t) ≥ 0 in R × [0,∞). In particular, we shall investigate how the interplay between the underlying motion(More)
In this paper we study mutual absolute continuity, finiteness of relative entropy and the possibility of their equivalence for probability measures on C([0,∞);Rd) induced by diffusion processes. We also determine explicit events which distinguish between two mutually singular measures in certain one-dimensional cases.
Let the random variable Zn,k denote the number of increasing subsequences of length k in a random permutation from Sn, the symmetric group of permutations of {1, ..., n}. We show that V ar(Zn,kn ) = o((EZn,kn ) ) as n → ∞ if and only if kn = o(n 2 5 ). In particular then, the weak law of large numbers holds for Zn,kn if kn = o(n 2 5 ); that is, lim n→∞(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)
Let X(t) be a positive recurrent diffusion process corresponding to an operator L on a domain D ⊆ Rd with oblique reflection at ∂D if D 6≡ Rd. For each x ∈ D, we define a volume-preserving norm that depends on the diffusion matrix a(x). We calculate the asymptotic behavior as 2 → 0 of the expected hitting time of the 2-ball centered at x and of the(More)
Consider a variant of the simple random walk on the integers, with the following transition mechanism. At each site x, the probability of jumping to the right is ω(x) ∈ [ 1 2 , 1), until the first time the process jumps to the left from site x, from which time onward the probability of jumping to the right is 1 2 . We investigate the transience/recurrence(More)