Unilateral small deviations of processes related to the fractional Brownian motion

Let x(s)$x\left(s\right)$, s∈R^d$s\in R^d$ be a Gaussian self-similar random process of index H$H$. We consider the problem of log-asymptotics for the probability _pT$p_T$ that x(s)$x\left(s\right)$, x(0)=0$x\left(0\right)=0$ does not exceed a fixed level in a star-shaped expanding domain T⋅Δ$T\cdot \Delta$ as T→∞$T\to \infty$. We solve the problem of the existence of the limit, θ?lim(−log_pT)/(logT)^D$\theta ?lim(-log{p}_T)/{(logT)}^D$, T→∞$T\to \infty$, for the fractional Brownian sheet x(s)$x\left(s\right)$, s∈[0,T]²$s\in {[0,T]}^2$ when D=2$D=2$, and we estimate θ$\theta$ for the integrated fractional Brownian motion when D=1$D=1$.     

Large deviations of infinite intersections of events in Gaussian processes

Consider events of the form {_Zs≥ζ(s),s∈S}$\{Z_s\ge \zeta \left(s\right),s\in S\}$, where Z$Z$ is a continuous Gaussian process with stationary increments, ζ$\zeta$ is a function that belongs to the reproducing kernel Hilbert space R$R$ of process Z$Z$, and S⊂R$S\subset \mathbb{R}$ is compact. The main problem considered in this paper is identifying the function β^∗∈R${\beta }^{\ast }\in R$ satisfying β^∗(s)≥ζ(s)${\beta }^{\ast }\left(s\right)\ge \zeta \left(s\right)$ on S$S$ and having minimal R$R$-norm. The smoothness (mean square differentiability) of Z$Z$ turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ(s)=s$\zeta \left(s\right)=s$ for s∈[0,1]$s\in [0,1]$ and Z$Z$ is either a fractional Brownian motion or an integrated Ornstein–Uhlenbeck process.     

The scaling limit of Poisson-driven order statistics with applications in geometric probability

Let _ηt${\eta }_t$ be a Poisson point process of intensity t≥1$t\ge 1$ on some state space Y$\mathbb{Y}$ and let f$f$ be a non-negative symmetric function on Y^k${\mathbb{Y}}^k$ for some k≥1$k\ge 1$. Applying f$f$ to all k$k$-tuples of distinct points of _ηt${\eta }_t$ generates a point process _ξt${\xi }_t$ on the positive real half-axis. The scaling limit of _ξt${\xi }_t$ as t$t$ tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the m$m$-th smallest point of _ξt${\xi }_t$ is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener–Itô chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen–Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as k$k$-flats, random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry.     

Rate of escape and central limit theorem for the supercritical Lamperti problem

The study of discrete-time stochastic processes on the half-line with mean drift at x$x$ given by _μ1(x)→0${\mu }_1\left(x\right)\to 0$ as x→∞$x\to \infty$ is known as Lamperti’s problem  . We give sharp almost-sure bounds for processes of this type in the case where _μ1(x)${\mu }_1\left(x\right)$ is of order x^−β$x^{-\beta }$ for some β∈(0,1)$\beta \in (0,1)$. The bounds are of order t^1/(1+β)$t^{1/(1+\beta )}$, so the process is super-diffusive but sub-ballistic (has zero speed). We make minimal assumptions on the moments of the increments of the process (finiteness of (2+2β+ε)$(2+2\beta +\epsilon )$-moments for our main results, so fourth moments certainly suffice) and do not assume that the process is time-homogeneous or Markovian. In the case where x^β_μ1(x)$x^{\beta }{\mu }_1\left(x\right)$ has a finite positive limit, our results imply a strong law of large numbers, which strengthens and generalizes earlier results of Lamperti and Voit. We prove an accompanying central limit theorem, which appears to be new even in the case of a nearest-neighbour random walk, although our result is considerably more general. This answers a question of Lamperti. We also prove transience of the process under weaker conditions than those that we have previously seen in the literature. Most of our results also cover the case where β=0$\beta =0$. We illustrate our results with applications to birth-and-death chains and to multi-dimensional non-homogeneous random walks.     

