We study Karhunen-Loeve expansions of the process $(X_t^{(α)})_{t\in[0,T)}$ given by the stochastic differential equation $dX_t^{(α)} = -\fracα{T-t} X_t^{(α)} dt+ dB_t,$ $t\in[0,T),$ with an initial condition $X_0^{(α)}=0,$ where $α>0,$ $T\in(0,\infty)$ and $(B_t)_{t\geq 0}$ is a standard Wiener process. This process is called an $α$-Wiener bridge or a scaled Brownian bridge, and in the special case of $α=1$ the usual Wiener bridge. We present weighted and unweighted Karhunen-Loeve expansions of $X^{(α)}$. As applications, we calculate the Laplace transform and the distribution function of the $L^2[0,T]$-norm square of $X^{(α)}$ studying also its asymptotic behavior (large and small deviation).
We provide a detailed derivation of the Karhunen-Loève expansion of a stochastic process. We also discuss briefly Gaussian processes, and provide a simple numerical study for the purpose of illustration.
We study a simple decision problem on the scaling parameter in the $α$-Brownian bridge $X^{(α)}$ on the interval $[0,1]$: given two values $α_0, α_1 \geq 0$ with $α_0 + α_1 \geq 1$ and some time $0 \leq T \leq 1$ we want to test $H_0: α= α_0$ vs. $H_1: α= α_1$ based on the observation of $X^{(α)}$ until time $T$. The likelihood ratio can be written as a functional of a quadratic form $ψ(X^{(α)})$ of $X^{(α)}$. In order to calculate the distribution of $ψ(X^{(α)})$ under the null hypothesis, we generalize the Karhunen-Loève Theorem to positive finite measures on $[0,1]$ and compute the Karhunen…
For $γ>-{1/2}$, we provide the Karhunen-Loève expansion of the weighted mean-centered Wiener process, defined by \[W _γ(t)=\frac{1}{\sqrt{1+2γ}}\Big\{W\big(t^{1+2γ}\big)- \int_0^1W\big(u^{1+2γ}\big)du\Big\},\] for $t\in(0,1]$. We show that the orthogonal functions in these expansions have simple expressions in term of Bessel functions. Moreover, we obtain that the $L^2[0,1]$ norm of $W_γ$ is identical in distribution with the $L^2[0,1]$ norm of the weighted Brownian bridge $t^γB(t)$.
This paper deals with the study, from a probabilistic point of view, of logistic-type differential equations with uncertainties. We assume that the initial condition is a random variable and the diffusion coefficient is a stochastic process. The main objective is to obtain the first probability density function, $f_1(p,t)$, of the solution stochastic process, $P(t,ω)$. To achieve this goal, first the diffusion coefficient is represented via a truncation of order $N$ of the Karhunen-Loève expansion, and second, the Random Variable Transformation technique is applied. In this manner, approximati…
We establish a Karhunen-Lo`eve expansion for generic centered, second order stochastic processes, which does not rely on topological assumptions. We further investigate in which norms the expansion converges and derive exact average rates of convergence for these norms. For Gaussian processes we additionally prove certain sharpness results in terms of the norm. Moreover, we show that the generic Karhunen-Lo`eve expansion can in some situations be used to construct reproducing kernel Hilbert spaces (RKHSs) containing the paths of a version of the process. As applications of the general theory, …
Karhunen-Loeve expansions (KLE) of stochastic processes are important tools in mathematics, the sciences, economics, and engineering. However, the KLE is primarily useful for those processes for which we can identify the necessary components, i.e., a set of basis functions, and the distribution of an associated set of stochastic coefficients. Our ability to derive these components explicitly is limited to a handful processes. In this paper we derive all the necessary elements to implement the KLE for a square-integrable Levy process. We show that the eigenfunctions are sine functions, identica…
We present a conjecture regarding the expectation of the maxima of $L^2$ norms of sub-vectors of a Gaussian vector; this has application to nonlinear reconstruction.
Given an orthonormal system of $L^{2}(D)$ consistent of continuous functions $(f_{n})_{n}$, with $D \subset \mathbb{R}^{d}$ compact, and given a sequence of strictly positive coefficients $(λ_{n})_{n}$ forming a convergent series, we prove that they consist in the eigenfunctions and eigenvectors of a covariance operator associated to a continuous positive-definite Kernel if and only if the sequence of partial sums $ \sum_{j \leq n} λ_{j} f_{j}^{2} $ is equicontinuous over $D$.
