Almost sure subexponential decay rates of scalar Ito-Volterra equations.

The paper studies the subexponential convergence of solutions of scalar Itˆo-Volterra equations. First, we consider linear equations with an instantaneous multiplicative noise term with intensity . If the kernel obeys lim t!1 k0(t)/k(t) = 0, and another nonexponential decay criterion, and the solution X tends to zero as t ! 1, then limsup t!1 log |X(t)| log(tk(t)) = 1 − (||), a.s. where the random variable (||) ! 0 as ! 1 a.s. We also prove a decay result for equations with a superlinear diffusion coefficient at zero. If the deterministic equation has solution which is uniformly asymptotically stable, and the kernel is subexponential, the decay rate of the stochastic problem is exactly the same as that of the underlying deterministic problem

Polynomial asymptotic stability of damped stochastic differential equations.

The paper studies the polynomial convergence of solutions of a scalar nonlinear Itˆo stochastic differential equation dX(t) = −f(X(t)) dt + (t) dB(t) where it is known, a priori, that limt!1 X(t) = 0, a.s. The intensity of the stochastic perturbation is a deterministic, continuous and square integrable function, which tends to zero more quickly than a polynomially decaying function. The function f obeys limx!0 sgn(x)f(x)/|x| = a, for some > 1, and a > 0. We study two asymptotic regimes: when tends to zero sufficiently quickly the polynomial decay rate of solutions is the same as for the deterministic equation (when 0). When decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established. Results which establish the necessity for to decay polynomially in order to guarantee the almost sure polynomial decay of solutions are also proven

On the almost sure running maxima of solutions of affine stochastic functional differential equations

This paper studies the large fluctuations of solutions of scalar and finite-dimensional affine stochastic functional differential equations with finite memory as well as related nonlinear equations. We find conditions under which the exact almost sure growth rate of the running maximum of each component of the system can be determined, both for affine and nonlinear equations. The proofs exploit the fact that an exponentially decaying fundamental solution of the underlying deterministic equation is sufficient to ensure that the solution of the affine equation converges to a stationary Gaussian process

Subexponential solutions of scalar linear integro-differential equations with delay

This paper considers the asymptotic behaviour of solutions of the scalar linear convolution integro-differential equation with delay x0(t) = − n Xi=1 aix(t − i) + Z t 0 k(t − s)x(s) ds, t > 0, x(t) = (t), − t 0, where = max1in i. In this problem, k is a non-negative function in L1(0,1)\C[0,1), i 0, ai > 0 and is a continuous function on [−, 0]. The kernel k is subexponential in the sense that limt!1 k(t)(t)−1 > 0 where is a positive subexponential function. A consequence of this is that k(t)et ! 1 as t ! 1 for every > 0

On the local dynamics of polynomial difference equations with fading stochastic perturbations

We examine the stability-instability behaviour of a polynomial difference equa- tion with state-independent, asymptotically fading stochastic perturbations. We find that the set of initial values can be partitioned into a stability region, an instability region, and a region of unknown dynamics that is in some sense \small". In the ¯rst two cases, the dynamic holds with probability at least 1 ¡ °, a value corresponding to the statistical notion of a confidence level. Aspects of an equation with state-dependent perturbations are also treated. When the perturbations are Gaussian, the difference equation is the Euler-Maruyama dis- cretisation of an It^o-type stochastic differential equation with solutions displaying global a.s. asymptotic stability. The behaviour of any particular solution of the difference equation can be made consistent with the corresponding solution of the differential equation, with probability 1 ¡ °, by choosing the stepsize parameter sufficiently small. We present examples illustrating the relationship between h, ° and the size of the stability region

Exponential asymptotic stability for linear volterra equations

This note studies the exponential asymptotic stability of the zero solution of the linear Volterra equation x˙ (t) = Ax(t) + t 0 K(t − s)x(s) ds by extending results in the paper of Murakami “Exponential Asymptotic Stability for scalar linear Volterra Equations”, Differential and Integral Equations, 4, 1991. In particular, when K isi ntegrable and has entries which do not change sign, and the equation has a uniformly asymptotically stable solution, exponential asymptotic stability can be identified by an exponential decay condition on the entries of K

Spurious oscillation in a uniform Euler discretisation of linear stochastic differential equations with vanishing delay

AbstractWe investigate the oscillatory behaviour of a random Euler-type difference equation, intended to serve as a discrete model of a linear Itô stochastic differential equation with vanishing delay. The oscillatory behaviour of the continuous process satisfying this differential equation was partially described in Appleby and Kelly [Asymptotic and oscillatory properties of linear stochastic delay differential equations with vanishing delay, Funct. Differential Equation 11(3–4) (2004) 235–265.] The construction of a discrete model that successfully mimics some of the properties of the continuous process would simplify the analysis, allowing the partial description to be completed. However, care must be taken; a uniform Euler discretisation yields spurious oscillatory behaviour. We present a complete analysis of the uniform scheme

Statistical estimation with Kronecker products in positron emission tomography

AbstractA method for linear statistical analysis of multidimensional imaging data is presented. It is applicable for a class of design and covariance matrices which involve Kronecker products. An efficient algorithm which allows for application of the method to large multidimensional data volumes is given. This has direct application to neuroimaging, and here the technique is applied to positron emission tomography (PET) data. PET is an in vivo functional imaging technique that measures biological processes such as blood flow and receptor concentrations. Here, the algorithm is used to correct for resolution degradation in these images. This process is typically referred to as PET partial volume correction. Examples involving both measured phantom and human data are given. This rapid algorithm leads to advances in the types of quantitative brain imaging studies that can be performed

On the Validity of the Imbert-Fick Law: Mathematical Modelling of Eye Pressure Measurement

YesOphthalmologists rely on a device known as the Goldmann applanation tonometer to make intraocular pressure (IOP) measurements. It measures the force required to press a flat disc against the cornea to produce a flattened circular region of known area. The IOP is deduced from this force using the Imbert-Fick principle. However, there is scant analytical justification for this analysis. We present a mathematical model of tonometry to investigate the relationship between the pressure derived by tonometry and the IOP. An elementary equilibrium analysis suggests that there is no physical basis for traditional tonometric analysis. Tonometry is modelled using a hollow spherical shell of solid material enclosing an elastic liquid core, with the shell in tension and the core under pressure. The shell is pressed against a rigid flat plane. The solution is found using finite element analysis. The shell material is anisotropic. Values for its elastic constants are obtained from literature except where data are unavailable, when reasonable limits are explored. The results show that the force measured by the Goldmann tonometer depends on the elastic constant values. The relationship between the IOP and the tonometer readings is complex, showing potentially high levels of inaccuracy that depend on IOP