Some mean first-passage time approximations for the Ornstein-Uhlenbeck process

Thomas, Marlin U.

doi:10.1017/s0021900200048439

Cited by 16 publications

(23 citation statements)

References 2 publications

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“…In particular, the mean first passage time provides a first apprehension of the phenomenon so that many stochastic oscillators are first studied by means of their mean first passage times [30][31][32][33]. The variance of the first passage time also reflects the range of the possible observed first passage times in real conditions and is therefore interesting in a direct simulation.…”

Section: X(t) + [1 + U(t)] X(t) = W(t)mentioning

confidence: 99%

Second-order moment of the first passage time of a quasi-Hamiltonian oscillator with stochastic parametric and forcing excitations

Vanvinckenroye

Denoël

2018

Journal of Sound and Vibration

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a b s t r a c tThis work focuses on the stochastic version of the linear Mathieu oscillator with both forced and parametric excitations of small intensity. In this quasi-Hamiltonian oscillator, the concept of energy stored in the oscillator plays a central role and is studied through the first passage time, which is the time required for the system to evolve from a given initial energy to a target energy. This time is a random variable due to the stochastic nature of the loading. The average first passage time has already been studied for this class of oscillator. However, the spread has only been studied under purely parametric excitation. Extending to combinations of both forcing and parametric excitations, this work provides a closed-form solution and a thorough analytical study of the coefficient of variation of the first passage time of the energy in this system. Simple asymptotic solutions are also derived in some particular ranges of parameters corresponding to different regimes.

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Section: X(t) + [1 + U(t)] X(t) = W(t)mentioning

confidence: 99%

Second-order moment of the first passage time of a quasi-Hamiltonian oscillator with stochastic parametric and forcing excitations

Vanvinckenroye

Denoël

2018

Journal of Sound and Vibration

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“…There exists a vast literature on the first passage time of a zero-mean OU process across a certain level x 0 [9], [10], [11]. In our context, x 0 is given byr 1 ( * ) = γ∆ .…”

Section: B First Passage Time Distributionmentioning

confidence: 99%

“…The error probability of the convolutional LDPC ensemble is then predicted using closed-form expressions to the survival time p.d.f. and its moments for the OU process [9], [10], [11]. The accuracy of the prediction is illustrated by comparing with simulated error probability curves.…”

Section: Introductionmentioning

confidence: 99%

A closed-form scaling law for convolutional LDPC codes over the BEC

Olmos

Urbanke

2013

2013 IEEE Information Theory Workshop (ITW)

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We propose a scaling law for the error probability of convolutional LDPC ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well with data derived from simulations over a wide range of parameters. I. INTRODUCTIONRecently, it has been proved that spatially coupled lowdensity parity-check (SC-LDPC) codes can achieve capacity over binary-input memoryless symmetric channels under belief propagation (BP) decoding [1]. SC-LDPC ensembles are constructed by coupling L (l, r)-regular LDPC codes, each one of length M bits, together with appropriate boundary conditions. When M tends to infinity and L is large, SC-LDPC codes exhibit a BP threshold arbitrarily close to the maximum-aposteriori (MAP) threshold of (l, r)-regular ensemble [1], [2]. In contrast, much less is known about the finite-length behavior of SC-LDPC codes [3]. This finite-length performance requires to relate the code parameters, namely the chain length L, the coupling pattern, and the code length n = M L, to the error probability. In this paper, we study a specific class of SC-LDPC codes, usually referred to as convolutional LPDC codes [2], when they are used for transmission over the binary erasure channel (BEC). In [4] we provided a first description of the scaling behavior of convolutional LDPC ensembles using an extensive set of simulations. However, no closed-form expression was provided to evaluate the error probability in the range of most interest, namely when we consider coupled codes above the BP threshold of (l, r)-regular LDPC code.In the present work we extend to convolutional LDPC codes the methodology proposed in [5] to analyze finite-length LDPC ensembles over the BEC. For the BEC the workings of the BP decoder can be cast in an alternative formulation, namely as peeling decoder (PD). In this formulation, any time a variable nodes gets decoded, it is removed from the graph along with all attached edges. The analysis then consists on studying the evolution of the graph. More precisely, it suffices to track the expected evolution of the graph and the variance around this expected behavior since errors in this framework correspond to large deviations from the expected behavior. If it is possible to derive expressions for the mean and the variance

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“…Indeed, the probability density of exit times is given by [21]: (2.9) and the mean exit time satisfies [22]: In addition, E t D = E t * D / and Var t D = Var t * D / 2 . By taking the derivative of E t * D with respect to and using the approximation for large that…”

Section: Changementioning

confidence: 99%

A Stochastic Analysis of Power Doubling Time for a Subcritical System

Allen

2013

Stochastic Analysis and Applications

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The power doubling time for a subcritical system is identified as a stochastic firstpassage time problem. Using stochastic point kinetics equations, relations for the mean doubling time and the standard deviation in doubling time are derived. It is shown that the power doubling time for a subcritical system is independent of whether the system is fast or thermal, weakly depends on source strength, and is approximately proportional to the inverse of the reactivity for small negative values of the reactivity.

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Some mean first-passage time approximations for the Ornstein-Uhlenbeck process

Abstract: This paper describes an accurate method of approximating the mean of the first-passage time distribution for an Ornstein-Uhlenbeck process with a single absorbing barrier. The accuracy of the approximation is demonstrated through some numerical comparisons.

Cited by 16 publications

References 2 publications

Second-order moment of the first passage time of a quasi-Hamiltonian oscillator with stochastic parametric and forcing excitations

Second-order moment of the first passage time of a quasi-Hamiltonian oscillator with stochastic parametric and forcing excitations

A closed-form scaling law for convolutional LDPC codes over the BEC

A Stochastic Analysis of Power Doubling Time for a Subcritical System

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