In this work, we consider the stochastic generalized Burgers–Huxley equation perturbed by space–time white noise and discuss the global solvability results. We show the existence of a unique global mild solution to such equation using a fixed point method and stopping time arguments. The existence of a local mild solution (up to a stopping time) is proved via contraction mapping principle. Then, establishing

In this paper, we consider an extension of Lie group theory to the class of constant delay autonomous stochastic differential equations of Itô form. The determining equations are deterministic even though they represent the stochastic process. The Lie algebras obtained are of low dimensions, and they form an Abelian group.

We show that \(\mathbb {P}( \ell _X(0,T] \le 1)=(c_X+o(1))T^{-(1-H)}\), where \(\ell _X\) is the local time measure at 0 of any recurrent H-self-similar real-valued process X with stationary increments that admits a sufficiently regular local time and \(c_X\) is some constant depending only on X. A special case is the Gaussian setting, i.e. when the underlying process is fractional Brownian motion

Let \( \{X, X_{n};~n \ge 1 \}\) be a sequence of independent and identically distributed Banach space valued random variables. This paper is devoted to providing a divergence criterion for a class of random series of the form \(\sum _{n=1}^{\infty } f_{n}\left( \left\| S_{n} \right\| \right) \) where \(S_{n} = X_{1} + \cdots + X_{n}, ~n \ge 1\) and \(\left\{ f_{n}(\cdot ); n \ge 1 \right\} \) is a

Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and correspond to (nonlinear in general) functionals that are linear on singly generated subalgebras or singly generated cones of functions. They lack subadditivity, and many standard techniques of measure theory and functional analysis do not apply

In this paper, we study the asymptotic properties of drift parameter estimations in reflected Ornstein–Uhlenbeck process, and establish their moderate deviations in both cases with one-sided barrier and two-sided barriers. The main methods consist of regenerative process techniques and the strong Markov property, as well as moderate deviations for martingales.

We study a process of generating random positive integer weight sequences \(\{ W_n \}\) where the gaps between the weights \(\{ X_n = W_n - W_{n-1} \}\) are i.i.d. positive integer-valued random variables. The main result of the paper is that if the gap distribution has a moment generating function with large enough radius of convergence, then the weight sequence is almost surely asymptotically m-complete

We consider the existence and Hölder continuity conditions for the k-th-order derivatives of self-intersection local time for d-dimensional fractional Brownian motion, where \(k=(k_1,k_2,\ldots , k_d)\). Moreover, we show a limit theorem for the critical case with \(H=\frac{2}{3}\) and \(d=1\), which was conjectured by Jung and Markowsky [7].

In this paper, we will study concentration inequalities for Banach space-valued martingales. Firstly, we prove that a Banach space X is linearly isomorphic to a p-uniformly smooth space (\(1

We develop Stein’s method for \(\alpha \)-stable approximation with \(\alpha \in (0,1]\), continuing the recent line of research by Xu (Ann Appl Probab 29(1):458–504, 2019) and Chen et al. (J Theor Probab, 2018. https://doi.org/10.1007/s10959-020-01004-1) in the case \(\alpha \in (1,2)\). The main results include an intrinsic upper bound for the error of the approximation in a variant of Wasserstein

This paper considers random processes of the form \(X_{n+1}=a_nX_n+b_n\pmod p\) where p is odd, \(X_0=0\), \((a_0,b_0), (a_1,b_1), (a_2,b_2),\ldots \) are i.i.d., and \(a_n\) and \(b_n\) are independent with \(P(a_n=2)=P(a_n=(p+1)/2)=1/2\) and \(P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3\). This can be viewed as a multiplicatively symmetrized version of a random process of Chung, Diaconis, and Graham. This paper

We consider a stationary sequence \((X_n)\) constructed by a multiple stochastic integral and an infinite-measure conservative dynamical system. The random measure defining the multiple integral is non-Gaussian and infinitely divisible and has a finite variance. Some additional assumptions on the dynamical system give rise to a parameter \(\beta \in (0,1)\) quantifying the conservativity of the system

Stephenson (2018) established annealed local convergence of Boltzmann planar maps conditioned to be large. The present work uses results on rerooted multi-type branching trees to prove a quenched version of this limit.

