Unbounded Drift Research Articles

The identification problem for complex-valued Ornstein–Uhlenbeck operators in $$L^p(\mathbb {R}^d,\mathbb {C}^N)$$ L p ( R d , C N )

In this paper we study perturbed Ornstein–Uhlenbeck operators $$\begin{aligned} \left[ \mathcal {L}_{\infty } v\right] (x)=A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle -B v(x),\,x\in \mathbb {R}^d,\,d\geqslant 2, \end{aligned}$$ for simultaneously diagonalizable matrices \(A,B\in \mathbb {C}^{N,N}\). The unbounded drift term is defined by a skew-symmetric matrix \(S\in \mathbb {R}^{d,d}\). Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. We prove under certain conditions that the maximal domain \(\mathcal {D}(A_p)\) of the generator \(A_p\) belonging to the Ornstein–Uhlenbeck semigroup coincides with the domain of \(\mathcal {L}_{\infty }\) in \(L^p(\mathbb {R}^d,\mathbb {C}^N)\) given by $$\begin{aligned} \mathcal {D}^p_{\mathrm {loc}}(\mathcal {L}_0)=\left\{ v\in W^{2,p}_{\mathrm {loc}}\cap L^p\mid A\triangle v + \left\langle S\cdot ,\nabla v\right\rangle \in L^p\right\} ,\,1<p<\infty . \end{aligned}$$ One key assumption is a new \(L^p\)-dissipativity condition $$\begin{aligned} |z|^2\mathrm {Re}\,\left\langle w,Aw\right\rangle + (p-2)\mathrm {Re}\,\left\langle w,z\right\rangle \mathrm {Re}\,\left\langle z,Aw\right\rangle \geqslant \gamma _A |z|^2|w|^2\;\forall \,z,w\in \mathbb {C}^N \end{aligned}$$ for some \(\gamma _A>0\). The proof utilizes the following ingredients. First we show the closedness of \(\mathcal {L}_{\infty }\) in \(L^p\) and derive \(L^p\)-resolvent estimates for \(\mathcal {L}_{\infty }\). Then we prove that the Schwartz space is a core of \(A_p\) and apply an \(L^p\)-solvability result of the resolvent equation for \(A_p\). In addition, we derive \(W^{1,p}\)-resolvent estimates. Our results may be considered as extensions of earlier works by Metafune, Pallara and Vespri to the vector-valued complex case.

Autonomous Underwater Vehicle Motion Response: A Nonacoustic Tool for Blue Water Navigation

AbstractAutonomous underwater vehicles (AUVs) use secondary velocity over ground measurements to aid the Inertial Navigation System (INS) to avoid unbounded drift in the point-to-point navigation solution. When operating in deep open ocean (i.e., in blue water—beyond the frequency-specific instrument range), the velocity measurements are either based on water column velocities or completely unavailable. In such scenarios, the velocity-relative-to-water measurements from an acoustic Doppler current profiler (ADCP) are often used for INS aiding. ADCPs have a blanking distance (typically ranging between 0.5 and 5 m) in proximity to the device in which the flow velocity data are undetectable. Hence, water velocities used to aid the INS solution can be significantly different from that near the vehicle and are subjected to significant noise. Previously, the authors introduced a nonacoustic method to calculate the water velocity components of a turbulent water column within the ADCP dead zone using the AUV motion response (referred to as the WVAM method). The current study analyzes the feasibility of incorporating the WVAM method within the INS by investigating the accuracy of it at different turbulence levels of the water column. Findings of this work demonstrate that the threshold limits of the method can be improved in the nonlinear ranges (i.e., at low and high levels of energy); however, by estimating a more accurate representation of vehicle hydrodynamic coefficients, this method has proven robust in a range of tidally induced flow conditions. The WVAM method, in its current state, offers significant potential to make a key contribution to blue water navigation when integrated within the vehicle's INS.

Large deviation principle for stochastic Boussinesq equations driven by Lévy noise

The current paper is devoted to dynamics of stochastic Boussinesq equation driven by Lévy pure jump noise. The large deviation principle of solution is established, and the effect of the highly nonlinear and unbounded drift is stated. Moreover, the large deviation principle for stochastic two dimensional hydrodynamical systems with Lévy noise is also obtained.

