ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

12. Probability and Statistics

1991 MS Classification: Mark Freidlin, University of Maryland, College Park, USA Random and Deterministic Perturbations of Nonlinear Oscillators

We consider perturbations of an oscillator with one degree of freedom:

q̈_{t}^{ϵ} + f(q_{t}^{ϵ}) = ϵb(q̇_{t}^{ϵ}, q_{t}^{ϵ}) + \sqrt{ϵ}σ(q̇_{t}^{ϵ}, q_{t}^{ϵ}) ∘ Ẇ_{t}, q_{0}^{ϵ} = q ∈ R^{1}.

Here W_{t} is the Wiener process in R^{1}, 0 < ϵ << 1. A typical example of the deterministic part of the perturbation is a small friction: b(q, q̇) = −βq̇. Let F (q) = ∫_{0}^{q} f(s)ds, q̇ = p, H(p, q) = p^{2}/2 + F (q). We assume that the potential F (q) is generic (this means that F (q) has a finite number of non-degenerate critical points) and lim_{|q|→∞} F (q) = ∞. Let Γ be the graph homeomorphic to the set of connected components of the level sets of the Hamiltonian H(p, q) provided with the natural topology: Γ = {I_{1}, ..., I_{n}; O_{1}, ...O_{m}}. Each vertex O_{i} belongs either to one edge (exterior vertex) or to three edges (interior vertex). A point y ∈ Γ can be characterized by two coordinates (H, k), where k is the number of the edge containing y, and H is the value of the Hamiltonian on the level-set component corresponding to y. Define a mapping Y : R^{2} → Γ so that Y (x) = (H(x), k(x)), x ∈ R^{2}, where k(x) is the number of the edge containing the point corresponding to the trajectory of the non-perturbed system (ϵ = 0) starting at x ∈ R^{2}. The function k(x), as well as H(x), is a first integral for the non-perturbed system. The pair X_{t}^{ϵ} = (p_{t}^{ϵ}, q_{t}^{ϵ}), p_{t}^{ϵ} = q̇_{t}^{ϵ}, forms a diffusion process. The long-time behavior of this process as ϵ ↓ 0 is determined by the averaging principle. The slow component of X_{t}^{ϵ} is the projection Y (X_{t}^{ϵ}) of X_{t}^{ϵ} on Γ. We prove that the processes Y_{t}^{ϵ} = Y (X_{t/ϵ}^{ϵ}) converge weakly in the space of continuous functions on Γ to a diffusion process Y_{t} on the graph Γ = {I_{1}, ..., I_{n}; O_{1}, ...O_{m}}. Diffusion processes on a graph Γ are defined by a family of second order differential operators L_{1}, ..L_{n} and by gluing conditions at the vertices. The operator L_{k} defines Y_{t} inside I_{k} ⊂ Γ. The gluing conditions at a vertex O_{i} ∈ Γ define, roughly speaking, what the trajectory is doing after is touches O_{i}. To describe the limiting process Y_{t}, denote by G_{k}(z) the domain in R^{2} bounded by C_{k}(z) = Y^{−1}(z, k) if (z, k) is not a vertex. Let S_{k}(z) be the area of G_{k}(z). One can consider S_{k}(z) as a function on Γ. This function has, in general, discontinuities at the interior vertices. If O_{i} is an interior vertex and I_{i_{1}}, I_{i_{2}}, I_{i_{3}} are the edges attached to O_{i}, denote S_{j}(O_{i}) the limit of S_{i_{j}}(z) as the point (z, i_{j}) ∈ I_{i_{j}} approaches O_{i}. Then in the case b(p, q) = 0, σ = unit matrix, the operators L_{k} defining the process on I_{k} have the form

L_{k} = \frac{1}{2S′_{k}(z)} \frac{d}{dz} (S_{k}(z)\frac{d}{dz}).

A smooth inside the edges and continuous on Γ function u(y), y ∈ Γ, belongs to the domain of the generator A of the limiting process Y_{t}, if and only if Au(y) is continuous on Γ and at any interior vertex O_{i} ∈ Γ the following gluing condition is satisfied

∑_{j=1}^{3} S_{j}(O_{i})D_{j}u(O_{i}) = 0.

Here D_{j} is the operator of differentiation in the coordinate H along I_{i_{j}} attached to O_{i}, j = 1, 2, 3. The operators L_{k} and the gluing conditions at the vertices define the limiting process in a unique way. The characteristics of the limiting process on Γ are calculated also for any σ(p, q) and b(p, q). Let now σ ≡ 0, so that just deterministic perturbations are considered. it turns out that the slow component Y (X_{t/ϵ}^{ϵ}) of the deterministically perturbed system in any reasonable sense converges as ϵ ↓ 0 to a stochastic process on Γ. In this case the limiting process Y_{t} = (z_{t}, k_{t}) is deterministic inside each edge I_{k}:

Ẏ_{t} =b_{k}(Y_{t}), Y_{t} ∈ I_{k},
b_{k}(z) =\frac{1}{S′_{k}(z)} ∫_{G_{k}(z)} \frac{∂b(p, q)}{∂p}dpdq.

At an interior vertex O_{i} the trajectory branches with certain probabilities between the edges attached to the vertex. The branching probabilities are expressed through the function b(p, q) and the curve Y^{−1}(O_{i}). To give rigorous sense to this statement, we consider random perturbations of the deterministic system Ẋ_{t/ϵ}^{ϵ} = (p_{t/ϵ}^{ϵ}, q_{t/ϵ}^{ϵ}) and then calculate the limit when first ϵ ↓ 0 and then the random perturbations tend to zero. It is important to underline that the limit is the same for a wide class of random perturbations of the equation or of the initial conditions. So that the structure of the limiting stochastic process is an intrinsic property of the oscillator and its deterministic perturbations, but not of the random perturbations which we use for regularization of the problem. We also will discuss some similar problems for systems with many degrees of freedom.