We prove a conjecture by Shannon Starr regarding the asymptotics for the number of tuples of commuting permutations with given number of joint orbits. These numbers generalize unsigned Stirling numbers of the first kind which count how many single permutations have a given number of cycles. In the case of pairs of permutations, these numbers are related to D'Arcais polynomials and the Nekrasov-Okounkov formula. As a consequence of the above asymptotics, we confirm a log-concavity conjecture in the regime of typical values for the number of joint orbits. As a result of possible indepentent interest in applied mathematics and mathematical physics, we also provide detailed asymptotics, using Mellin transform techniques, for certain multiple series or multivariate Ramanujan sums which are related to ordinary generating functions of Dirichlet convolutions of power laws. Besides these multiple sums asymptotics, our proofs use bivariate saddle point analysis related to the Meinardus theorem in the delicate case of multiple poles for the associated Dirichlet series.
We formulate the Schrödinger problem for interacting particle systems in the hydrodynamical regime thus extending the standard setting of independent particles. This involves the large deviations rate function for the empirical measure which is in fact a richer observable than the hydrodynamic observables density and current. In the case in which the constraints are the initial and final density, we characterize the optimal measure for the Schrödinger problem. We also introduce versions of the Schrödinger problem in which the constraints are related to the current and analyze the corresponding optimal measures.
We consider a diffusion process on $\mathbb R^n$ and prove a large deviation principle for the empirical process in the joint limit in which the time window diverges and the noise vanishes. The corresponding rate function is given by the expectation of the Freidlin-Wentzell functional per unit of time. As an application of this result, we obtain a variational representation of the rate function for the Gallavotti-Cohen observable in the small noise and large time limits.
Non-uniqueness in law for three-dimensional Navier-Stokes equations forced by random noise was established recently in Hofmanov$\acute{\mathrm{a}}$ et al. (2019, arXiv:1912.11841 [math.PR]). The purpose of this work is to prove non-uniqueness in law for the Boussinesq system forced by random noise. Diffusion within the equation of its temperature scalar field has a full Laplacian and the temperature scalar field can be initially smooth.
We study the two-dimensional Navier-Stokes equations forced by random noise with a diffusive term generalized via a fractional Laplacian that has a positive exponent strictly less than one. Because intermittent jets are inherently three-dimensional, we instead adapt the theory of intermittent form of the two-dimensional stationary flows to the stochastic approach presented by Hofmanov$\acute{\mathrm{a}}$, Zhu $\&$ Zhu (2019, arXiv:1912.11841 [math.PR]) and prove its non-uniqueness in law.
In the paper we present simple examples of linear random fields defined on $\ZZ^2$ and $\ZZ^3$ which exhibit the scaling transition phenomenon. These examples lead to more general definition of the scaling transition and allow to understand the mechanism of appearance of this phenomenon better. In previous papers devoted to the scaling transition it was proved mainly for random fields with finite variance and long-range dependence. We consider random fields with finite and infinite variance. Our examples show that the scaling transition phenomenon can be observed for linear random fields with the so-called negative dependence, which is part of short-range dependence. Relation of the scaling transition with Lamperti type theorems for random fields is discussed.
Consider a spectrally positive Stable($1+α$) process whose jumps we interpret as lifetimes of individuals. We mark the jumps by continuous excursions assigning "sizes" varying during the lifetime. As for Crump-Mode-Jagers processes (with "characteristics"), we consider for each level the collection of individuals alive. We arrange their "sizes" at the crossing height from left to right to form an interval partition. We study the continuity and Markov properties of the interval-partition-valued process indexed by level. From the perspective of the Stable($1+α$) process, this yields new theorems of Ray-Knight-type. From the perspective of branching processes, this yields new, self-similar models with dense sets of birth and death times of (mostly short-lived) individuals. This paper feeds into projects resolving conjectures by Feng and Sun (2010) on the existence of certain measure-valued diffusions with Poisson--Dirichlet stationary laws, and by Aldous (1999) on the existence of a continuum-tree-valued diffusion.
