Homogeneous White Noise Research Articles

A Feynman–Kac approach for the spatial derivative of the solution to the Wick stochastic heat equation driven by time homogeneous white noise

We consider the (unique) mild solution [Formula: see text] of a one-dimensional stochastic heat equation on [Formula: see text] driven by time-homogeneous white noise in the Wick–Skorokhod sense. The main result of this paper is the computation of the spatial derivative of [Formula: see text], denoted by [Formula: see text], and its representation as a Feynman–Kac type closed form. The chaos expansion of [Formula: see text] makes it possible to find its (optimal) Hölder regularity especially in space.

Thermal Stresses in Bodies with Random Initial Temperature Distribution

This paper describes a stochastic analysis in the one-dimensional uncoupled linear thermoelasticity. The autocorrelation functions of transient temperature and thermal stresses are analytically obtained in seven simple geometries: infinite plate, infinite strip, hollow sphere, infinite body with a spherical cavity, infinite hollow cylinder, annular disk and infinite body with a cylindrical hole, which have a random initial temperature distribution. Numerical calculations are performed for some geometries under the condition that the randomness in the initial temperature is given as a homogeneous white noise. The transient behavior of the mean square temperature and thermal stresses is illustrated and it is observed in objects confined to a finite region that whereas the maximum value of the mean square thermal stresses occurs within the objects shortly after the heat flow begins, it appears at a lateral surface after a length of time has elapsed.

Turbulence, raindrops and the l1/2 number density law

Using a unique data set of three-dimensional drop positions and masses (the HYDROP experiment), we show that the distribution of liquid water in rain displays a sharp transition between large scales which follow a passive scalar-like Corrsin–Obukhov (k-5/3) spectrum and a small-scale statistically homogeneous white noise regime. We argue that the transition scale lc is the critical scale where the mean Stokes number (= drop inertial time/turbulent eddy time) Stl is unity. For five storms, we found lc in the range 45–75 cm with the corresponding dissipation scale Stη in the range 200–300. Since the mean interdrop distance was significantly smaller (≈ 10 cm) than lc we infer that rain consists of ‘patches’ whose mean liquid water content is determined by turbulence with each patch being statistically homogeneous. For l>lc, we have Stl<1 and due to the observed statistical homogeneity for l<lc, we argue that we can use Maxey's relations between drop and wind velocities at coarse grained resolution lc. From this, we derive equations for the number and mass densities (n and ρ) and their variance fluxes (ψ and χ). By showing that χ is dissipated at small scales (with lρ, diss≈lc) and ψ over a wide range, we conclude that ρ should indeed follow Corrsin–Obukhov k-5/3 spectra but that n should instead follow a k-2 spectrum corresponding to fluctuations scaling as Δρ∝l1/3 and Δn∝l1/2. While the Corrsin–Obukhov law has never been observed in rain before, its discovery is perhaps not surprising; in contrast the Δn≈l1/2 number density law is quite new. The key difference between the Δρ, Δn laws is the fact that the microphysics (coalescence, breakup) conserves drop mass, but not numbers of particles. This implies that the timescale for the transfer of the density variance flux χ is determined by the strongly scale-dependent turbulent velocity whereas the timescale for the transfer of the number variance flux ψ is determined by the weakly scale-dependent drop coalescence speed. We argue that the l1/2 law may also hold (although in a slightly different form) for cloud drops. Because they are consequences of symmetries, we expect the l1/3, l1/2 laws to be robust. Since the large-scale turbulence determines the n and ρ fields which are the 0th and 1st moments of the drop-size distribution, they constrain the microphysics: dimensional analysis shows that the cumulative probability distribution of nondimensional drop mass should be a universal function dependent only on scale; we confirm this empirically. The combination of number and mass density laws can be used to develop stochastic compound multifractal Poisson processes which are useful new tools for studying and modelling rain. We discuss the implications of this for the rain rate statistics including a simplified model, which can explain the observed rain rate spectra.

Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: itô's case

In this article we prove new results concerning the long-time behaviour of random fields that are solutions in some generalized sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise. Our main results state that these random fields eventually converge with probability. one to a global attractor represented by a single random variable whose properties we investigate in detail. We analyze the partial differential equations of this article in light Itô's stochastic calculus and thereby obtain stabilization and stability results which are substantially different from our earlier results concerning their interpretation in the sense of Stratonovitch. In particular, the asymptotic properties of the random fields that we investigate here exhibit no recurrence and oscillatory properties

Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Stratonovitch's case

In this article we prove new results concerning the long-time behavior of random fields that are solutions in some sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise. Our main results state that these random fields eventually homogeneize with respect to the spatial variable and finally converge to a non-random global attractor which consists of two spatially and temporally homogeneous asymptotic states. More precisely, we prove that the random fields either stabilize exponentially rapidly with probability one around one of the asymptotic states, or that they set out to oscillate between them. In the first case we can also determine exactly the corresponding Lyapunov exponents. In the second case we prove that the random fields are in fact recurrent in that they can reach every point between the two asymptotic states in a finite time with probability one. In both cases we also interpret our results in terms of stability properties of the global attractor and we provide estimates for the average time that the random fields spend in small neighborhoods of the asymptotic states. Our methods of proof rest upon the use of a suitable regularization of the Brownian motion along with a related Wong-Zakai approximation procedure.

On the large-time dynamics of a class of parabolic equations subjected to homogeneous white noise: Itô's case

We present new results concerning the long-time behavior of random fields that are solutions in some variational sense to a class of semilinear parabolic equations subjected to homogeneous white noise. Our main results state that these random fields eventually converge to a random global attractor. This attractor is represented by a single random variable which, with probability one, takes on its values in a two element-set consisting of spatially and temporally homogeneous asymptotic states. We can then completely elucidate the behavior of the random fields around it.

An Integral Inequality and Its Application to a Problem of Stabilization by Noise

We prove the integral inequality [formula] for nonnegative functions b and g with g(∞) = 0 as well as various extensions. We apply it to g( x) = exp(−α x 2), α > 0, thereby answering the question of whether an explosive scalar equation dy/ dt = b( y( t)) can be made nonexplosive by adding homogeneous white noise in the negative.

Noise and array configuration considerations for matched‐field processing in phone and mode space

The conditions under which mode space matched‐field processing offers performance gains relative to phone space matched‐field processing are investigated. Particular emphasis is placed on determining the effects of spatially correlated versus spatially uncorrelated noise fields and choice of array configuration on signal detection. Varying amounts of homogeneous white noise and spatially correlated noise as calculated by a normal mode noise model were combined with the field due to a submerged source to produce simulated phone space covariance matrices for a range of array configuration parameters. A phone‐to‐mode spacing mapping was then used to obtain the corresponding cross‐amplitude correlation matrices. The results of both conventional and MLM processing were analyzed and the trade‐off between the number of phones and percent of the water column spand was investigated.