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- On Nonlinear Markov Processes in the Sense of McKean
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Abstract: We study nonlinear time-inhomogeneous Markov processes in the sense of McKean’s (Proc Natl Acad Sci USA 56(6):1907–1911, 1966) seminal work. These are given as families of laws $$\mathbb {P}_{s,\zeta }$$, $$s\ge 0$$, on path space, where $$\zeta $$ runs through a set of admissible initial probability measures on $$\mathbb {R}^d$$. In this paper, we concentrate on the case where every $$\mathbb {P}_{s,\zeta }$$ is given as the path law of a solution to a McKean–Vlasov stochastic differential equation (SDE), where the latter is allowed to have merely measurable coefficients, which in particular are not necessarily weakly continuous in the measure variable. Our main result is the identification of general and checkable conditions on such general McKean–Vlasov SDEs, which imply that the path laws of their solutions form a nonlinear Markov process. Our notion of nonlinear Markov property is in McKean’s spirit, but more general in order to include processes whose one-dimensional time marginal densities solve a nonlinear parabolic partial differential equation, more precisely, a nonlinear Fokker–Planck–Kolmogorov equation, such as Burgers’ equation, the porous media equation and variants thereof with transport-type drift, and also the very recently studied two-dimensional vorticity Navier–Stokes equation and the p-Laplace equation. In all these cases, the associated McKean–Vlasov SDEs are such that both their diffusion and drift coefficients singularly depend (i.e., Nemytskii type) on the one-dimensional time marginals of their solutions. We stress that for our main result the nonlinear Fokker–Planck–Kolmogorov equations do not have to be well posed. Thus, we establish a one-to-one correspondence between solution flows of a large class of nonlinear parabolic PDEs and nonlinear Markov processes.
PubDate: 2025-06-24
- Limit Theorems for Stochastic Exponentials of Matrix-Valued Lévy
Processes-
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Abstract: We study the long-time behaviour of matrix-valued stochastic exponentials of Lévy processes, i.e. of multiplicative Lévy processes in the general linear group. In particular, we prove laws of large numbers as well as central limit theorems for the logarithmized norm, logarithmized entries and the logarithmized determinant of the stochastic exponential. Where possible, Berry–Esseen bounds are also stated.
PubDate: 2025-06-24
- Central Limit Theorem for Crossings in Randomly Embedded Graphs
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Abstract: We consider the number of crossings in a random embedding of a graph, G, with vertices in convex position. We give explicit formulas for the mean and variance of the number of crossings as a function of various subgraph counts of G. Using Stein’s method and size-biased coupling, we establish an upper bound on the Kolmogorov distance between the distribution of the number of crossings and a standard normal random variable. We also consider the case where G is a random graph and obtain a Kolmogorov bound between the distribution of crossings and a Gaussian mixture distribution. As applications, we obtain central limit theorems with convergence rates for the number of crossings in random embeddings of matchings, path graphs, cycle graphs, disjoint union of triangles, random d-regular graphs, and mixtures of random graphs.
PubDate: 2025-06-20
- On Weak Convergence of Stochastic Wave Equation with Colored Noise on
$$\mathbb {R}$$-
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Abstract: In this paper, we study the following stochastic wave equation on the real line: $$\partial _t^2 u_{\alpha }=\partial _x^2 u_{\alpha }+b\left( u_\alpha \right) +\sigma \left( u_\alpha \right) \eta _{\alpha }$$. The noise $$\eta _\alpha $$ is white in time and colored in space with a covariance structure $$\mathbb {E}[\eta _\alpha (t,x)\eta _\alpha (s,y)]=\delta (t-s)f_\alpha (x-y)$$ where $$f_\alpha $$ is continuous with respect to $$\alpha $$ in Fourier mode, see Assumption 1.2. We prove the continuity of the probability measure induced by the solution $$u_\alpha $$, in terms of $$\alpha $$, with respect to the convergence in law in the topology of continuous functions with uniform metric on compact sets. We also give several examples of $$f_{\alpha }$$ to which our theorem applies.
PubDate: 2025-06-20
- Gaussian Generating Functionals on Easy Quantum Groups
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Abstract: We describe all Gaussian generating functionals on several easy quantum groups given by non-crossing partitions. This includes in particular the free unitary, orthogonal and symplectic quantum groups. We further characterize central Gaussian generating functionals and describe a centralization procedure yielding interesting (non-Gaussian) generating functionals.
PubDate: 2025-06-16
- A Law of Large Numbers for Local Patterns in Schur Measures and a Schur
Process-
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Abstract: The aim of this note is to prove a law of large numbers for local patterns in discrete point processes. We investigate two different situations: a class of point processes on the one-dimensional lattice including certain Schur measures, and a model of random plane partitions, introduced by Okounkov and Reshetikhin. The results state in both cases that the linear statistic of a function, weighted by the appearance of a fixed pattern in the random configuration and conveniently normalized, converges to the deterministic integral of that function weighted by the expectation with respect to the limit process of the appearance of the pattern.
