1,314 research outputs found
Almost Sure Convergence of Solutions to Non-Homogeneous Stochastic Difference Equation
We consider a non-homogeneous nonlinear stochastic difference equation
X_{n+1} = X_n (1 + f(X_n)\xi_{n+1}) + S_n, and its important special case
X_{n+1} = X_n (1 + \xi_{n+1}) + S_n, both with initial value X_0, non-random
decaying free coefficient S_n and independent random variables \xi_n. We
establish results on \as convergence of solutions X_n to zero. The necessary
conditions we find tie together certain moments of the noise \xi_n and the rate
of decay of S_n. To ascertain sharpness of our conditions we discuss some
situations when X_n diverges. We also establish a result concerning the rate of
decay of X_n to zero.Comment: 22 pages; corrected more typos, fixed LaTeX macro
Summability of solutions of the heat equation with inhomogeneous thermal conductivity in two variables
We investigate Gevrey order and 1-summability properties of the formal
solution of a general heat equation in two variables. In particular, we give
necessary and sufficient conditions for the 1-summability of the solution in a
given direction. When restricted to the case of constants coefficients, these
conditions coincide with those given by D.A. Lutz, M. Miyake, R. Schaefke in a
1999 article, and we thus provide a new proof of their result.Comment: 16 page
Borel summability and Lindstedt series
Resonant motions of integrable systems subject to perturbations may continue
to exist and to cover surfaces with parametric equations admitting a formal
power expansion in the strength of the perturbation. Such series may be,
sometimes, summed via suitable sum rules defining functions of the
perturbation strength: here we find sufficient conditions for the Borel
summability of their sums in the case of two-dimensional rotation vectors with
Diophantine exponent (e. g. with ratio of the two independent
frequencies equal to the golden mean).Comment: 17 pages, 1 figur
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