Regularity of strong solutions of one-dimensional SDE’s with discontinuous and unbounded drift

In this paper we develop a method for constructing strong solutions of one-dimensional Stochastic Differential Equations where the drift may be discontinuous and unbounded. The driving noise is the Brownian Motion and we show that the solution is Sobolev-differentiable in the initial condition and Malliavin differentiable. This method is not based on a pathwise uniqueness argument. We will apply these results to the stochastic transport equation. More specifically, we obtain a continuously differentiable solution of the stochastic transport equation when the driving function is a step function.

Kernel estimates for nonautonomous Kolmogorov equations

Using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernels of some nonautonomous Kolmogorov operators with possibly unbounded drift and diffusion coefficients.

Exponentially weighted resolvent estimates for complex Ornstein–Uhlenbeck systems

In this paper we study differential operators of the form \begin{align*} \left[\mathcal{L}_\infty v \right](x) = A\triangle v(x) + \left\langle Sx,\nabla v(x) \right\rangle - Bv(x), \,x \in \mathbb{R}^d, \,d \geqslant 2, \end{align*} for matrices $A,B\in\mathbb{C}^{N,N}$, where the eigenvalues of $A$ have positive real parts. The sum $A\triangle v(x) + \left\langle Sx, \nabla v(x) \right\rangle$ is known as the Ornstein-Uhlenbeck operator with an unbounded drift term defined by a skew-symmetric matrix $S\in\mathbb{R}^{d,d}$. Differential operators such as $\mathcal{L}_{\infty}$ arise as linearizations at rotating waves in time-dependent reaction diffusion systems. The results of this paper serve as foundation for proving exponential decay of such waves. Under the assumption that $A$ and $B$ can be diagonalized simultaneously we construct a heat kernel matrix $H(x,\xi,t)$ of $\mathcal{L}_{\infty}$ that solves the evolution equation $v_t=\mathcal{L}_{\infty}v$. In the following we study the Ornstein-Uhlenbeck semigroup \begin{align*} \left[ T(t)v\right](x) = \int_{\mathbb{R}^d} H(x,\xi,t) v(\xi) d\xi,\,x \in \mathbb{R}^d,\, t>0, \end{align*} in exponentially weighted function spaces. This is used to derive resolvent estimates for $\mathcal{L}_{\infty}$ in exponentially weighted $L^p$-spaces $L^p_{\theta} (\mathbb{R}^d,\mathbb{C}^N)$, $1\leqslant p<\infty$, as well as in exponentially weighted $C_{\mathrm{b}}$-spaces $C_{\mathrm{b},\theta}(\mathbb{R}^d,\mathbb{C}^N)$.

Quantitative uniqueness estimates for second order elliptic equations with unbounded drift

In this paper we derive quantitative uniqueness estimates at infinity for solutions to an elliptic equation with unbounded drift in the plane. More precisely, let $u$ be a real solution to $\Delta u+W\cdot\nabla u=0$ in ${\mathbf R}^2$, where $W$ is real vector and $\|W\|_{L^p({\mathbf R}^2)}\le K$ for $2\le p<\infty$. Assume that $\|u\|_{L^{\infty}({\mathbf R}^2)}\le C_0$ and satisfies certain a priori assumption at $0$. Then $u$ satisfies the following asymptotic estimates at $R\gg 1$ \[ \inf_{|z_0|=R}\sup_{|z-z_0|<1}|u(z)|\ge \exp(-C_1R^{1-2/p}\log R)\quad\text{if}\quad 2<p<\infty \] and \[ \inf_{|z_0|=R}\sup_{|z-z_0|<1}|u(z)|\ge R^{-C_2}\quad\text{if}\quad p=2, \] where $C_1>0$ depends on $p, K, C_0$, while $C_2>0$ depends on $K, C_0$ . Using the scaling argument in [BK05], these quantitative estimates are easy consequences of estimates of the maximal vanishing order for solutions of the local problem. The estimate of the maximal vanishing order is a quantitative form of the strong unique continuation property.

