Convolution semigroups, canonical processes, and Brownian motion
نویسنده
چکیده
the product σ-algebra. Let A : E × E → E be A(x1, x2) = x1 + x2. For ν1, ν2 ∈ P(E), the convolution of ν1 and ν2 is the pushforward of the product measure ν1 × ν2 by A: ν1 ∗ ν2 = A∗(ν1 × ν2). The convolution ν1 ∗ ν2 is an element of P(E). Let I = R≥0. A convolution semigroup is a family (νt)t∈I of elements of P(E) such that for s, t ∈ I, νs+t = νs ∗ νt. From this, it turns out that μ0 = δ0. 2 A convolution semigroup is called continuous when the map t 7→ νt is continuous I →P(E). 1See http://individual.utoronto.ca/jordanbell/notes/narrow.pdf 2http://individual.utoronto.ca/jordanbell/notes/markovkernels.pdf, Theorem 3.
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