In arXiv:1609.05666v1 [math.PR] a functional limit theorem was proved. It states that symmetric processes associated with resistance metric measure spaces converge when the underlying spaces converge with respect to the Gromov-Hausdorff-vague topology, and a certain uniform recurrence condition is satisfied. Such a theorem finds particularly nice applications if the resistance metric measure space is a metric measure tree. To illustrate this, we state functional limit theorems in old and new examples of suitably rescaled random walks in random environment on trees. First, we take a critical Galton-Watson tree conditioned on its total progeny and a non-lattice branching random walk on $\mathbb{R}^d$ indexed by it. Then, conditional on that, we consider a biased random walk on the range of the preceding. Here, by non-lattice we mean that distinct branches of the tree do not intersect once mapped in $\mathbb{R}^d$. This excludes the possibility that the random walk on the range may jump from one branch to the other without returning to the recent common ancestor. We prove, after introducing the bias parameter $β^{n^{-1/4}}$, for some $β>1$, that the biased random walk on the range of a large critical non-lattice branching random walk converges to a Brownian motion in a random Gaussian potential on Aldous' continuum random tree (CRT). Our second new result introduces the scaling limit of the edge-reinforced random walk on a size-conditioned Galton-Watson tree with finite variance as a Brownian motion in a random Gaussian potential on the CRT with a drift proportional to the distance to the root.
In this paper, we consider a fixed delay Cox-Ingersoll-Ross process (CIR process) on the regime where it does not hit zero, the aim is to determine a positive preserving implicit Euler Scheme. On a time grid with constant stepsize our scheme extends the scheme proposed by Alfonsi in 2005 for the classical CIR model. Furthermore, we consider its piecewise linear interpolation, and, under suitable conditions, we establish the order of strong convergence in the uniform norm, thus extending the results of Dereich et al. in 2011.
We introduce and study the inhomogeneous exponential jump model - an integrable stochastic interacting particle system on the continuous half line evolving in continuous time. An important feature of the system is the presence of arbitrary spatial inhomogeneity on the half line which does not break the integrability. We completely characterize the macroscopic limit shape and asymptotic fluctuations of the height function (= integrated current) in the model. In particular, we explain how the presence of inhomogeneity may lead to macroscopic phase transitions in the limit shape such as shocks or traffic jams. Away from these singularities the asymptotic fluctuations of the height function around its macroscopic limit shape are governed by the GUE Tracy-Widom distribution. A surprising result is that while the limit shape is discontinuous at a traffic jam caused by a macroscopic slowdown in the inhomogeneity, fluctuations on both sides of such a traffic jam still have the GUE Tracy-Widom distribution (but with different non-universal normalizations). The integrability of the model comes from the fact that it is a degeneration of the inhomogeneous stochastic higher spin six vertex models studied earlier in arXiv:1601.05770 [math.PR]. Our results on fluctuations are obtained via an asymptotic analysis of Fredholm determinantal formulas arising from contour integral expressions for the q-moments in the stochastic higher spin six vertex model. We also discuss "product-form" translation invariant stationary distributions of the exponential jump model which lead to an alternative hydrodynamic-type heuristic derivation of the macroscopic limit shape.
We define a Markov process on the partitions of $[n]=\{1,\ldots,n\}$ by drawing a sample in $[n]$ at each time of a Poisson process, by merging blocks that contain one of these points and by leaving all other blocks unchanged. This coalescent process appears in the study of the connected components of random graph processes in which connected subgraphs are added over time with probabilities that depend only on their size. First, we determine the asymptotic distribution of the coalescent time. Then, we define a Bienayme-Galton-Watson (BGW) process such that its total population size dominates the block size of an element. We compute a bound for the distance between the total population size distribution and the block size distribution at a time proportional to $n$. As a first application of this result, we establish the coagulation equations associated with this coalescent process. As a second application, we estimate the size of the largest block in the subcritical and supercritical regimes as well as in the critical window.
