Geometric singular perturbation theory for stochastic differential equations Nils Berglund

Slow–fast systems: heuristics In fast time s: x ′ = f(x, y) x ∈ R n, fast variable y ′ = εg(x, y) y ∈ Rm, slow variable • Perturbation of x ′ = f(x, λ), with slowly moving parameter λ • Simplest case: x?(λ) asympt. stable equilibrium point In slow time t = εs: εx ̇ = f(x, y) x ∈ R n, fast variable y ̇ = g(x, y) y ∈ Rm, slow variable • Slow manifold: f(x?(y), y) = 0 (for all y in some domain) • Reduced equation: y ̇ = g(x?(y), y) 1 Geometric singular perturbation theory εx ̇ = f(x, y) x ∈ R n, fast variable y ̇ = g(x, y) y ∈ Rm, slow variable • Slow manifold: f(x?(y), y) = 0 (for all y in some domain) • Stability: Eigenvalues of ∂xf(x?(y), y) have negative real parts Theorem [Tihonov ’52, Fenichel ’79] ∃ adiabatic manifold x = x̄(y, ε) s.t. • x̄(y, ε) is invariant • x̄(y, ε) attracts nearby solutions • x̄(y, ε) = x?(y) +O(ε) x y1y2 x = x?(y) x = x̄(y, ε) 2 Stochastic perturbation: one-dimensional case dxt = 1 ε f(xt, t) dt+ σ√ ε dWt Slow–fast system with yt = t Stable equil. branch: f(x?(t), t) = 0, a?(t) = ∂xf(x?(t), t) 6 −a0 Adiabatic solution: x̄(t, ε) = x?(t) +O(ε) ā(t, ε) = ∂xf(x̄(t, ε), t) = a?(t) +O(ε) B(h): strip of width ' h/|ā(t, ε) | around x̄(t, ε). x̄(t, ε) xt x?(t) B(h) 3 Stochastic perturbation: one-dimensional case dxt = 1 ε f(xt, t) dt+ σ