Random perturbations of Hamiltonian systems.

*(English)*Zbl 0804.60070
Mem. Am. Math. Soc. 523, 82 p. (1994).

Hamiltonian systems in \(R^ 2\), \(\dot X_ t= \overline {\nabla} H(X_ t)\), \(X_ 0= x= (p,q)\in R^ 2\) where
\[
\overline {\nabla} H(x)= \biggl( {{\partial H(p,q)} \over {\partial q}}, -{{\partial H(p,q)} \over {\partial p}}\biggr),
\]
are under consideration. Random perturbations of the kind \(d \widetilde{X}^ \varepsilon_ t =\overline {\nabla} H(\widetilde{X}^ \varepsilon_ t)dt+ \varepsilon^{1/2} d\widetilde{W}_ t, \widetilde {X}^ \varepsilon_ 0=x\), are studied, where \(\widetilde {W}_ t\) is a Wiener process in \(R^ 2\), \(0< \varepsilon\to 0\). After rescaling one gets \(dX^ \varepsilon_ t= \varepsilon^{-1} \overline{\nabla} H(X^ \varepsilon_ t)dt+ d\widetilde {W}_ t, X^ \varepsilon_ 0=x\), which may be considered as a fast rotation along the non-perturbed paths and a slow motion “across” them.

For many cases the behavior of \(X^ \varepsilon_ t, t\leq T/\varepsilon\), may be described via a rate function and a quasi- potential in the problem of large deviations. The points of the state space may be identified with the help of this quasi-potential. But the Hamiltonian systems provide, in fact, a very important example when all points are equivalent from this point of view, at least, in the “non- degenerate” case in some sense.

The averaging principle is proved in a region covered by closed paths \(X_ t\), i.e. for all small \(\varepsilon>0\) the motions of \(X_{t/\varepsilon}\) and \(X_ t^ \varepsilon\) along these paths are approximately the same for \(t\leq T\). The motion “across” the non- perturbed paths is proved to be approximately equal to a diffusion whose characteristics are obtained by averaging with respect to the invariant measure on each path; this invariant measure is proportional to \(dl/ | b(x)|\), \(b(x)= \overline {\nabla} H(x)\). Thus, it makes sense to use \(H\) as a new “coordinate” for the process. Denote by \(Y\) the mapping \(x \mapsto H(x)\). Then the generating differential operator of the diffusion \(Y (X_ t^ \varepsilon)\), \(t\leq T\), should be approximately equal to \(L f(H)= {1\over 2} A(H) f''(H)+ B(H) f'(H)\), where \[ A(H)= {{\oint (|\nabla H(x)|^ 2/| b(x)|)dl} \over {\oint (1/| b(x)|)dl}}, \qquad B(H)= {{\oint (2^{-1} H(x)/ | b(x)|)dl} \over {\oint (1/| b(x)|)dl}} \] and all integrals are calculated along the non-perturbed closed path \(x\): \(H(x)=H\).

If one identifies all points \(x\in R^ 2\) by the mapping \(Y\), one will obtain a graph which consists of several segements and vertices. (Here it is absolutely necessary for the reader of this review to imagine a picture.) General conditions for a diffusion on the graph may include second order operators at each vertex (“gluing conditions”). The non- degenerate case for the perturbed system in the sense of the second derivative matrix of the Hamiltonian at critical points is considered. Moreover, it is assumed that only a finite number of such points exists, and some additional requirements on the Hamiltonian hold true. Then only first order operators appear in gluing conditions.

Main results establish the existence and uniqueness of the limiting diffusion \(Y(H)\) on the graph and the weak convergence of perturbed distributions of the processes \(Y (X^ \varepsilon_ t)\) to the limiting one as \(\varepsilon\to 0\) in the space of continuous functions on the graph. The proof uses about twenty technical lemmas.

For many cases the behavior of \(X^ \varepsilon_ t, t\leq T/\varepsilon\), may be described via a rate function and a quasi- potential in the problem of large deviations. The points of the state space may be identified with the help of this quasi-potential. But the Hamiltonian systems provide, in fact, a very important example when all points are equivalent from this point of view, at least, in the “non- degenerate” case in some sense.

The averaging principle is proved in a region covered by closed paths \(X_ t\), i.e. for all small \(\varepsilon>0\) the motions of \(X_{t/\varepsilon}\) and \(X_ t^ \varepsilon\) along these paths are approximately the same for \(t\leq T\). The motion “across” the non- perturbed paths is proved to be approximately equal to a diffusion whose characteristics are obtained by averaging with respect to the invariant measure on each path; this invariant measure is proportional to \(dl/ | b(x)|\), \(b(x)= \overline {\nabla} H(x)\). Thus, it makes sense to use \(H\) as a new “coordinate” for the process. Denote by \(Y\) the mapping \(x \mapsto H(x)\). Then the generating differential operator of the diffusion \(Y (X_ t^ \varepsilon)\), \(t\leq T\), should be approximately equal to \(L f(H)= {1\over 2} A(H) f''(H)+ B(H) f'(H)\), where \[ A(H)= {{\oint (|\nabla H(x)|^ 2/| b(x)|)dl} \over {\oint (1/| b(x)|)dl}}, \qquad B(H)= {{\oint (2^{-1} H(x)/ | b(x)|)dl} \over {\oint (1/| b(x)|)dl}} \] and all integrals are calculated along the non-perturbed closed path \(x\): \(H(x)=H\).

If one identifies all points \(x\in R^ 2\) by the mapping \(Y\), one will obtain a graph which consists of several segements and vertices. (Here it is absolutely necessary for the reader of this review to imagine a picture.) General conditions for a diffusion on the graph may include second order operators at each vertex (“gluing conditions”). The non- degenerate case for the perturbed system in the sense of the second derivative matrix of the Hamiltonian at critical points is considered. Moreover, it is assumed that only a finite number of such points exists, and some additional requirements on the Hamiltonian hold true. Then only first order operators appear in gluing conditions.

Main results establish the existence and uniqueness of the limiting diffusion \(Y(H)\) on the graph and the weak convergence of perturbed distributions of the processes \(Y (X^ \varepsilon_ t)\) to the limiting one as \(\varepsilon\to 0\) in the space of continuous functions on the graph. The proof uses about twenty technical lemmas.

Reviewer: A.Yu.Veretennikov (Moskva)

##### MSC:

60J60 | Diffusion processes |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60F17 | Functional limit theorems; invariance principles |

35B40 | Asymptotic behavior of solutions to PDEs |

34C29 | Averaging method for ordinary differential equations |

34F05 | Ordinary differential equations and systems with randomness |