Almost-Invariant and Finite-Time Coherent Sets: Directionality, Duration, and Diffusion

The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation (SDE) with globally Lipschitz continuous drift and diffusion coefficients. Recent results extend this convergence to coefficients that grow, at most, linearly. For superlinearly growing coefficients, finite-time convergence in the strong mean-square sense remains. In this article, we answer this question to the negative and prove, for a large class of SDEs with non-globally Lipschitz continuous coefficients, that Euler’s approximation converges neither in the strong mean-square sense nor in the numerically weak sense to the exact solution at a finite time point. Even worse, the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean-square sense and in the numerically weak sense.

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Extension of Some Asymptotic Results on Finite Time Ruin in Compound Poisson Model with Constant Interest Force and Diffusion

2010 International Conference on Management and Service Science ◽

10.1109/icmss.2010.5576887 ◽

2010 ◽

Author(s):

Tao Jiang ◽

Congyan Tong

Keyword(s):

Finite Time ◽

Poisson Model ◽

Asymptotic Results ◽

Compound Poisson ◽

Compound Poisson Model ◽

Interest Force ◽

Constant Interest Force ◽

Constant Interest ◽

And Diffusion

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Arbitrary-order corrections for finite-time drift and diffusion coefficients

Physical Review E ◽

10.1103/physreve.80.031103 ◽

2009 ◽

Vol 80 (3) ◽

Cited By ~ 11

Author(s):

C. Anteneodo ◽

R. Riera

Keyword(s):

Diffusion Coefficients ◽

Finite Time ◽

Arbitrary Order ◽

Time Drift ◽

And Diffusion

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Meridional Circulation, Turbulence, and Diffusion Processes in Stars

International Astronomical Union Colloquium ◽

10.1017/s0252921100080477 ◽

1976 ◽

Vol 32 ◽

pp. 109-116 ◽

Cited By ~ 1

Author(s):

S. Vauclair

Keyword(s):

Convection Zone ◽

Diffusion Processes ◽

Meridional Circulation ◽

Diffusion Time ◽

Work In Progress ◽

First Results ◽

Stellar Envelopes ◽

And Diffusion ◽

Turbulence And Diffusion ◽

Hr Diagram

This paper gives the first results of a work in progress, in collaboration with G. Michaud and G. Vauclair. It is a first attempt to compute the effects of meridional circulation and turbulence on diffusion processes in stellar envelopes. Computations have been made for a 2 Mʘstar, which lies in the Am - δ Scuti region of the HR diagram.Let us recall that in Am stars diffusion cannot occur between the two outer convection zones, contrary to what was assumed by Watson (1970, 1971) and Smith (1971), since they are linked by overshooting (Latour, 1972; Toomre et al., 1975). But diffusion may occur at the bottom of the second convection zone. According to Vauclair et al. (1974), the second convection zone, due to He II ionization, disappears after a time equal to the helium diffusion time, and then diffusion may happen at the bottom of the first convection zone, so that the arguments by Watson and Smith are preserved.

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