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This article deals with the asymptotic behaviour as $t\to +\infty$ of the survival function $P[T > t],$ where $T$ is the first passage time above a non negative level of a random process starting from zero. In many cases of physical…
A coagulation process is studied in a set of random masses, in which two randomly chosen masses and the smallest mass of the set multiplied by some fixed parameter $\omega\in [-1,1]$ are iteratively added. Besides masses (or primary…
The aim of this paper is to study the laws of the exponential functionals of the processes $X$ with independent increments, namely $$I_t= \int _0^t\exp(-X_s)ds, \,\, t\geq 0,$$ and also $$I_{\infty}= \int _0^{\infty}\exp(-X_s)ds.$$ Under…
Reformulated uniform asymptotic expansions are derived for ordinary differential equations having a large parameter and a simple turning point. These involve Airy functions, but not their derivatives, unlike traditional asymptotic…
A large class of evolutionary processes can be modeled by a rule which involves self-replication of some physical quantity with a non local rescaling. I show that a class of such models are exactly solvable -- in the discrete as well as…
In this article the almost semi-continuous step-process $\xi (t)$ is considered. The conditional characteristic functions of the jumps of $\xi (t)$ have the form $\mathrm{E} [ e^{i\alpha \xi_k}/\xi_k>0 ]=c(c-i\alpha)^{-1}$. For such…
In this article, we study the asymptotic behaviour of L\'evy processes with no positive jumps conditioned to stay positive. We establish integral tests for the lower envelope at 0 and at $+\infty$ and an analogue of Khintchin's law of the…
In this paper we establish functional Erd\H{o}s-Renyi laws for L\'evy processes, i.e. limit theorems for sets of functions on [0,1] associated to their increments. First, we determine precise conditions under which, in a general framework,…
We discuss some surprising phenomena from basic calculus related to oscillating functions and to the theorem on the differentiability of inverse functions. Among other things, we see that a continuously differentiable function with a strict…
The time evolution of correlation functions in statistical systems is described by an exact functional differential equation for the corresponding generating functionals. This allows for a systematic discussion of non-equilibrium physics…
We show that a substantial portion of stochastic calculus can be developed along similar lines to ordinary calculus, with derivative-based concepts driving the development. We define a notion of stopping derivative, which is a form of right…
We prove that two fixed univariate functions, namely, an arbitrary continuous non-affine function and a concrete affine function, are sufficient to approximate continuous functions of one variable under the operations of addition and…
Let \xi_t, t\in[0,T], be a strong Markov process with values in a complete separable metric space (X,\rho) and with transition probability function P_{s,t}(x,dy), 0\le s\le t\le T, x\in X. For any h\in[0,T] and a>0, consider the function…
In this paper, we will extend the falling and rising factorial transforms \cite{ref. 1} which in this case every arbitrary function can be applied. Then, the properties of these transforms will be investigated and some corollaries will be…
We establish limit theorems for the maxima and minima of Airy$_1$ and Airy$_2$ processes (denoted by $\mathcal{A}_1(\cdot)$ and $\mathcal{A}_2(\cdot)$ respectively) over growing intervals. In particular, we identify the finite non-zero…
We obtain approximation results for general positive linear operators satisfying mild conditions, when acting on discontinuous functions and absolutely continuous functions having discontinuous derivatives. The upper bounds, given in terms…
In computational practice, most attention is paid to rational approximations of functions and approximations by the sum of exponents. We consider a wide enough class of nonlinear approximations characterized by a set of two required…
We consider a random walk with a negative drift and with a jump distribution which under Cram\'er's change of measure belongs to the domain of attraction of a spectrally positive stable law. If conditioned to reach a high level and suitably…
We establish an It\^o-type formula for finite $p$-variation paths with jumps for arbitrary $p\geq 1$. The formula is stated in a fully pathwise form and separates the reduced rough integral from explicit left- and right-jump correction…
Let $S_n =X_1+\cdots +X_n$ be an irreducible random walk (r.w.) on the one dimensional integer lattice with zero mean, infinite variance and i.i.d. increments $X_n$. We obtain an upper and lower bounds of the potential function, $a(x)$, of…