We derive a Karhunen-Loève expansion of the Gauss process $B_t - g(t)\int_0^1 g'(u)\,d B_u$, $t\in[0,1]$, where $(B_t)_{t\in[0,1]}$ is a standard Wiener process and $g:[0,1]\to R$ is a twice continuously differentiable function with $g(0) = 0$ and $\int_0^1 (g'(u))^2\,d u =1$. This process is an important limit process in the theory of goodness-of-fit tests. We formulate two special cases with the function $g(t)=\frac{\sqrt{2}}π\sin(πt)$, $t\in[0,1]$, and $g(t)=t$, $t\in[0,1]$, respectively. The latter one corresponds to the Wiener bridge over $[0,1]$ from $0$ to $0$.
This article introduces adaptive Fourier decomposition (AFD) type methods, emphasizing on those that can be applied to stochastic processes and random fields, mainly including stochastic adaptive Fourier decomposition and stochastic pre-orthogonal adaptive Fourier decomposition. We establish their algorithms based on the covariant function and prove that they enjoy the same convergence rate as the Karhunen-Loève (KL) decomposition. The AFD type methods are compared with the KL decomposition. In contrast with the latter, the AFD type methods do not need to compute eigenvalues and eigenfunctions…
A quantum Karhunen-Loeve expansion and quadratic-exponential functionals for linear quantum stochastic systems
Research School of Electrical, Energy and Materials Engineering, College of Engineering and Computer Science, Australian National University（澳大利亚国立大学工程与计算机科学学院电气、能源与材料工程研究学院）
This paper considers one-mode open quantum harmonic oscillators with a pair of conjugate position and momentum variables driven by vacuum bosonic fields according to a linear quantum stochastic differential equation. Such systems model cavity resonators in quantum optical experiments. Assuming that the quadratic Hamiltonian of the oscillator is specified by a positive definite energy matrix, we consider a modified version of the quantum Karhunen-Loeve expansion of the system variables proposed recently. The expansion employs eigenvalues and eigenfunctions of the two-point commutator kernel for…
We use a conditional Karhunen-Loève (KL) model to quantify and reduce uncertainty in a stochastic partial differential equation (SPDE) problem with partially-known space-dependent coefficient, $Y(x)$. We assume that a small number of $Y(x)$ measurements are available and model $Y(x)$ with a KL expansion. We achieve reduction in uncertainty by conditioning the KL expansion coefficients on measurements. We consider two approaches for conditioning the KL expansion: In Approach 1, we condition the KL model first and then truncate it. In Approach 2, we first truncate the KL expansion and then condi…
Hierarchical Gaussian Random Fields for Multilevel Markov Chain Monte Carlo: Coupling Stochastic Partial Differential Equation and The Karhunen-Loève Decomposition
In this article, we develop a new approach to functional quantization, which consists in discretizing only a finite subset of the Karhunen-Loève coordinates of a continuous Gaussian semimartingale $X$. Using filtration enlargement techniques, we prove that the conditional distribution of $X$ knowing its first Karhunen-Loève coordinates is a Gaussian semimartingale with respect to a bigger filtration. This allows us to define the partial quantization of a solution of a stochastic differential equation with respect to $X$ by simply plugging the partial functional quantization of $X$ in the SDE. …
Fluctuation theorem is one of the major achievements in the field of nonequilibrium statistical mechanics during the past two decades. Steady-state fluctuation theorem of sample entropy production rate in terms of large deviation principle for diffusion processes have not been rigorously proved yet due to technical difficulties. Here we give a proof for the steady-state fluctuation theorem of a diffusion process in magnetic fields, with explicit expressions of the free energy function and rate function. The proof is based on the Karhunen-Loéve expansion of complex-valued Ornstein-Uhlenbeck pro…
We present two linear relations between an arbitrary (real tempered second order) generalized stochastic process over $\mathbb{R}^{d}$ and White Noise processes over $\mathbb{R}^{d}$. The first is that any generalized stochastic process can be obtained as a linear transformation of a White Noise. The second indicates that, under dimensional compatibility conditions, a generalized stochastic process can be linearly transformed into a White Noise. The arguments rely on the regularity theorem for tempered distributions, which is used to obtain a mean-square continuous stochastic process which is …