PubDate: 2025-06-02
- On a Generalisation of the Coupon Collector Problem
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Abstract: We consider a generalisation of the classical coupon collector problem. We define a super-coupon to be any s-subset of a universe of n coupons. In each round, a random r-subset from the universe is drawn and all its s-subsets are marked as collected. We show that the time to collect all super-coupons is $$\left( {\begin{array}{c}r\\ s\end{array}}\right) ^{-1}\left( {\begin{array}{c}n\\ s\end{array}}\right) \log \left( {\begin{array}{c}n\\ s\end{array}}\right) [(1 + o(1))]$$ on average and has a Gumbel limit after a suitable normalisation. In a similar vein, we show that for any $$\alpha \in (0, 1)$$, the expected time to collect $$(1 - \alpha )$$-proportion of all super-coupons is $$\left( {\begin{array}{c}r\\ s\end{array}}\right) ^{-1}\left( {\begin{array}{c}n\\ s\end{array}}\right) \log \big (\frac{1}{\alpha }\big )[(1 + o(1))]$$. The $$r = s$$ case of this model is equivalent to the classical coupon collector model. We also consider a temporally dependent model where the r-subsets are drawn according to the following Markovian dynamics: the r-subset at round $$k + 1$$ is formed by replacing a random coupon from the r-subset drawn at round k with another random coupon from outside this r-subset. We link the time it takes to collect all super-coupons in the $$r = s$$ case of this model to the cover time of random walk on a certain finite regular graph and conjecture that in general, it takes $$\frac{r}{s} \left( {\begin{array}{c}r\\ s\end{array}}\right) ^{-1}\left( {\begin{array}{c}n\\ s\end{array}}\right) \log \left( {\begin{array}{c}n\\ s\end{array}}\right) [(1 + o(1))]$$ time on average to collect all super-coupons.
PubDate: 2025-05-28
- Pathwise Blowup of Space-Time Fractional Stochastic Partial Differential
Equations-
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Abstract: The finite time blowup in the almost sure sense of a class of space-time fractional stochastic partial differential equations is discussed. Both the cases of white noise and colored noise are considered. The sufficient and necessary condition between the blowup and Osgood condition is obtained when the spatial domain is bounded. In addition, a sufficient condition for the blowup is obtained when the spatial domain is the whole space.
PubDate: 2025-05-28
- Poisson–Dirichlet Scaling Limits of Kemp’s Supertrees
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Abstract: We determine the Gromov–Hausdorff–Prokhorov scaling limits and local limits of Kemp’s d-dimensional binary trees and other models of supertrees. The limits exhibit a root vertex with infinite degree and are constructed by rescaling infinitely many independent stable trees or other spaces according to a function of a two-parameter Poisson–Dirichlet process and gluing them together at their roots. We discuss universality aspects of random spaces constructed in this fashion and sketch a phase diagram.
PubDate: 2025-05-26
- Non-Instantaneous Impulsive Fractional Stochastic Differential Systems
with Damping: Optimal Controls and Trajectory Controllability-
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Abstract: We investigate a new class of impulsive fractional stochastic differential systems with damping effects in Banach spaces, where the abrupt changes occur suddenly at specific points and extend over finite time intervals. Initially, we explore the solvability of the system by applying stochastic analysis, fractional calculus, and the Banach contraction principle. Next, we utilize Balder’s theorem to establish the existence of optimal controls. Additionally, under certain conditions, we establish the trajectory controllability of the system by employing generalized Grönwall’s inequality. An example is provided to demonstrate the validity of the results.
PubDate: 2025-05-24
- Discretization of integrals driven by multifractional Brownian motions
with discontinuous integrands-
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Abstract: We establish the rate of convergence in the $$L^1$$-norm for equidistant approximations of stochastic integrals with discontinuous integrands driven by multifractional Brownian motion. Our findings extend the known results for the case when the driver is a fractional Brownian motion.
PubDate: 2025-05-23
- Cluster Random Fields and Random-Shift Representations
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Abstract: Cluster random fields (CRFs) play a crucial role in the study of extremes of stationary regularly varying random fields (RFs). In particular, they appear in the Rosiński representation of max-stable and $$\alpha $$-stable RFs. In this contribution we introduce CRFs in an abstract setting proving that they are crucial for the construction of shift-generated classes of $$\alpha $$-homogeneous RFs. Further, we investigate the relations between CRFs, tail RFs and spectral tail RFs. Applications discussed in this contribution include new representations of extremal functional indices and purely dissipative max-stable RFs.
PubDate: 2025-05-21
- Polynomial Convergence Rates for Markov Kernels Under Nested Modulated
Drift Conditions-
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Abstract: When a Markov kernel P satisfies a minorization condition and nested modulated drift conditions, Jarner and Roberts provided an asymptotic polynomial convergence rate in weighted total variation norm of $$P^n(x,\cdot )$$ to the invariant probability measure $$\pi $$ of P. In connection with this polynomial asymptotic, we propose explicit and simple estimates on series of such weighted total variation norms, from which an estimate for the total variation norm of $$P^n(x,\cdot )-\pi $$ is deduced. The proofs are self-contained and based on the residual kernel and the Nummelin-type representation of $$\pi $$. No coupling technique is used.
