SIGLEAvailable from British Library Lending Division - LD:3463.021(125) / BLDSC - British Library... more SIGLEAvailable from British Library Lending Division - LD:3463.021(125) / BLDSC - British Library Document Supply CentreGBUnited Kingdo
A system of M (≥ 2) coupled singularly perturbed linear reaction-diffusion equations is considere... more A system of M (≥ 2) coupled singularly perturbed linear reaction-diffusion equations is considered on the unit square. Under certain hypotheses on the coupling, a maximum principle is established for the differential operator. The relationship between compatibility conditions at the corners of the square and the smoothness of the solution on the closed domain is fully described. A decomposition of the solution of the system is constructed. A finite difference method for the solution of the system on a Shishkin mesh is presented and it is proved that the computed solution is second-order accurate (up to a logarithmic factor). Numerical results are given to support this result and to investigate the effect of weaker compatibility assumptions on the data.
Based on vertical movement data derived from electronic storage tags (DST) attached to fish, we c... more Based on vertical movement data derived from electronic storage tags (DST) attached to fish, we construct a stochastic model that aims at capturing the main characteristics of the observations over one year. We use a mixed Ornstein-Uhlenbeck process to model attraction to a limited number of concentration points on the depth axis. A methodology for determining states and transition probabilities between them, and for setting model parameters for the process, is discussed. We show some examples of simulations using the model, and compare the simulations to the original data. In general, the model appears to capture the main characteristics of the vertical dynamics, except in cases where the data is characterized by a longlasting transient state at the start of the time series.
Fokker-Planck Equation in x and v for Harmonic Oscillator 167 15 Velocity Replacement Echoes 16 R... more Fokker-Planck Equation in x and v for Harmonic Oscillator 167 15 Velocity Replacement Echoes 16 Rate Equations for Discrete Processes 17 Generalized Moment Expansion 18 Curve Crossing in a Protein: Coupling of the Elementary Quantum Process to Motions of the Protein ∂ t x(t) = A[x(t), t]. (2.5) describes a deterministic drift of particles along the field A[x(t), t]. Equations like (2.5) can actually describe a wide variety of phenomena, like chemical kinetics or the firing of neurons. Since such systems are often subject to random perturbations, noise is added to the deterministic equations to yield associated stochastic differential equations. In such cases as well as in the case of classical Brownian particles, the noise term B[x(t), t] • η(t) needs to be specified on the basis of the underlying origins of noise. We will introduce further below several mathematical models of noise and will consider the issue of constructing suitable noise terms throughout this book. For this purpose, one often adopts a heuristic approach, analysing the noise from observation or from a numerical simulation and selecting a noise model with matching characteristics. These characteristics are introduced below. Before we consider characteristics of the noise term η(t) in (2.3) we like to demonstrate that the one-dimensional Langevin equation (2.2) of a classical particle, written here in the form µq = f (q) − γq + σ ξ(t)
The existence of strong and weak càdlàg versions of a solution to a linear equation in a Hilbert ... more The existence of strong and weak càdlàg versions of a solution to a linear equation in a Hilbert space H, driven by a Lévy process taking values in a Hilbert space U ← H is established. The so-called cylindrical càdlàg property is investigated as well. A special emphasis is put on infinite systems of linear equations driven by independent Lévy processes.
We prove exponential convergence to the invariant measure, in the total variation norm, for solut... more We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by α-stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Harris theorem, and the second on Doeblin's coupling argument [10]. Irreducibility, Lyapunov function techniques, and uniform strong Feller property play an essential role in both approaches. We concentrate on two classes of Markov processes: solutions of finite-dimensional equations, introduced in [29], with Hölder continuous drift and a general, non-degenerate, symmetric α-stable noise, and infinite-dimensional parabolic systems, introduced in [32], with Lipschitz drift and cylindrical α-stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. This improves, in particular, an earlier result established in [30] using the weak convergence induced by the Kantorovich-Wasserstein metric.
A method from stochastic flow theory is used to obtain smoothing properties of the transition sem... more A method from stochastic flow theory is used to obtain smoothing properties of the transition semigroup Pt of a class of stochastic differential equations on Hilbert space. The equations considered may have unbounded coefficients and include such stochastic partial differential equations as for Xt in L (0,π) In certain cases a formula for the Fréchet derivative of Ptf is given,
A stochastic system described by a semilinear equation with a small noise is considered. Under su... more A stochastic system described by a semilinear equation with a small noise is considered. Under suitable hypotheses, the rate functionals for the family of distributions associated to the solution and the exit time and exit place of the solution are computed.