In this paper, we study strong solutions of some non-local difference–differential equations linked to a class of birth–death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed

We consider decoupling inequalities for random variables taking values in a Banach space X. We restrict the class of distributions that appear as conditional distributions while decoupling and show that each adapted process can be approximated by a Haar-type expansion in which only the pre-specified conditional distributions appear. Moreover, we show that in our framework a progressive enlargement

We obtain a strong renewal theorem with infinite mean beyond regular variation, when the underlying distribution belongs to the domain of geometric partial attraction of a semistable law with index \(\alpha \in (1/2,1]\). In the process we obtain local limit theorems for both finite and infinite mean, that is, for the whole range \(\alpha \in (0,2)\). We also derive the asymptotics of the renewal function

In this paper, we explore the generalized mixed fractional Brownian motion in the set-indexed framework and generalize several recent results from Miao et al. (Lecture Notes and Math, Springer, New York, 2008), Zili (J. Appl. Math. Stoch. Anal. 30:1–9, 2006) and Thale (Appl. Math. Sci. 3(28):1885–1901, 2009). We present the characterization of generalized mixed set-indexed fractional Brownian motion

In this paper, we prove the existence of strong solutions to an stochastic differential equation with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters \(H<\frac{1}{2}.\) Here, the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case

Consider \(n\) nodes \(\{X_i\}_{1 \le i \le n}\) independently distributed in the unit square \(S,\) each according to a density \(f\), and let \(K_n\) be the complete graph formed by joining each pair of nodes by a straight line segment. For every edge \(e\) in \(K_n\), we associate a weight \(w(e)\) that may depend on the individual locations of the endvertices of \(e\) and is not necessarily a power

The aim of this paper is to develop tractable large deviation approximations for the empirical measure of a small noise diffusion. The starting point is the Freidlin–Wentzell theory, which shows how to approximate via a large deviation principle the invariant distribution of such a diffusion. The rate function of the invariant measure is formulated in terms of quasipotentials, quantities that measure

We construct graphs (trees of bounded degree) on which the contact process has critical rate (which will be the same for both global and local survival) equal to any prescribed value between zero and \(\lambda _c({\mathbb {Z}})\), the critical rate of the one-dimensional contact process. We exhibit both graphs in which the process at this target critical value survives (locally) and graphs where it

We prove local convergence results for rerooted conditioned multi-type Galton–Watson trees. The limit objects are multitype variants of the random sin-tree constructed by Aldous (1991), and differ according to which types recur infinitely often along the backwards growing spine.

The Half-Plane Half-Comb walk is a random walk on the plane, when we have a square lattice on the upper half-plane and a comb structure on the lower half-plane, i.e., horizontal lines below the x-axis are removed. We prove that the probability that this walk returns to the origin in 2N steps is asymptotically equal to \(2/(\pi N).\) As a consequence, we prove strong laws and a limit distribution for

The present paper investigates the effects of tempering the power law kernel of the moving average representation of a fractional Brownian motion (fBm) on some local and global properties of this Gaussian stochastic process. Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) are the processes that are considered in order to investigate the

In this paper, we introduce a specific kind of doubly reflected backward stochastic differential equations (in short DRBSDEs), defined on probability spaces equipped with general filtration that is essentially non quasi-left continuous, where the barriers are assumed to be predictable processes. We call these equations predictable DRBSDEs. Under a general type of Mokobodzki’s condition, we show the

The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power-law shape function and non-stationary noises with a power-law variance function. In this paper, we study sample path properties of the generalized fractional Brownian motion, including Hölder

Consider a collection of particles whose state evolution is described through a system of interacting diffusions in which each particle is driven by an independent individual source of noise and also by a small amount of noise that is common to all particles. The interaction between the particles is due to the common noise and also through the drift and diffusion coefficients that depend on the state

This paper concerns the density of the Hartman–Watson law. Yor (Z Wahrsch Verw Gebiete 53:71–95, 1980) obtained an integral formula that gives a closed-form expression of the Hartman–Watson density. In this paper, based on Yor’s formula, we provide alternative integral representations for the density. As an immediate application, we recover in part a result of Dufresne (Adv Appl Probab 33:223–241,

Under the scenario of high-frequency data, a consistent estimator of the realized Laplace transform of volatility is proposed by Todorov and Tauchen (Econometrica 80:1105–1127, 2012) and a related central limit theorem has been well established. In this paper, we investigate the asymptotic tail behaviour of the empirical realized Laplace transform of volatility (ERLTV). We establish both a large deviation

We study a class of stationary determinantal processes on configurations of N labeled objects that may be present or absent at each site of \({\mathbb {Z}}^d\). Our processes, which include the uniform spanning forest as a principal example, arise from the block Toeplitz matrices of matrix-valued functions on the d-torus. We find the maximum level of uniform insertion tolerance for these processes

We investigate a Belinschi–Nica-type semigroup for free and Boolean max-convolutions. We prove that this semigroup at time one connects limit theorems for freely and Boolean max-infinitely divisible distributions. Moreover, we also construct a max-analogue of Boolean-classical Bercovici–Pata bijection, establishing the equivalence of limit theorems for Boolean and classical max-infinitely divisible

