Abstract

Gill and Myers [GM] proved that every separable infinite-dimensional Banach space, denoted B, has an isomorphic, isometric embedding in R ∞ = R × R × · · · . They used this result and a method due to Yamasaki [YA] to construct a sigma-finite Lebesgue measure λB for B and defined the associated integral R B · dλB in a way that equals a limit of finite-dimensional Lebesgue integrals. The objective of this talk is to apply this theory to developing a constructive solution to the Ornstein-Uhlenbech equation : ∂u(x, t) ∂t = 4u(x, t) + x · 5u(x, t) , u(x, 0) = φ(x) (1) where x ∈ B, φ ∈ C2 0 [B], and t ∈ [0, ∞). Our approach is constructive in the sense that the solution u(x, t) of equation (1) is expressible as an integral R B · dλB which, by the aforementioned definition, equals a limit of Lebesgue integrals on Euclidean space as the dimension n → ∞. Thus with this theory we may evaluate infinite-dimensional quantities, such as the solution u(x, t), by means of finite-dimensional approximation.

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A Constructive Solution to the Ornstein-Uhlenbech Equation on a Separable Banach Space of Infinite Dimension

Gill and Myers [GM] proved that every separable infinite-dimensional Banach space, denoted B, has an isomorphic, isometric embedding in R ∞ = R × R × · · · . They used this result and a method due to Yamasaki [YA] to construct a sigma-finite Lebesgue measure λB for B and defined the associated integral R B · dλB in a way that equals a limit of finite-dimensional Lebesgue integrals. The objective of this talk is to apply this theory to developing a constructive solution to the Ornstein-Uhlenbech equation : ∂u(x, t) ∂t = 4u(x, t) + x · 5u(x, t) , u(x, 0) = φ(x) (1) where x ∈ B, φ ∈ C2 0 [B], and t ∈ [0, ∞). Our approach is constructive in the sense that the solution u(x, t) of equation (1) is expressible as an integral R B · dλB which, by the aforementioned definition, equals a limit of Lebesgue integrals on Euclidean space as the dimension n → ∞. Thus with this theory we may evaluate infinite-dimensional quantities, such as the solution u(x, t), by means of finite-dimensional approximation.