Harnack inequality for degenerate and singular elliptic equations with unbounded drift

We prove a Harnack inequality for functions which, at points of large gradient, are solutions of elliptic equations with unbounded drift.

Harnack-Type Inequalities and Applications for SDE Driven by Fractional Brownian Motion

For stochastic differential equation driven by fractional Brownian motion with Hurst parameter H > 1/2, Harnack-type inequalities are established by constructing a coupling with unbounded time-dependent drift. These inequalities are applied to the study of existence and uniqueness of invariant measure for a discrete Markov semigroup constructed in terms of the distribution of the solution. Furthermore, we show that entropy-cost inequality holds for the invariant measure.

6DOF Wireless Tracking Wand Using MARG and Vision Sensor Fusion

We present a low cost battery-powered 6-degree-of-freedom wireless wand for 3D modeling in free space by tri-axis Magnetic, Angular Rate, Gravity (MARG) and vision sensor fusion. Our approach has two stages of sensor fusion, each with different algorithms for finding 3D orientation and position. The first stage fusion algorithm, a complementary filter, utilizes MARG sensors to compute 3D orientation relative to the direction of gravity and earth's magnetic field in a quaternion format, which was adjusted with compensations for magnetic distortion. The second stage fusion algorithm, a Kalman filter, utilizes accelerometer data and IR marker velocity to compute 3D position. In order to compute the IR marker linear velocity along the optical axis (the z-axis), we present a simple and efficient image-based technique to find the distance of the object from the camera using blob area pixels in the image. Our fusion (inside-in and outside-in) approach efficiently solves short time occlusion, needs of frequent calibration, and unbounded drift problems involved in numerical integration of inertial sensors data and improves the degrees of freedom at low cost without compromising accuracy. The results are compared with a leading commercial magnetic motion tracking system to demonstrate the performance of the wand.

Symmetry results for nonlinear elliptic operators with unbounded drift

We prove the one-dimensional symmetry of solutions to elliptic equations of the form −div(eG(x)a(|∇u|)∇u) = f(u) eG(x), under suitable energy conditions. Our results holds without any restriction on the dimension of the ambient space.

A new approach to stochastic evolution equations with adapted drift

In this paper we develop a new approach to stochastic evolution equations with an unbounded drift A which is dependent on time and the underlying probability space in an adapted way. It is well-known that the semigroup approach to equations with random drift leads to adaptedness problems for the stochastic convolution term. In this paper we give a new representation formula for the stochastic convolution which avoids integration of non-adapted processes. Here we mainly consider the parabolic setting. We establish connections with other solution concepts such as weak solutions. The usual parabolic regularity properties are derived and we show that the new approach can be applied in the study of semilinear problems with random drift. At the end of the paper the results are illustrated with two examples of stochastic heat equations with random drift.

The Oseen–Navier–Stokes flow in the exterior of a rotating obstacle: The non-autonomous case

Abstract Consider the incompressible Navier–Stokes flow past a rotating obstacle with a general time-dependent angular velocity and a time-dependent outflow condition at infinity – sometimes called an Oseen condition. By a suitable change of coordinates the problem is transformed to a non-autonomous problem with unbounded drift terms on a fixed exterior domain Ω ⊂ ℝ d $\Omega \subset {\mathbb {R}}^d$ . It is shown that the solution to the linearized problem is governed by a strongly continuous evolution system { T Ω ( t , s ) } t ≥ s ≥ 0 $\lbrace T_\Omega (t,s)\rbrace _{t\ge s\ge 0}$ on L σ p ( Ω ) $L^p_\sigma (\Omega )$ for 1 &lt; p &lt; ∞ $1&amp;lt;p&amp;lt;\infty $ . Moreover, Lp -Lq smoothing properties and gradient estimates of T Ω ( t , s ) $T_\Omega (t,s)$ , 0 ≤ s ≤ t $0\le s \le t$ , are obtained. These results are the key ingredients to show local in time existence of mild solutions to the full nonlinear problem for initial value in L σ p ( Ω ) $L^p_\sigma (\Omega )$ , p ≥ d.