PubDate: 2025-05-10
- The Extinction Rate of a Branching Random Walk with a Barrier in a
Time-Inhomogeneous Random Environment-
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Abstract: Consider a supercritical branching random walk in a time-inhomogeneous random environment. We impose a selection (called barrier) on survival in the following way. The position of the barrier may depend on the generation and the environment. In each generation, only the individuals born below the barrier can survive and reproduce. When the barrier causes the extinction of the system, we give the extinction rate in the sense of $$L^p~(p\ge 1)$$. Moreover, we show the $$L^p$$ convergence of the small deviation probability for a random walk with random environment in time.
PubDate: 2025-05-10
- A Note on the Convergence of the Extreme Eigenvalues of a
Large-Dimensional Sample Covariance Matrix-
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Abstract: In this study, we explore both weak and strong convergence properties of extreme eigenvalues in a large-dimensional sample covariance matrix, specifically in cases where the data matrix comprises independent, though not identically distributed, elements. Our findings reveal that, provided there exists a uniform boundedness condition on the $$(2+\delta )$$-th moment for some $$\delta>0$$ and the proper Lindeberg condition is satisfied, the established convergence results in Yin, Bai and Krishnaiah, (Probab. Theory Relat. Fields 78:509–521, 1988) and Bai, and Yin (Ann. Probab. 21:1275–1294, 1993) remain applicable.
PubDate: 2025-04-22
- Conformable Fractional Stochastic Differential Inclusions Driven by
Poisson Jumps with Optimal Control and Clarke Subdifferential-
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Abstract: This manuscript is devoted to analysing the solvability and optimal control of a conformable fractional stochastic differential inclusion with Clarke subdifferential and deviated argument. The proposed conformable fractional impulsive inclusion system’s solvability in Hilbert space is established by employing fractional calculus, multivalued analysis, stochastic analysis, semigroup theory and a multivalued fixed point theorem. Furthermore, under some suitable assumptions, the existence of optimal control is derived by employing Balder’s theorem. Lastly, an application is provided to validate the developed theoretical results.
PubDate: 2025-03-28
- Almost Sure Central Limit Theorems for Parabolic/Hyperbolic Anderson
Models with Gaussian Colored Noises-
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Abstract: This short note is devoted to establishing the almost sure central limit theorem for the parabolic/hyperbolic Anderson models driven by colored-in-time Gaussian noises, completing recent results on quantitative central limit theorems for stochastic partial differential equations. We combine the second-order Gaussian Poincaré inequality with the method of characteristic functions of Ibragimov and Lifshits, effectively overcoming the challenge from the lack of Itô tools in this colored-in-time setting, and achieving results that are inaccessible with previous methods.
PubDate: 2025-03-28
- Zero–One Laws for Events with Positional Symmetries
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Abstract: We use an information-theoretic argument due to O’Connell (2000) to prove that every sufficiently symmetric event concerning a countably infinite family of independent and identically distributed random variables is deterministic (i.e., has a probability of either 0 or 1). The i.i.d. condition can be relaxed. This result encompasses the Hewitt–Savage zero–one law and the ergodicity of the Bernoulli process, but also applies to other scenarios such as infinite random graphs and simple renormalization processes.
PubDate: 2025-03-24
- Regular Diffusion and Stochastic Differential Equation with Generalized
Drift-
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Abstract: In this paper, we shall discuss some relations between the one-dimensional regular diffusion and stochastic differential equation with generalized drift. We give a necessary and sufficient condition for a regular diffusion to satisfy a stochastic differential equation with measure-valued drift. We also give a sufficient condition for a regular diffusion to satisfy a stochastic differential equation with distributional drift. Several examples are presented.
PubDate: 2025-03-24
- Joint Extremes of Inversions and Descents of Random Permutations
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Abstract: We provide asymptotic theory for the joint distribution of $$X_\textrm{inv}$$ and $$X_\textrm{des}$$, the numbers of inversions and descents of random permutations. Recently, [14] proved that $$X_\textrm{inv}$$, respectively, $$X_\textrm{des}$$, is in the maximum domain of attraction of the Gumbel distribution. To tackle the dependency between these two permutation statistics, we use Hájek projections and a suitable quantitative Gaussian approximation. We show that $$(X_\textrm{inv}, X_\textrm{des})$$ is in the maximum domain of attraction of the two-dimensional Gumbel distribution with independent margins. This result can be stated in the broader combinatorial framework of finite Coxeter groups, on which our method also yields the central limit theorem for $$(X_\textrm{inv}, X_\textrm{des})$$ and various other permutation statistics as a novel contribution. In particular, signed permutation groups with random biased signs and products of classical Weyl groups are investigated.
PubDate: 2025-03-18
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