Existence and uniqueness of solution of some function-space differential equations, II

AbstractLet ϱ(τ, s) be a symmetric and positive definite function, 0 ⩽ τ ⩽ t, 0 ⩽ s ⩽ t, satisfying a Hölder condition in τ, uniformly in s. Let V(x) be a polynomial of degree N such that Re V(x) is bounded from below for x real and such that V(0) = 0. We consider functionals u(q, x, λ) defined for q ϵ C([0, t], R), − ∞ < x < ∞, 0 < λ < ∞, which satisfy the following system of equations. 1.(a) δuδq(τ) = (λ2 ∝0t ϱ(τ, s) q(s) ds) u − λ2 ∝0t ϱ(τ, s) V′ (δδq(s)) u ds − λ2ϱ(t, τ)∂u∂x 0 < τ <t,2.(b) limτ→t−δuδq(τ)3.(c) λ∂u∂λ = ∝0t q(σ)δuδq(σ) dσ − ∝0tδδq(s) V′ (δδq(s)u ds − u − x∂u∂x4.(d) lim uλ→0(O,X,λ)=δ(x) The above system is shown to have a unique solution in the class of functionals DN(ϱ), which will be defined below. We first note that there exists a Gaussian process {z(τ) ¦ 0 ⩽ τ ⩽ t} with mean zero and covariance ϱ(τ, s). We will denote expectation with respect to such a process by Ezϱ{ }. We denote by ϱq the function given by ∝0t ϱ(·, s) q(s) ds. We will denote ∝0t z(σ) q(σ) dσ by (z, q).Definition. A functional u(q, x, λ) on C([0, t], R) × (−∞, ∞) × (0, ∞) will be said to be a member of DN(ϱ) if, for all τi, 0 ⩽ τi ⩽ t, i = 1 ··· m, m ⩽ 3(N − 1), q ϵ C([0, t], R), −∞ < x < ∞, 0 < λ < ∞, 1.(i) [δmu(q, x, λ)δq(τ1) ··· δq(τm)] exists, is jointly continuous in q and τi, i = 1 ··· m, and is continuously differentiable in x and λ, and2.(ii) there exists a function F(·, ·) continuous on [0, ∞) × (0, ∞), and absolute constants, M, K, α, β, 0 ⩽ α < 1, 1 < β such that ¦δmu(q, x, λ)δq(τ1 ··· δq(τm)¦ ⩽ F(∥ ϱq ∥, λ) exp (λ22 (q, ϱq) − ¦ x ¦β), ¦∝−∞∞ u(q, x, λ) dx ¦ ⩽ M exp (K2α + K ∥ λ2ϱq ∥α + λ22 (q, ϱq)), ¦∝−∞∞ δmu(q, x, λ) dxδq(τ1 ··· δq(τm)¦ ⩽ M exp (Kλ2α + K∥ λ2ϱq ∥2α + α22(q, ϱq)). Theorem 1. There exists one and only one solution to the system (a)–(d) in the class DN(ϱ). The solution is given by u∗(q, x, λ) = 12π ∝−∞∞ e−iμxEzλ2ϱ {exp (− ∝0t V(z(σ)) dσ + (z,q) + iμz(t))} dμ