We study the eigenvalues of a Laplace–Beltrami operator defined on the set of the symmetric polynomials, where the eigenvalues are expressed in terms of partitions of integers. To study the behaviors of these eigenvalues, we assign partitions with the restricted uniform measure, the restricted Jack measure, the uniform measure, or the Plancherel measure. We first obtain a new limit theorem on the restricted

The paper deals with some properties of set-valued functions having bounded Riesz p-variation. Set-valued integrals of Young type for such multifunctions are introduced. Selection results and properties of such set-valued integrals are discussed. These integrals contain as a particular case set-valued stochastic integrals with respect to a fractional Brownian motion, and therefore, their properties

We study concentration properties of vertex degrees of n-dimensional Erdős–Rényi random graphs with edge probability \(\rho /n\) by means of high moments of these random variables in the limit when n and \(\rho \) tend to infinity. These moments are asymptotically close to one-variable Bell polynomials \({{\mathcal {B}}}_k(\rho ), k\in {{\mathbb {N}}}\), that represent moments of the Poisson probability

We study semi-martingale obliquely reflected Brownian motion with drift in the first quadrant of the plane in the transient case. Our main result determines a general explicit integral expression for the moment generating function of Green’s functions of this process. To that purpose we establish a new kernel functional equation connecting moment generating functions of Green’s functions inside the

The distribution of the sum of 1-dependent lattice vectors with supports on coordinate axes is approximated by a multivariate compound Poisson distribution and by signed compound Poisson measure. The local and \(\ell _\alpha \)-norms are used to obtain the error bounds. The Heinrich method is used for the proofs.

In this paper, we consider the one-sided and the two-sided first exit problem for a jump diffusion process with semimartingale local time. Denote this process by \(X=\{X_{t},t\ge 0\}\) and set \(\tau _{l}=\inf \{t\ge 0, X_{t}\le l\}\) and \(\tau _{l,u}=\inf \{t\ge 0, X_{t}\notin (l,u)\}\) with \(l

It is well known that, on a purely algebraic level, a simplified version of the central limit theorem (CLT) can be proved in the framework of a non-commutative probability space, under the hypotheses that the sequence of non-commutative random variables we consider is exchangeable and obeys a certain vanishing condition of some of its joint moments. In this approach (which covers versions for both

In this paper, we study a type of stochastic McKean–Vlasov equations with non-Lipschitz coefficients. Firstly, by an Euler–Maruyama approximation the existence of its weak solutions is proved. Then we observe the pathwise uniqueness of its weak solutions. Finally, it is shown that the Euler–Maruyama approximation has an optimal strong convergence rate.

In this paper, we study the reflected backward stochastic differential equations driven by G-Brownian motion with two reflecting obstacles, which means that the solution lies between two prescribed processes. A new kind of approximate Skorohod condition is proposed to derive the uniqueness and existence of the solutions. The uniqueness can be proved by a priori estimates and the existence is obtained

We prove an upper large deviation bound on the scale of the mean for a symmetric random walk in the plane satisfying certain moment conditions. This result complements the study by Phetpradap for the random walk range, which is restricted to dimension three and higher, and of van den Berg, Bolthausen and den Hollander, for the volume of the Wiener sausage.

We define and study the three-dimensional windings along Brownian paths in the quaternionic Euclidean, projective and hyperbolic spaces. In particular, the asymptotic laws of these windings are shown to be Gaussian for the flat and spherical geometries while the hyperbolic winding exhibits a different long time-behavior. The corresponding asymptotic law seems to be new and is related to the Cauchy

Consider a truncated circular unitary matrix which is a \(p_n\) by \(p_n\) submatrix of an n by n circular unitary matrix after deleting the last \(n-p_n\) columns and rows. Jiang and Qi (J Theor Probab 30:326–364, 2017) and Gui and Qi (J Math Anal Appl 458:536–554, 2018) study the limiting distributions of the maximum absolute value of the eigenvalues (known as spectral radius) of the truncated matrix

Here, we study stochastic differential equations with a reflecting boundary condition. We provide sufficient conditions for pathwise uniqueness and non-explosion property of solutions in a framework admitting non-Lipschitz continuous coefficients and non-smooth domains.

We present sufficient conditions, in terms of the jumping kernels, for two large classes of conservative Markov processes of pure-jump type to be purely discontinuous martingales with finite second moment. As an application, we establish the law of the iterated logarithm for sample paths of the associated processes.

