We introduce the detachment process, a novel, time-inhomogeneous Markov process inspired by I. P. Tóth’s problem [4, 13] concerning the number of “lonely passengers” (those without companions) when n passengers are seated independently and uniformly in k initially empty buses. Tóth showed that this number is stochastically non-decreasing in k for fixed n. We complement [4, 13] by treating the number of buses k as a time parameter. Specifically, for a fixed number of passengers n, the state of our Markov process at time k≥1 is exactly Tóth’s configuration (n,k). (We extend the process’s definition for all t∈[1,∞).) These processes can be coupled for all n≥1, and this larger coupled process is what we dub the detachment process. Our investigation focuses on properties related to detachment, clumping, the number of lonely passengers and of non-empty buses. The central notion is detachment, which occurs at time k if each passenger occupies a distinct bus; the process is in a state of detachment at k. At a detachment time k the process transitions from a non-detached state at k−1 to a detached state at k. Four critical time-scales are identified — linear, quadratic, and log-corrected linear or quadratic in the number of passengers, n. For example, we provide a scaling limit result for the {0,1}-valued process X(n) on (0,∞), defined by Xt(n):=𝟙(state of detachment at timetn2), in the sense of finite-dimensional distributions. On the other hand, it is shown that if, for n≫1,y>0, one defines the time scale k(n)=k(n,y):=n2[2(2logn−2loglogn+y)]−1, and e(n,k) is the expected number of times the process is in a state of detachment up to time k, then limn→∞e(n,k(n,y))=c(y):=e−y∕8. We also investigate (relative) clumping and explore why modeling the number of passengers with a Poisson distribution simplifies the analysis of Tóth’s original model. In the latter derivation, a comparison theorem for binomial distributions, originally obtained by J. Najnudel [9], is presented with a novel proof. A comparison with the literature on occupancy problems is given in Subsection 1.6.
Motivated by probabilistic representations of the Navier–Stokes equations, we introduce a new class of stochastic differential equations whose coefficients depend on the entire flow of time marginals. Under one-sided Lipschitz conditions and in the presence of singular drifts, we establish the existence and uniqueness of both strong and weak solutions. These distribution-flow dependent stochastic differential equations are closely related to quasilinear backward Kolmogorov equations and Fokker–Planck equations. We also study distribution-flow dependent SDEs associated with generalized two-dimensional Navier–Stokes equations and driven by fractional Brownian noise. When the Hurst parameter H∈(0,1 2) and the initial vorticity ν0∈M(R2), we prove global well-posedness and smoothness of solutions. In the case H=1 2, we establish global strong well-posedness and smoothness for the corresponding SDEs associated with the backward Navier–Stokes equations, and clarify their relation to the Navier–Stokes equations.
In this paper, we establish Wong-Zakai approximation and support theorem for quasilinear parabolic stochastic partial differential equations. Since this class of equations is not locally monotone, we implement a regularization strategy using a sequence of heat kernel to modify the matrix-valued diffusion term, thereby recovering the monotonicity. As a result, the problem is reduced to prove two results of further layer approximations. To this end, we further employ a kinetic formulation of the approximating solutions in the L1 setting and fully use the doubling variables techniques.
In this paper, we study the asymptotic behaviors of supercritical branching Markov processes {X-t, t > 0} with spatial motions being L & eacute;vy processes having regularly renormalized by some appropriate function h(t), {X-t, t > 0} converges weakly to some random measure. In this paper, we first establish the large deviations of {X-t, t > 0}. More precisely, we prove that, (i) when renormalized by a function A(t) which grows faster than h(t), {X-t, t > 0}, conditioned on {R-t > A(t)}, converges weakly; (ii) when renormalized by a function A(t) which grows slower than h(t), {X-t, t > 0}, conditioned on {Rt < A(t)}, converges weakly, where Rt is the maximum of X-t. We also characterize the almost sure envelopes of Rt.
We study the ℓ∞→ℓ∞ operator norm of products of independent random matrices with independent and identically distributed entries. For n-by-n matrices whose entries are centered, have unit variance, and have a finite moment of order 4α for some α>1, we find that the operator norm of the product of p matrices behaves asymptotically like np+1 2 2∕π. The case of products of possibly non-square matrices with possibly non-centered entries is also covered.
