Probability Theory and Related Fields最新文献

Abstract

We consider the limit of solutions of scaled linear kinetic equations with a reflection-transmission-killing condition at the interface. Both the coefficient describing the probability of killing and the scattering kernel degenerate. We prove that the long-time, large-space limit is the unique solution of a version of the fractional in space heat equation that corresponds to the Kolmogorov equation for a symmetric stable process, which is reflected, or transmitted while crossing the interface and is killed upon the first hitting of the interface. The results of the paper are related to the work in Komorowski et al. (Ann Prob 48:2290–2322, 2020), where the case of a non-degenerate probability of killing has been considered.

Associated to two given sequences of eigenvalues (lambda _1 ge cdots ge lambda _n) and (mu _1 ge cdots ge mu _n) is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis of random hives, we show that if the eigenvalues are drawn from the GUE ensemble, then the associated augmented hives exhibit concentration as (n rightarrow infty ). Our main ingredients include a representation due to Speyer of augmented hives involving a supremum of linear functions applied to a product of Gelfand–Tsetlin polytopes; known results by Klartag on the KLS conjecture in order to handle the aforementioned supremum; covariance bounds of Cipolloni–Erdős–Schröder of eigenvalue gaps of GUE; and the use of the theory of determinantal processes to analyze the GUE minor process.

We explore the relationship between recursive distributional equations and convergence results for finite difference schemes of parabolic partial differential equations (PDEs). We focus on a family of random processes called symmetric cooperative motions, which generalize the symmetric simple random walk and the symmetric hipster random walk introduced in Addario-Berry et al. (Probab Theory Related fields 178(1–2):437–473, 2020). We obtain a distributional convergence result for symmetric cooperative motions and, along the way, obtain a novel proof of the Bernoulli central limit theorem. In addition, we prove a PDE result relating distributional solutions and viscosity solutions of the porous medium equation and the parabolic p-Laplace equation, respectively, in one dimension.

We prove a non-extinction result for Fleming–Viot-type systems of two particles with dynamics described by an arbitrary symmetric Hunt process under the assumption that the reference measure is finite. Additionally, we describe an invariant measure for the system, we discuss its ergodicity, and we prove that the reference measure is a stationary measure for the embedded Markov chain of positions of the surviving particle at successive branching times.

In this paper, we provide relations among the following properties:

1. (a)

   the tail triviality of a probability measure (mu ) on the configuration space ({varvec{Upsilon }});

2. (b)

   the finiteness of a suitable (L^2)-transportation-type distance (bar{textsf {d} }_{varvec{Upsilon }});

3. (c)

   the irreducibility of local ({mu })-symmetric Dirichlet forms on ({varvec{Upsilon }}).

As an application, we obtain the ergodicity (i.e., the convergence to the equilibrium) of interacting infinite diffusions having logarithmic interaction and arising from determinantal/permanental point processes including (text {sine}_{2}), (text {Airy}_{2}), (text {Bessel}_{alpha , 2}) ((alpha ge 1)), and (text {Ginibre}) point processes. In particular, the case of the unlabelled Dyson Brownian motion is covered. For the proof, the number rigidity of point processes in the sense of Ghosh–Peres plays a key role.

Given a probability measure (mu ) on ({{mathbb {R}}}^n), Tukey’s half-space depth is defined for any (xin {{mathbb {R}}}^n) by (varphi _{mu }(x)=inf {mu (H):Hin {{{mathcal {H}}}}(x)}), where (mathcal{H}(x)) is the set of all half-spaces H of ({{mathbb {R}}}^n) containing x. We show that if (mu ) is a non-degenerate log-concave probability measure on ({{mathbb {R}}}^n) then

where (L_{mu }) is the isotropic constant of (mu ) and (c_1,c_2>0) are absolute constants. The proofs combine large deviations techniques with a number of facts from the theory of (L_q)-centroid bodies of log-concave probability measures. The same ideas lead to general estimates for the expected measure of random polytopes whose vertices have a log-concave distribution.