# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

All Issues

## Discrete & Continuous Dynamical Systems - B

July 2006 , Volume 6 , Issue 4

Special Issue
Stochastic Dynamics in Finite and Infinite Dimensions:
Theory and Applications

Guest Editors:
Vadim Kaloshin, Sergey Lototsky and Michael Roeckner

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2006, 6(4): i-ii doi: 10.3934/dcdsb.2006.6.4i +[Abstract](1949) +[PDF](25.5KB)
Abstract:
This special issue is a collection of papers on a wide range of topics in stochastic dynamics, including asymptotic problems, numerical methods, and applications in biology, finance, physics, and other areas. A stochastic dynamical system is often modelled by a stochastic differential equation; the papers in this issue deal with the three main classes of stochastic equations: ordinary, partial, and equations with memory. All papers went through a rigorous peer-review process.

2006, 6(4): 651-666 doi: 10.3934/dcdsb.2006.6.651 +[Abstract](2015) +[PDF](274.4KB)
Abstract:
We derive a large deviation principle for a stochastic Navier-Stokes equation for the vorticity of a two-dimensional fluid when the magnitude of the random term tends to zero. The key is the verification of the exponential tightness for the stochastic vorticity.
2006, 6(4): 667-696 doi: 10.3934/dcdsb.2006.6.667 +[Abstract](1810) +[PDF](375.7KB)
Abstract:
This paper studies the pathwise asymptotic stability of the zero solution of scalar stochastic differential equation of Itô type. Specifically, we provide conditions for solutions to converge to zero at given non-exponential rates. The results completely classify the rates of decay of many parameterised families of stochastic differential equations.
2006, 6(4): 697-709 doi: 10.3934/dcdsb.2006.6.697 +[Abstract](2088) +[PDF](261.5KB)
Abstract:
We prove an Oseledets-type theorem for differential equations with a right-hand side that depends on the history of the solution via a random linear operator. This result is applied then to a linear system with memory obtained from the linearized Stochastic Navier--Stokes system on the 2D torus.
2006, 6(4): 711-734 doi: 10.3934/dcdsb.2006.6.711 +[Abstract](1753) +[PDF](400.9KB)
Abstract:
Molecular motors are protein structures that play a central role in accomplishing mechanical work inside a cell. While chemical reactions fuel this work, it is not exactly known how this chemical-mechanical conversion occurs. Recent advances in microbiological techniques have enabled at least indirect observations of molecular motors which in turn have led to significant effort in the mathematical modeling of these motors in the hope of shedding light on the underlying mechanisms involved in intracellular transport. Kinesin which moves along microtubules that are spread throughout the cell is a prime example of the type of motors that are studied in this work. The motion is linked to the presence of a chemical, ATP, but how the ATP is involved in motion is not clearly understood. One commonly used model for the dynamics of kinesin in the biophysics literature is the Brownian ratchet mechanism. In this work, we give a precise mathematical formulation of a Brownian ratchet (or more generally a diffusion ratchet) via an infinite system of stochastic differential equations with reflection. This formulation is seen to arise in the weak limit of a natural discrete space model that is often used to describe motor dynamics in the literature. Expressions for asymptotic velocity and effective diffusivity of a biological motor modeled via a Brownian ratchet are obtained. Linearly progressive biomolecular motors often carry cargos via an elastic linkage. A two-dimensional coupled stochastic dynamical system is introduced to model the dynamics of the motor-cargo pair. By proving that an associated two dimensional Markov process has a unique stationary distribution, it is shown that the asymptotic velocity of a motor pulling a cargo is well defined as a certain Law of Large Number limit, and finally an expression for the asymptotic velocity in terms of the invariant measure of the Markov process is obtained.