New Drift Reset Method for Pedestrian Tracking with a Low-Cost IMU

In this work, we study the pedestrian position tracking problem using a foot-mounted inertial measurement unit (IMU). The IMU consists of a tri-axis rate-gyro, a tri-axis accelerometer and a tri-axis magnetometer. The magnetometer is used for constructing a global reference frame at the initial phase. The pedestrian orientation and position are estimated by integrating signals of the rate-gyro and the accelerometer. Since the integration operation usually introduces unbounded drift errors, a new drift reset method derived from a constant-ground-normal condition is proposed to compensate for the accumulated drift error. Furthermore, a temperature compensation (TC) method is described which can alleviate the rate-gyro error due to temperature variation. Real experiments have been conducted with six subjects to test the performance of the proposed method and the results show the promising performance of the proposed tracking system.

One-dimensional symmetry for semilinear equations with unbounded drift

We consider semilinear equations with unbounded drift in the whole of $R^n$ and we show that monotone solutions with finite energy are one-dimensional.

Homogenization with an oscillating drift: from L 2-bounded to unbounded drifts, 2D compactness results, and 3D nonlocal effects

This paper extends results obtained by Tartar (1977, 1986) and revisited in Briane and Gérard (Ann Scuola Norm Sup Pisa, to appear), on the homogenization of a Stokes equation perturbed by an oscillating drift. First, a N-dimensional scalar equation, for N ≥ 3, and a tridimensional Stokes equation are considered in the periodic framework only assuming the L 2-boundedness of the drift and so relaxing the equi-integrability condition of Briane and Gérard (Ann Scuola Norm Sup Pisa, to appear). Then, it is proved that the L 2-boundedness can be removed in dimension two, provided that the divergence of the drift has a sign. On the contrary, nonlocal effects are derived in dimension three with a free divergence drift that is only bounded in L 1.

Data Fusion Of Stereo Vision and Gyroscope For Estimation Of Indoor Mobile Robot Orientation

In this paper a Kalman filter based data fusion method uses a novel combination of stereo vision and gyroscope angular velocity information. The proposed solution helps to eliminate the gyroscope unbounded drift errors. Since the gyroscope offers a higher bandwidth and availability of angular velocity it is corrected with the stereo vision system which has lower bandwidth and availability but bounded errors. A successful set of experiments were conducted to evaluate the integrated angle calculated from gyroscope and the proposed sensor fusion method.

Boundary Harnack principle for second order elliptic equations with unbounded drift

We prove the boundary Harnack principle for ratios of solutions u/v of non-divergence second order elliptic equations Lu = a ij D ij u + b i D i u = 0 in a bounded domain Ω ⊂ $ {\mathbb R} $ n . We assume that b i ∈ L n (Ω) and Ω is a twisted Holder domain of order α ∈ (1/2, 1]. Based on this result, we derive the Holder regularity of u/v for uniform domains. Bibliography: 27 titles.

Carleson type estimates for second order elliptic equations with unbounded drift

We derive a Carleson type estimate for positive solutions of non-divergence second order elliptic equations Lu = a ij D ij u + b i D i u = 0 in a bounded domain Ω ⊂ ℝ n . We assume that b i ∈ L n (Ω) and Ω is a twisted Holder domain of order α ∈ (1/2, 1] which satisfies a weak regularity condition. We also provide an example which shows that the main result fails in general if α ∈ (0, 1/2]. Bibliography: 18 titles.

A non-autonomous model problem for the Oseen-Navier-Stokes flow with rotating effects

Consider the Navier-Stokes flow past a rotating obstacle with a general time-dependent angular velocity and a time-dependent outflow condition at infinity. After rewriting the problem on a fixed domain, one obtains a non-autonomous system of equations with unbounded drift terms. It is shown that the solution to a model problem in the whole space case Rd is governed by a strongly continuous evolution system on Lpσ(Rd) for 1 < p < ∞. The strategy is to derive a representation formula, similar to the one known in the case of non-autonomous Ornstein-Uhlenbeck equations. This explicit formula allows to prove Lp-Lq estimates and gradient estimates for the evolution system. These results are key ingredients to obtain (local) mild solutions to the full nonlinear problem by a version of Kato's iteration scheme.