ABSTRACT
This chapter is concerned with some parabolic Itoˆ equations in Rd. In Chapter 3 we considered such equations in a bounded domain D. The analysis was based mainly on the method of eigenfunction expansion for the associated elliptic boundary-valued problem. In Rd the spectrum of such an elliptic operator may be continuous and the method of eigenfunction expansion is no longer suitable. For an elliptic operator with constant coefficients, the method of Fourier transform is a natural substitution. Consider the initial-value or Cauchy problem for the heat equation on the real line:
∂u
∂t =
∂2u
∂x2 + f(x, t), −∞ < x <∞, t > 0,
u(x, 0) = h(x),
(4.1)
where f and g are given smooth functions. Let uˆ(λ, t) denote a Fourier transform of u in the space variable defined by
uˆ(λ, t) =
√ 1
2π
∫ e−iλxu(x, t)dx
with i = √−1, where λ ∈ R is a parameter and the integration is over the real
line. By applying a Fourier transform to the heat equation, it yields a simple ordinary differential equation
duˆ
dt = −λ2uˆ+ fˆ(λ, t), uˆ(λ, 0) = hˆ(λ), (4.2)
which can be easily solved to give
uˆ(λ, t) = hˆ(λ)e−λ 2t +
∫ t
e−λ 2(t−s)fˆ(λ, s)ds.