A definition of d-dimensional n-Meixner random vectors is given first. This definition involves the commutators of their semi-quantum operators. After that we focus on the 1-Meixner random vectors and derive a system of d partial differential equations satisfied by their Laplace transform. We provide a set of necessary conditions for this system to be integrable. We use these conditions to give a complete

We prove under general conditions that a trimmed subordinator satisfies a self-standardized central limit theorem (SSCLT). Our basic tool is a powerful distributional approximation result of Zaitsev (Probab Theory Relat Fields 74:535–566, 1987). Among other results, we obtain as special cases of our subordinator result the recent SSCLTs of Ipsen et al. (Stoch Process Appl 130:2228–2249, 2020) for trimmed

For hidden Markov models, one of the most popular estimates of the hidden chain is the Viterbi path—the path maximising the posterior probability. We consider a more general setting, called the pairwise Markov model (PMM), where the joint process consisting of finite-state hidden process and observation process is assumed to be a Markov chain. It has been recently proven that under some conditions

In this paper we show the Hörmander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking, under general Hörmander’s Lie bracket conditions, we show the regularization effect of discontinuous Lévy noises for possibly degenerate stochastic differential equations with jumps. To treat the large jumps, we use the perturbation argument together

Given a sequence \((X_n)\) of symmetrical random variables taking values in a Hilbert space, an interesting open problem is to determine the conditions under which the series \(\sum _{n=1}^\infty X_n\) is almost surely convergent. For independent random variables, it is well known that if \(\sum _{n=1}^\infty \mathbb {E}(\Vert X_n\Vert ^2) <\infty \), then \(\sum _{n=1}^\infty X_n\) converges almost

Assume that \(X_{\Sigma } \in \mathbb {R}^{n}\) is a centered random vector following a multivariate normal distribution with positive definite covariance matrix \(\Sigma \). Let \(g : \mathbb {R}^{n} \rightarrow \mathbb {C}\) be measurable and of moderate growth, say \(|g(x)| \lesssim (1 + |x|)^{N}\). We show that the map \(\Sigma \mapsto \mathbb {E}[g(X_{\Sigma })]\) is smooth, and we derive convenient

In this paper, we prove multilevel concentration inequalities for bounded functionals \(f = f(X_1, \ldots , X_n)\) of random variables \(X_1, \ldots , X_n\) that are either independent or satisfy certain logarithmic Sobolev inequalities. The constants in the tail estimates depend on the operator norms of k-tensors of higher order differences of f. We provide applications for both dependent and independent

Path decomposition is performed to characterize the law of the pre-/post-supremum, post-infimum and the intermediate processes of a spectrally negative Lévy process taken up to an independent exponential time T. As a result, mainly the distributions of the supremum of the post-infimum process and the maximum drawdown of the pre-/post-supremum, post-infimum processes and the intermediate processes are

In this note, we correct claims made in.

In this paper, we show that the mixed fractional Poisson process (MFPP) exhibits the long-range dependence property. It is proved by establishing an asymptotic result for the covariance of inverse mixed stable subordinator. Also, it is shown that the increment process of the MFPP, namely the mixed fractional Poissonian noise, has the short-range dependence property.

For a continuous-time catalytic branching random walk (CBRW) on \({\mathbb {Z}}\), with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include regular variation of the jump distribution tail for underlying random walk and the well-known \(L\log L\) condition for the offspring numbers

We construct a probabilistic model for the number of divisors of a random uniform integer that converges in the mod-Poisson sense to the same limiting function as its original counterpart, the one arising in the Sathé–Selberg theorem. This construction involves a conditioning and gives an alternative perspective to the usual paradigm of “hybrid product” models developed by Gonek, Hughes and Keating

In this paper, we prove the Girsanov theorem for G-Brownian motion without the non-degenerate condition. The proof is based on the perturbation method in the nonlinear setting by constructing a product space of the G-expectation space and a linear space that contains a standard Brownian motion. The estimates for exponential martingales of G-Brownian motion are important for our arguments.

We study the motion of the hypersurface \((\gamma _t)_{t\ge 0}\) evolving according to the mean curvature perturbed by \(\dot{w}^Q\), the formal time derivative of the Q-Wiener process \({w}^Q\), in a two-dimensional bounded domain. Namely, we consider the equation describing the evolution of \(\gamma _t\) as a stochastic partial differential equation (SPDE) with a multiplicative noise in the Stratonovich

We consider a branching random walk \(S_nX(t)\) on a supercritical random Galton–Watson tree. We compute the Hausdorff and packing dimensions of the level set \(E(\alpha )\) of infinite branches in the boundary of tree endowed with random metric along which the average of \(S_n X(t)/n\) have a given limit point.