By M. I. Freidlin, A. D. Wentzell

Random perturbations of Hamiltonian platforms in Euclidean areas result in stochastic techniques on graphs, and those graphs are outlined by way of the Hamiltonian. when it comes to white-noise variety perturbations, the restricting strategy should be a variety procedure at the graph. Its features are expressed during the Hamiltonian and the features of the noise. Freidlin and Wentzell calculate the method at the graph below sure stipulations and increase a strategy which permits attention of a few asymptotic difficulties. The Dirichlet challenge for corresponding elliptic equations with a small parameter are hooked up with boundary difficulties at the graph.

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27) 1 f(Y(XC )) + I e A t [Xf(Y(xf)) - Af(Y(xf))] dt] x1 JQ t t -XT *u(z) = E e [e 2 z - f(Y(z)). 28) is arbitrarily small for sufficiently small e, uniformly in z € U C (<5). As for ^ ( z ) , for z * U D (±6) we have x k,i k l k k ^T(z) = 0. -0, x I k | H ( y ) - H(0, ) I < s \ K J < (X||f|| + ||Xf - Af|| + 2 ) K . 26), except the Oth, the argument in ki - p.. )| Hcil i k ' RANDOM PERTURBATIONS OF HAMILTONIAN SYSTEMS < (X||f|| + ||Xf - Af|| + 2 + V |q(0k)|)ic5.

Between the lines « - 5)- Let Then, if 6 = 5(e) \ = raln{t: is such that K * Dx>: \ = min{t: 6(e) —>0, 6(e)/e-—>» ( e — > 0 ) , E e x G = 0(£n(6/e)) x x uniformly in x € D ; *t * a 2out U a 2inK MARK I. FREIDLIN AND ALEXANDER D. WENTZELL 46 PG{xe < ox x x uniformly for Proof. } ->1, ~ 2out lin* introduce new coordinates Let or such that the point co-shaped curve are the axes be the branch along which the point dynamical system. is small enough one can x, x ~1 x is the ~2 x. } ~ 2out o* x so that + E x x x We will also change the scale B x 1 ^ 4 = 1, Ix1!

1 < (1 - min A[ 4 ) ) m . n, but not on 6 or 5'. 6). The Theorem is proved. • K. So taking f, A > 0, K= §4. 4 In this section there will be many estimates with various constants, which we will denote by letter A with some subscript. 2) of going out of any bounded region. 3) E€H(XC) = H(x) + E G ^AH(Xe) ds. 2. Introducing the graph, we did not describe the metric on it. ) - H(x. _1) i= lie on the same segment of the graph. In order to prove the pre-compactness it is sufficient to prove that: 37 38 MARK I.

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