SIGLEAvailable from British Library Lending Division - LD:3463.021(125) / BLDSC - British Library... more SIGLEAvailable from British Library Lending Division - LD:3463.021(125) / BLDSC - British Library Document Supply CentreGBUnited Kingdo
A system of M (≥ 2) coupled singularly perturbed linear reaction-diffusion equations is considere... more A system of M (≥ 2) coupled singularly perturbed linear reaction-diffusion equations is considered on the unit square. Under certain hypotheses on the coupling, a maximum principle is established for the differential operator. The relationship between compatibility conditions at the corners of the square and the smoothness of the solution on the closed domain is fully described. A decomposition of the solution of the system is constructed. A finite difference method for the solution of the system on a Shishkin mesh is presented and it is proved that the computed solution is second-order accurate (up to a logarithmic factor). Numerical results are given to support this result and to investigate the effect of weaker compatibility assumptions on the data.
Based on vertical movement data derived from electronic storage tags (DST) attached to fish, we c... more Based on vertical movement data derived from electronic storage tags (DST) attached to fish, we construct a stochastic model that aims at capturing the main characteristics of the observations over one year. We use a mixed Ornstein-Uhlenbeck process to model attraction to a limited number of concentration points on the depth axis. A methodology for determining states and transition probabilities between them, and for setting model parameters for the process, is discussed. We show some examples of simulations using the model, and compare the simulations to the original data. In general, the model appears to capture the main characteristics of the vertical dynamics, except in cases where the data is characterized by a longlasting transient state at the start of the time series.
Fokker-Planck Equation in x and v for Harmonic Oscillator 167 15 Velocity Replacement Echoes 16 R... more Fokker-Planck Equation in x and v for Harmonic Oscillator 167 15 Velocity Replacement Echoes 16 Rate Equations for Discrete Processes 17 Generalized Moment Expansion 18 Curve Crossing in a Protein: Coupling of the Elementary Quantum Process to Motions of the Protein ∂ t x(t) = A[x(t), t]. (2.5) describes a deterministic drift of particles along the field A[x(t), t]. Equations like (2.5) can actually describe a wide variety of phenomena, like chemical kinetics or the firing of neurons. Since such systems are often subject to random perturbations, noise is added to the deterministic equations to yield associated stochastic differential equations. In such cases as well as in the case of classical Brownian particles, the noise term B[x(t), t] • η(t) needs to be specified on the basis of the underlying origins of noise. We will introduce further below several mathematical models of noise and will consider the issue of constructing suitable noise terms throughout this book. For this purpose, one often adopts a heuristic approach, analysing the noise from observation or from a numerical simulation and selecting a noise model with matching characteristics. These characteristics are introduced below. Before we consider characteristics of the noise term η(t) in (2.3) we like to demonstrate that the one-dimensional Langevin equation (2.2) of a classical particle, written here in the form µq = f (q) − γq + σ ξ(t)
The existence of strong and weak càdlàg versions of a solution to a linear equation in a Hilbert ... more The existence of strong and weak càdlàg versions of a solution to a linear equation in a Hilbert space H, driven by a Lévy process taking values in a Hilbert space U ← H is established. The so-called cylindrical càdlàg property is investigated as well. A special emphasis is put on infinite systems of linear equations driven by independent Lévy processes.
We prove exponential convergence to the invariant measure, in the total variation norm, for solut... more We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by α-stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Harris theorem, and the second on Doeblin's coupling argument [10]. Irreducibility, Lyapunov function techniques, and uniform strong Feller property play an essential role in both approaches. We concentrate on two classes of Markov processes: solutions of finite-dimensional equations, introduced in [29], with Hölder continuous drift and a general, non-degenerate, symmetric α-stable noise, and infinite-dimensional parabolic systems, introduced in [32], with Lipschitz drift and cylindrical α-stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. This improves, in particular, an earlier result established in [30] using the weak convergence induced by the Kantorovich-Wasserstein metric.
A method from stochastic flow theory is used to obtain smoothing properties of the transition sem... more A method from stochastic flow theory is used to obtain smoothing properties of the transition semigroup Pt of a class of stochastic differential equations on Hilbert space. The equations considered may have unbounded coefficients and include such stochastic partial differential equations as for Xt in L (0,π) In certain cases a formula for the Fréchet derivative of Ptf is given,
A stochastic system described by a semilinear equation with a small noise is considered. Under su... more A stochastic system described by a semilinear equation with a small noise is considered. Under suitable hypotheses, the rate functionals for the family of distributions associated to the solution and the exit time and exit place of the solution are computed.
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Papers by J. Zabczyk