Strong Feller Property of Sticky Reflected Distorted Brownian Motion

Using Girsanov transformations we construct from sticky reflected Brownian motion on \([0,\infty )\) a conservative diffusion on \(E:=[0,\infty )^n\), \(n \in \mathbb {N}\), and prove that its transition semigroup possesses the strong Feller property for a specified general class of drift functions. By identifying the Dirichlet form of the constructed process we characterize it as sticky reflected distorted Brownian motion. In particular, the relations of the underlying analytic Dirichlet form methods to the probabilistic methods of random time changes and Girsanov transformations are presented. Our studies of the mathematical model are motivated by its applications to the dynamical wetting model with \(\delta \)-pinning and repulsion.     

The Weak Convergence Theorem for the Distribution of the Maximum of a Gaussian Random Walk and Approximation Formulas for its Moments

In this study, asymptotic expansions of the moments of the maximum (M(β)) of Gaussian random walk with negative drift (???β), β?>?0, are established by using Bell Polynomials. In addition, the weak convergence theorem for the distribution of the random variable Y(β)?≡?2?β?M(β) is proved, and the explicit form of the limit distribution is derived. Moreover, the approximation formulas for the first four moments of the maximum of a Gaussian random walk are obtained for the parameter β?∈?(0.5, 3.2] using meta-modeling.     

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In this paper the local functional limit theorem for incrementsof a Brownian motion is derived with large and small deviations, and the local functionalconvergence rate for increments of Brownian motion in Holder norm with respect to(r,p)capacity is estimated.     

Rates of Convergence for Random Walk Summation

We prove a Marcinkiewicz-Zygmund type strong law of large numbersfor random walk summation methods. We show that the rate ofconvergence of this type of sums is equivalent to the existenceof moments of the summands.     

Exponential Convergence in Probability for Empirical Means of Brownian Motion and of Random Walks

Given a Brownian motion (B _t) _t0 in R ^d and a measurable real function f on R ^d belonging to the Kato class, we show that 1/t ₀ ^t f(B _s ) ds converges to a constant z with an exponential rate in probability if and only if f has a uniform mean z. A similar result is also established in the case of random walks.     

Weak Convergence of a Planar Random Evolution to the Wiener Process

The weak convergence of the distributions of a symmetrical random evolution in a plane controlled by a continuous-time homogeneous Markov chain with n, n3, states to the distribution of a two-dimensional Brownian motion, as the intensity of transitions tends to infinity, is proved.     

A Darling-Erdos-Type Theorem for Standardized Random Walk Summation

In this paper we consider the random walk summation method whichincludes, for example, the Borel, Euler, Meyer-König andValiron methods, and obtain a Darling-Erds-type limit theoremfor the maximum of normalized sums defined by random walk summation.     

A Functional Limit Theorem for Random Walk Conditioned to Stay Non-Negative

In this paper we consider an aperiodic integer-valued randomwalk S and a process S* that is a harmonic transform of S killedwhen it first enters the negative half; informally, S* is ‘Sconditioned to stay non-negative’. If S is in the domainof attraction of the standard normal law, without centring,a suitably normed and linearly interpolated version of S convergesweakly to standard Brownian motion, and our main result is thatunder the same assumptions a corresponding statement holds forS*, the limit of course being the three-dimensional Bessel process.As this process can be thought of as Brownian motion conditionedto stay non-negative, in essence our result shows that the interchangeof the two limit operations is valid. We also establish somerelated results, including a local limit theorem for S*, anda bivariate renewal theorem for the ladder time and height process,which may be of independent interest.     

Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time

We consider a branching random walk on \({\mathbb {R}}\) with a stationary and ergodic environment \(\xi =(\xi _n)\) indexed by time \(n\in {\mathbb {N}}\). Let \(Z_n\) be the counting measure of particles of generation n and \(\tilde{Z}_n(t)=\int \mathrm{e}^{tx}Z_n(\mathrm{d}x)\) be its Laplace transform. We show the \(L^p\) convergence rate and the uniform convergence of the martingale \(\tilde{Z}_n(t)/{\mathbb {E}}[\tilde{Z}_n(t)|\xi ]\), and establish a moderate deviation principle for the measures \(Z_n\).     

Invariant Measures for Transient Reflected Brownian Motion in a Wedge: Existence and Uniqueness

Our main result is the existence and uniqueness of an invariant measure for reflected Brownian motion (RBM) in a wedge that is transient to ∞. We also consider this question for RBM that has been killed at the corner of the wedge.     

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??The local limit theorems for the minimum of a random walk with Markovian increments is given, with using Presman's factorization theory. This result implies the asymptotic behaviour of the survival probability for a critical branching process in Markovian depended random environment.     

On the Exit Distribution of Partially Reflected Brownian Motion in Planar Domains

We show that the dimension of the exit distribution of planar partially reflected Brownian motion can be arbitrarily close to 2.     

Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds

Let be a one-parameter family of positive integral operators on a locally compact space . For a possibly non-uniform partition of define a finite measure on the path space by using a) for the transition between any two consecutive partition times of distance and b) a suitable continuous interpolation scheme (e.g. Brownian bridges or geodesics). If necessary normalize the result to get a probability measure. We prove a version of Chernoff's theorem of semigroup theory and tightness results which yield convergence in law of such measures as the partition gets finer. In particular let be a closed smooth submanifold of a manifold . We prove convergence of Brownian motion on , conditioned to visit at all partition times, to a process on whose law has a density with respect to Brownian motion on which contains scalar, mean and sectional curvatures terms. Various approximation schemes for Brownian motion on are also given.      

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We establish a quenched central limit theorem (CLT) for the branching Brownian motion with random immigration in dimension $d\geq4$. The limit is a Gaussian random measure, which is the same as the annealed central limit theorem, but the covariance kernel of the limit is different from that in the annealed sense when d=4.     

A Random Matrix Approximation for the Non-commutative Fractional Brownian Motion

A functional limit theorem for the empirical measure-valued process of eigenvalues of a matrix fractional Brownian motion is obtained. It is shown that the limiting measure-valued process is the non-commutative fractional Brownian motion recently introduced by Nourdin and Taqqu (J Theor Probab 27:220–248, 2014). Young and Skorohod stochastic integral techniques and fractional calculus are the main tools used.