2006, 6(4): 735-749 doi: 10.3934/dcdsb.2006.6.735 +[Abstract](1959) +[PDF](236.4KB)
Abstract:
Large-time asymptotic properties of solutions to a semilinear stochastic beam equation with damping in a bounded domain is considered. First an energy inequality and the exponential bound for a linear stochastic beam equation is established. Under appropriate conditions, the existence and uniqueness theorem for the nonlinear stochastic beam equation is proved. Next the main results on the boundedness of global solutions and the exponential stability of an equilibrium solution, in the mean-square sense, are given. Two examples are presented to illustrate some applications of the theorems.
2006, 6(4): 751-760 doi: 10.3934/dcdsb.2006.6.751 +[Abstract](1746) +[PDF](220.2KB)
Abstract:
We consider a degenerate elliptic Kolmogorov--type operator arising from second order stochastic differential equations in $\mathbb R^{n}$ perturbed by noise. We study a realization of such an operator in $L^2$ spaces with respect to an explicit invariant measure, and we prove that it is $m$-dissipative.
2006, 6(4): 761-781 doi: 10.3934/dcdsb.2006.6.761 +[Abstract](2229) +[PDF](322.8KB)
Abstract:
In this article, we prove the convergence of a semi-discrete scheme applied to the stochastic Korteweg--de Vries equation driven by an additive and localized noise. It is the Crank--Nicholson scheme for the deterministic part and is implicit. This scheme was used in previous numerical experiments on the influence of a noise on soliton propagation [8, 9]. Its main advantage is that it is conservative in the sense that in the absence of noise, the $L^2$ norm is conserved. The proof of convergence uses a compactness argument in the framework of $L^2$ weighted spaces and relies mainly on the path-wise uniqueness in such spaces for the continuous equation. The main difficulty relies in obtaining a priori estimates on the discrete solution. Indeed, contrary to the continuous case, Ito formula is not available for the discrete equation.
2006, 6(4): 783-802 doi: 10.3934/dcdsb.2006.6.783 +[Abstract](1772) +[PDF](291.6KB)
Abstract:
We consider propagation and time reversal of wave pulses in a random environment. The focus of our analysis is the development of an expression for the two frequency mutual coherence function for the harmonic wave field. This quantity plays a crucial role in the analysis of many wave propagation phenomena and we illustrate by explicitly considering time reversal in the context of time pulses with a high carrier frequency. In a time-reversal experiment the wave received by an active transducer or antenna (receiver-emitter) array, is recorded in a finite time window and then re-emitted into the medium time reversed, that is, the tails of the recorded signals are sent first. The re-emitted wave pulse will focus approximately on the original source location. We use explicit expressions for the mutual coherence functions and their asymptotic approximations in the regime of long or short propagation distance and a high carrier frequency to analyze the refocusing of the wave pulse in the time reversal experiment. A novel aspect of our analysis is that we are able to characterize precisely the decoherence length in temporal frequency. This allows us to analyze for instance the time reversal experiment when the mirror has a finite aperture in time.
2006, 6(4): 803-828 doi: 10.3934/dcdsb.2006.6.803 +[Abstract](1725) +[PDF](389.8KB)
Abstract:
Taking into account some likeness of moderate deviations (MD) and central limit theorems (CLT), we develop an approach, which made a good showing in CLT, for MD analysis of a family

