Invariant Measures for a Random Evolution Equation with Small Perturbations

2001 ◽  
Vol 17 (4) ◽  
pp. 631-642 ◽  
Author(s):  
Fu Bao Xi
Keyword(s):  
Evolution Equation ◽  
Invariant Measures ◽  
Random Evolution ◽  
Small Perturbations

scholarly journals Invariant measures for some one-dimensional attractors

1982 ◽  
Vol 2 (3-4) ◽  
pp. 317-337 ◽  
Author(s):  
M. V. Jacobson
Keyword(s):  
Invariant Measures ◽  
Small Perturbations ◽  
One Dimensional ◽  
Quadratic Maps ◽  
Hyperbolic Attractors

AbstractWe consider certain non-invertible maps of the square which are extensions of the quadratic maps of the interval and their small perturbations. We show that several maps of the type possess attractors which are not hyperbolic but have invariant measures similar to Bowen-Ruelle measures for hyperbolic attractors.

Large deviations for random evolution equation in Besov-Orlicz space

2020 ◽  
pp. 357-387
Author(s):  
Dina Miora Rakotonirina ◽  
Jocelyn Hajaniaina Andriatahina ◽  
Rado Abraham Randrianomenjanahary ◽  
Toussaint Joseph Rabeherimanana
Keyword(s):  
Evolution Equation ◽  
Large Deviations ◽  
Orlicz Space ◽  
Evolution Equations ◽  
Random Evolution ◽  
Young Function ◽  
Large Deviations Principle

In this paper, we develop a large deviations principle for random evolution equations to the Besov-Orlicz space $\mathcal{B}_{M_2, w}^{v, 0}$ corresponding to the Young function $M_2(x)=\exp(x^2)-1$.

Downstream and upstream influence in river meandering. Part 2. Planimetric development

2001 ◽  
Vol 438 ◽  
pp. 213-230 ◽  
Author(s):  
G. SEMINARA ◽  
G. ZOLEZZI ◽  
M. TUBINO ◽  
D. ZARDI
Keyword(s):  
Evolution Equation ◽  
Numerical Solutions ◽  
Landau Equation ◽  
Fourier Series Expansion ◽  
Equilibrium Solutions ◽  
Ginzburg Landau ◽  
Small Perturbations ◽  
Channel Pattern ◽  
River Meandering ◽  
Spatial Modulations

The exact solution of the problem of river morphodynamics derived in Part 1 is employed to formulate and solve the problem of planimetric evolution of river meanders. A nonlinear integrodifferential evolution equation in intrinsic coordinates is derived. An exact periodic solution of such an equation is then obtained in terms of a modified Fourier series expansion such that the wavenumbers of the various Fourier modes are time dependent. The amplitudes of the Fourier modes and their wavenumbers satisfy a nonlinear system of coupled ordinary differential equations of the Landau type. Solutions of this system display the occurrence of two possible scenarios. In the sub-resonant regime, i.e. when the aspect ratio of the channel is smaller than the resonant value, meandering evolves according to the classical picture: a periodic train of small-amplitude sine-generated meanders migrating downstream evolve into the classical, upstream skewed, train of meanders of Kinoshita type. Evolution displays all the experimentally observed features: the meander growth rate increases up to a maximum and then decreases, while the migration speed decreases monotonically. No equilibrium solutions are found. In the super-resonant regime the picture is essentially reversed: downstream skewing develops while meanders migrate upstream.Numerical solutions of the planimetric evolution equation are obtained for the case when the initial channel pattern exhibits random small perturbations of the straight configuration. Under these conditions, the evolution displays the typical features of solutions of the Ginzburg–Landau equation, in particular, the occurrence of spatial modulations of the meandering pattern which organizes itself in the form of wavegroups. Furthermore, multiple loops develop in the advanced stage of meander growth.

scholarly journals Stroock-Varadhan support theorem for random evolution equation in Besov-Orlicz spaces

2020 ◽  
pp. 187-203
Author(s):  
Jocelyn Hajaniaina Andriatahina ◽  
Dina Miora Rakotonirina ◽  
Toussaint Joseph Rabeherimanana
Keyword(s):  
Differential Equation ◽  
Evolution Equation ◽  
Stochastic Differential Equation ◽  
Stochastic Processes ◽  
Orlicz Space ◽  
Modulus Of Continuity ◽  
Orlicz Spaces ◽  
Random Evolution ◽  
Support Theorem ◽  
The Family

We consider the family of stochastic processes $X=\{X_t, t\in [0;1]\}\,,$ where $X$ is the solution of the It\^{o} stochastic differential equation \[dX_t = \sigma(X_t, Z_t)dW_t + b(X_t,Y_t) dt \hspace*{2cm}\] whose coefficients Lipschitzian depend on $Z=\{Z_t, t\in [0;1]\} $ and $Y=\{Y_t, t\in [0;1]\}$. We prove that the trajectories of $X$ a.s. belong to the Besov-Orlicz space defined by the f nction $M(x)=e^{x^2}-1$ and the modulus of continuity $\omega(t)=\sqrt{t\log(1/t)}$. The aim of this work is to characterize the support of the law $X$ in this space.

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