Hwang's quasi-power theorem asserts that a sequence of random variables whose moment generating functions are approximately given by powers of some analytic function is asymptotically normally distributed. This theorem is generalised to higher dimensional random variables. To obtain this result, a higher dimensional analogue of the Berry-Esseen inequality is proved, generalising a two-dimensional version of Sadikova.
Suppose $d\ge 2$ and $0<β<α<2$. We consider the non-local operator $\mathcal{L}^{b}=Δ^{α/2}+\mathcal{S}^{b}$, where $$\mathcal{S}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-β)\int_{|z|>\varepsilon}\left(f(x+z)-f(x)\right)\frac{b(x,z)}{|z|^{d+β}}\,dy.$$ Here $b(x,z)$ is a bounded measurable function on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ that is symmetric in $z$, and $\mathcal{A}(d,-β)$ is a normalizing constant so that when $b(x, z)\equiv 1$, $\mathcal{S}^{b}$ becomes the fractional Laplacian $Δ^{β/2}:=-(-Δ)^{β/2}$. In other words, $$\mathcal{L}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-β)\int_{|z|>\varepsilon}\left(f(x+z)-f(x)\right) j^b(x, z)\,dz,$$ where $j^b(x, z):= \mathcal{A}(d,-α) |z|^{-(d+α)}+ \mathcal{A}(d,-β) b(x, z)|z|^{-(d+β)}$. It is recently established in Chen and Wang [arXiv:1312.7594 [math.PR]] that, when $j^b(x, z)\geq 0$ on $\mathbb{R}^d\times \mathbb{R}^d$, there is a conservative Feller process $X^{b}$ having $\mathcal{L}^b$ as its infinitesimal generator. In this paper we establish, under certain conditions on $b$, a uniform boundary Harnack principle for harmonic functions of $X^b$ (or equivalently, of $\mathcal{L}^b$) in any $κ$-fat open set. We further establish uniform gradient estimates for non-negative harmonic functions of $X^{b}$ in open sets.
For $d\ge 2$ and $0<β<α<2$, consider a family of non-local operators $\mathcal{L}^{b}=Δ^{α/2}+\mathcal{S}^{b}$ on $\mathbb{R}^d$, where $$ \mathcal{S}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-β)\int_{ \{z\in \mathbb{R}^d: |z|>\varepsilon\}} (f(x+z)-f(x))\frac{b(x,z)}{|z|^{d+β}}\,dz, $$ and $b(x,z)$ is a bounded measurable function on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ with $b(x,z)=b(x,-z)$ for every $x,z\in\mathbb{R}^{d}$. Here ${\cal A}(d, -β)$ is a normalizing constant so that $\mathcal{S}^b=-(-Δ)^{β/2}$ when $b(x, z)\equiv 1$. It was recently shown in Chen and Wang [arXiv:1312.7594 [math.PR]] that when $b(x, z) \geq -\frac{\mathcal{A}(d, -α)} {\mathcal{A}(d, -β)}\, |z|^{β-α}$, then $\mathcal{L}^b$ admits a unique fundamental solution $p^b(t, x, y)$ which is strictly positive and continuous. The kernel $p^b(t, x, y)$ uniquely determines a conservative Feller process $X^b$, which has strong Feller property. The Feller process $X^b$ is also the unique solution to the martingale problem of $(\mathcal{L}^b, \mathcal{S}(\mathbb{R}^d))$, where $\mathcal{S}(\mathbb{R}^d)$ denotes the space of tempered functions on $\mathbb{R}^d$. In this paper, we are concerned with the subprocess $X^{b,D}$ of $X^{b}$ killed upon leaving a bounded $C^{1,1}$ open set $D\subset \mathbb{R}^d$. We establish explicit sharp two-sided estimates for the transition density function of $X^{b, D}$.
This is an appendix containing further examples to S. Janson, Moments of Gamma type and the Brownian supremum process area, arXiv:1002.4135 [math.PR] and Probability Surveys 7 (2010), 1-52.