A limit theorem for the time of ruin in a Gaussian ruin problem

For certain Gaussian processes X(t)$X\left(t\right)$ with trend −ct^β$-c{t}^{\beta }$ and variance V²(t)$V^2\left(t\right)$, the ruin time is analyzed where the ruin time is defined as the first time point t$t$ such that X(t)−ct^β≥u$X\left(t\right)-c{t}^{\beta }\ge u$. The ruin time is of interest in finance and actuarial subjects. But the ruin time is also of interest in other applications, e.g. in telecommunications where it indicates the first time of an overflow. We derive the asymptotic distribution of the ruin time as u→∞$u\to \infty$ showing that the limiting distribution depends on the parameters β$\beta$, V(t)$V\left(t\right)$ and the correlation function of X(t)$X\left(t\right)$.     

A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation

We consider the semilinear parabolic equation _ut=Δu+u^p$u_t=\mathrm{\Delta }u+u^p$ on R^N${\mathbb{R}}^N$, where the power nonlinearity is subcritical. We first address the question of existence of entire solutions, that is, solutions defined for all x∈R^N$x\in {\mathbb{R}}^N$ and t∈R$t\in \mathbb{R}$. Our main result asserts that there are no positive radially symmetric bounded entire solutions. Then we consider radial solutions of the Cauchy problem. We show that if such a solution is global, that is, defined for all t?0$t?0$, then it necessarily converges to 0, as t→∞$t\to \infty$, uniformly with respect to x∈R^N$x\in {\mathbb{R}}^N$.     

Local independence of fractional Brownian motion

Let _σ(t,t′)${\sigma }_{(t,t^{\prime })}$ be the sigma-algebra generated by the differences _Xs−_Xs′$X_s-X_{s^{\prime }}$ with s,s^′∈(t,t^′)$s,s^{\prime }\in (t,t^{\prime })$, where _{(Xt)−∞<t<∞}${\left(X_t\right)}_{-\infty <t<\infty }$ is the fractional Brownian motion with Hurst index H∈(0,1)$H\in (0,1)$. We prove that for any two distinct timepoints _t1$t_1$ and _t2$t_2$ the sigma-algebras _{σ(t1−ε,t1+ε)}${\sigma }_{(t_1-\epsilon ,t_1+\epsilon )}$ and _{σ(t2−ε,t2+ε)}${\sigma }_{(t_2-\epsilon ,t_2+\epsilon )}$ are asymptotically independent as ε↘0$\epsilon \searrow 0$. We show the independence in the strong sense that Shannon’s mutual information between the two σ$\sigma$-algebras tends to zero as ε↘0$\epsilon \searrow 0$. Some generalizations and quantitative estimates are also provided.     

Transformation formulas for fractional Brownian motion

We derive a Molchan–Golosov-type integral transform which changes fractional Brownian motion of arbitrary Hurst index K$K$ into fractional Brownian motion of index H$H$. Integration is carried out over [0,t]$[0,t]$, t>0$t>0$. The formula is derived in the time domain. Based on this transform, we construct a prelimit which converges in L²(P)$L^2\left(\mathbb{P}\right)$-sense to an analogous, already known Mandelbrot–Van Ness-type integral transform, where integration is over (−∞,t]$(-\infty ,t]$, t>0$t>0$.     

A remark on least-squares Galerkin procedures for pseudohyperbolic equations

In this paper, we introduce two split least-squares Galerkin finite element procedures for pseudohyperbolic equations arising in the modelling of nerve conduction process. By selecting the least-squares functional properly, the procedures can be split into two sub-procedures, one of which is for the primitive unknown variable and the other is for the flux. The convergence analysis shows that both the two methods yield the approximate solutions with optimal accuracy in L²(Ω)$L^2\left(\Omega \right)$ norm for u$u$ and _ut$u_t$ and (L²(Ω))²${\left(L^2\left(\Omega \right)\right)}^2$ norm for the flux σ$\sigma$. Moreover, the two methods get approximate solutions with first-order and second-order accuracy in time increment, respectively. A numerical example is given to show the efficiency of the introduced schemes.     

Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes

We discuss joint temporal and contemporaneous aggregation of N$N$ independent copies of AR(1) process with random-coefficient a∈[0,1)$a\in [0,1)$ when N$N$ and time scale n$n$ increase at different rate. Assuming that a$a$ has a density, regularly varying at a=1$a=1$ with exponent −1<β<1$-1<\beta <1$, different joint limits of normalized aggregated partial sums are shown to exist when N^1/(1+β)/n$N^{1/(1+\beta )}/n$ tends to (i) ∞$\infty$, (ii) 0$0$, (iii) 0<μ<∞$0<\mu <\infty$. The limit process arising under (iii) admits a Poisson integral representation on (0,∞)×C(R)$(0,\infty )\times C\left(\mathbb{R}\right)$ and enjoys ‘intermediate’ properties between fractional Brownian motion limit in (i) and sub-Gaussian limit in (ii).     

Ruin probability in the presence of risky investments

We consider an insurance company in the case when the premium rate is a bounded non-negative random function _ct$c_t$ and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brownian motion with mean return a   and volatility σ>0$\sigma >0$. If β?2a/σ²-1>0$\beta ?2a/{\sigma }^2-1>0$ we find exact the asymptotic upper and lower bounds for the ruin probability Ψ(u)$\Psi (u)$ as the initial endowment u   tends to infinity, i.e. we show that _C*u^-β?Ψ(u)?C^*u^-β$C_{*}{u}^{-\beta }?\Psi (u)?C^{*}{u}^{-\beta }$ for sufficiently large u  . Moreover if _ct=c^*e^γt$c_t=c^{*}{\mathrm{e}}^{\gamma t}$ with γ?0$\gamma ?0$ we find the exact asymptotics of the ruin probability, namely Ψ(u)∼u^-β$\Psi (u)\sim u^{-\beta }$. If β?0$\beta ?0$, we show that Ψ(u)=1$\Psi (u)=1$ for any u?0$u?0$.     

Blow-up and propagation of disturbances for fast diffusion equations

This paper is concerned with the Cauchy problem for the fast diffusion equation _ut−Δu^m=αu^_p1$u_t-\Delta u^m=\alpha u^{p_1}$ in R^N${\mathbb{R}}^N$ (N≥1$N\ge 1$), where m∈(0,1)$m\in (0,1)$, _p1>1$p_1>1$ and α>0$\alpha >0$. The initial condition _u0$u_0$ is assumed to be continuous, nonnegative and bounded. Using a technique of subsolutions, we set up sufficient conditions on the initial value _u0$u_0$ so that u(t,x)$u(t,x)$ blows up in finite time, and we show how to get estimates on the profile of u(t,x)$u(t,x)$ for small enough values of t>0$t>0$.     

Heat-kernel estimates for random walk among random conductances with heavy tail

We study models of discrete-time, symmetric, Z^d${\mathbb{Z}}^d$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances _ωxy∈[0,1]${\omega }_{xy}\in [0,1]$, with polynomial tail near 0 with exponent γ>0$\gamma >0$. We first prove for all d≥5$d\ge 5$ that the return probability shows an anomalous decay (non-Gaussian) that approaches (up to sub-polynomial terms) a random constant times n⁻²$n^{-2}$ when we push the power γ$\gamma$ to zero. In contrast, we prove that the heat-kernel decay is as close as we want, in a logarithmic sense, to the standard decay n^−d/2$n^{-d/2}$ for large values of the parameter γ$\gamma$.     

Asymptotics of a Brownian ratchet for protein translocation

Protein translocation in cells has been modelled by Brownian ratchets  . In such models, the protein diffuses through a nanopore. On one side of the pore, ratcheting molecules bind to the protein and hinder it to diffuse out of the pore. We study a Brownian ratchet by means of a reflected Brownian motion _(Xt)t≥0${\left(X_t\right)}_{t\ge 0}$ with a changing reflection point _(Rt)t≥0${\left(R_t\right)}_{t\ge 0}$. The rate of change of _Rt$R_t$ is γ(_Xt−_Rt)$\gamma (X_t-R_t)$ and the new reflection boundary is distributed uniformly between _Rt−$R_{t-}$ and _Xt$X_t$. The asymptotic speed of the ratchet scales with γ^1/3${\gamma }^{1/3}$ and the asymptotic variance is independent of γ$\gamma$.