$S^\kappa_t=\frac{1}{t^\kappa}\int_0^tH(X_s)ds, t\to\infty$

for an ergodic diffusion process $X_t$, provided that $0.5<\kappa<1$, and appropriate $H$. We use a well known decomposition with "corrector'':

$\frac{1}{t^\kappa}\int_0^tH(X_s)ds=$corrector$+\frac{1}{t^\kappa}$ ${M_t}$martingale.

and show that, as in the CLT analysis, the corrector is negligible, and the main contribution in the MD brings the family "$\frac{1}{t^\kappa}M_t, \ t\to\infty.$'' Starting from Freidlin, [7], and finishing by Wu's papers [33]-[37], in the MD study Laplace's transform dominates. In the paper, we replace this technique by "Stochastic exponential'' one, enabling to formulate the MDP conditions in terms of "drift-diffusion'' parameters and $H$. However, a verification of these conditions heavily depends on a specificity of a diffusion model. That is why the paper is named "Examples ...''.

2006, 6(4): 829-834 doi: 10.3934/dcdsb.2006.6.829 +[Abstract](1729) +[PDF](179.0KB)
Abstract:
We prove that if the Black-Scholes formula holds with the spot volatility for call options with all strikes, then the volatility parameter is constant.
2006, 6(4): 835-866 doi: 10.3934/dcdsb.2006.6.835 +[Abstract](1896) +[PDF](280.2KB)
Abstract:
We discuss the Cauchy problem for the stochastic Burgers equation with a nonlinear term of polynomial growth in the whole real line. We also establish the existence of an invariant measure when the equation has an additional zero order dissipation. Many authors have discussed similar issues for the stochastic Burgers equation in various different contexts. But our results for the whole real line are new. Also, our method is different from those of the previous works on the stochastic Burgers equation. In particular, our result on the existence of an invariant measure relies on the author's recent work on a certain class of stochastic evolution equations.
2006, 6(4): 867-880 doi: 10.3934/dcdsb.2006.6.867 +[Abstract](1791) +[PDF](315.8KB)
Abstract:
We consider a stochastic flow in an interval $[-a,b]$, where $a,b>0$. Each point of the interval is driven by the same Brownian path and jumps to zero when it reaches the boundary of the interval. Assuming that $a/b$ is irrational we study the long term behavior of a random measure $\mu_t$, the image of a finite Borel measure $\mu_0$ under the flow. We show that if $\mu_0$ is absolutely continuous with respect to the Lebesgue measure then the time averages of the variance of $\mu_t$ converge to zero almost surely. We also prove that for an arbitrary finite Borel measure $\mu_0$ the Lebesgue measure of the support of $\mu_t$ decreases to zero as $t\to\infty$ with probability one.
2006, 6(4): 881-893 doi: 10.3934/dcdsb.2006.6.881 +[Abstract](2237) +[PDF](227.4KB)
Abstract:
In this paper, we develop a large deviations principle for stochastic delay equations driven by small multiplicative white noise. Both upper and lower large deviations estimates are established.
2006, 6(4): 895-910 doi: 10.3934/dcdsb.2006.6.895 +[Abstract](2312) +[PDF](251.9KB)
Abstract:
We study the time-regularity of the paths of solutions to stochastic partial differential equations (SPDE) driven by additive infinite-dimensional fractional Brownian noise. Sharp sufficient conditions for almost-sure Hölder continuity, and other, more irregular levels of uniform continuity, are given when the space parameter is fixed. Additionally, a result is included on time-continuity when the solution is understood as a spatially Hölder-continuous-function-valued stochastic process. Tools used for the study include the Brownian representation of fractional Brownian motion, and sharp Gaussian regularity results.
2006, 6(4): 911-926 doi: 10.3934/dcdsb.2006.6.911 +[Abstract](1769) +[PDF](315.2KB)
Abstract:
The paper is devoted to studying the problem of ergodicity for the complex Ginzburg--Landau (CGL) equation perturbed by an external random force. We show that the conditions of a simple general result established in [22] are fulfilled for the equation in question. As a consequence, we prove that the corresponding family of Markov processes has a unique stationary distribution, which possesses a mixing property. The result of this paper was announced in the joint work with Sergei Kuksin [14].
2006, 6(4): 927-940 doi: 10.3934/dcdsb.2006.6.927 +[Abstract](1923) +[PDF](224.7KB)
Abstract:
In this paper we investigate the wellposedness of a class of Forward-Backward SDEs. Compared to the existing methods in the literature, our result has the following features: (i) arbitrary time duration; (ii) random coefficients; (iii) (possibly) degenerate forward diffusion; and (iv) no monotonicity condition. As a trade off, we impose some assumptions on the derivatives of the coefficients. A comparison theorem is also proved under the same conditions. This work is motivated by studying numerical methods for FBSDEs.
2006, 6(4): 941-955 doi: 10.3934/dcdsb.2006.6.941 +[Abstract](1905) +[PDF](292.8KB)
Abstract:
An equation that arises in mathematical studies of the transport of pollutants in groundwater and of oil recovery processes is of the form: $-\nabla_{x}\cdot(\kappa(x,\cdot)\nabla_{x}u(x,\omega))=f(x)$, for $x\in D$, where $\kappa(x,\cdot)$, the permeability tensor, is random and models the properties of the rocks, which are not know with certainty. Further, geostatistical models assume $\kappa(x,\cdot)$ to be a log-normal random field. The use of Monte Carlo methods to approximate the expected value of $u(x,\cdot)$, higher moments, or other functionals of $u(x,\cdot)$, require solving similar system of equations many times as trajectories are considered, thus it becomes expensive and impractical. In this paper, we present and explain several advantages of using the White Noise probability space as a natural framework for this problem. Applying properly and timely the Wiener-Itô Chaos decomposition and an eigenspace decomposition, we obtain a symmetric positive definite linear system of equations whose solutions are the coefficients of a Galerkin-type approximation to the solution of the original equation. Moreover, this approach reduces the simulation of the approximation to $u(x,\omega)$ for a fixed $\omega$, to the simulation of a finite number of independent normally distributed random variables.

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