Cameron–Martin theorem

Cameron–Martin theorem In mathematics, the Cameron–Martin theorem or Cameron–Martin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain elements of the Cameron–Martin Hilbert space.

Contents 1 Motivation 2 Statement of the theorem 3 Integration by parts 4 An application 5 See also 6 References Motivation The standard Gaussian measure γn on n-dimensional Euclidean space Rn is not translation-invariant. (In fact, there is a unique translation invariant Radon measure up to scale by Haar's theorem: the n-dimensional Lebesgue measure, denoted here dx.) Instead, a measurable subset A has Gaussian measure {displaystyle gamma _{n}(A)={frac {1}{(2pi )^{n/2}}}int _{A}exp left(-{tfrac {1}{2}}langle x,xrangle _{mathbf {R} ^{n}}right),dx.} Here {displaystyle langle x,xrangle _{mathbf {R} ^{n}}} refers to the standard Euclidean dot product in Rn. The Gaussian measure of the translation of A by a vector h ∈ Rn is {displaystyle {begin{aligned}gamma _{n}(A-h)&={frac {1}{(2pi )^{n/2}}}int _{A}exp left(-{tfrac {1}{2}}langle x-h,x-hrangle _{mathbf {R} ^{n}}right),dx\[4pt]&={frac {1}{(2pi )^{n/2}}}int _{A}exp left({frac {2langle x,hrangle _{mathbf {R} ^{n}}-langle h,hrangle _{mathbf {R} ^{n}}}{2}}right)exp left(-{tfrac {1}{2}}langle x,xrangle _{mathbf {R} ^{n}}right),dx.end{aligned}}} So under translation through h, the Gaussian measure scales by the distribution function appearing in the last display: {displaystyle exp left({frac {2langle x,hrangle _{mathbf {R} ^{n}}-langle h,hrangle _{mathbf {R} ^{n}}}{2}}right)=exp left(langle x,hrangle _{mathbf {R} ^{n}}-{tfrac {1}{2}}|h|_{mathbf {R} ^{n}}^{2}right).} The measure that associates to the set A the number γn(A−h) is the pushforward measure, denoted (Th)∗(γn). Here Th : Rn → Rn refers to the translation map: Th(x) = x + h. The above calculation shows that the Radon–Nikodym derivative of the pushforward measure with respect to the original Gaussian measure is given by {displaystyle {frac {mathrm {d} (T_{h})_{*}(gamma ^{n})}{mathrm {d} gamma ^{n}}}(x)=exp left(leftlangle h,xrightrangle _{mathbf {R} ^{n}}-{tfrac {1}{2}}|h|_{mathbf {R} ^{n}}^{2}right).} The abstract Wiener measure γ on a separable Banach space E, where i : H → E is an abstract Wiener space, is also a "Gaussian measure" in a suitable sense. How does it change under translation? It turns out that a similar formula to the one above holds if we consider only translations by elements of the dense subspace i(H) ⊆ E.

Statement of the theorem Let i : H → E be an abstract Wiener space with abstract Wiener measure γ : Borel(E) → [0, 1]. For h ∈ H, define Th : E → E by Th(x) = x + i(h). Then (Th)∗(γ) is equivalent to γ with Radon–Nikodym derivative {displaystyle {frac {mathrm {d} (T_{h})_{*}(gamma )}{mathrm {d} gamma }}(x)=exp left(langle h,xrangle ^{sim }-{tfrac {1}{2}}|h|_{H}^{2}right),} where {displaystyle langle h,xrangle ^{sim }=I(h)(x)} denotes the Paley–Wiener integral.

The Cameron–Martin formula is valid only for translations by elements of the dense subspace i(H) ⊆ E, called Cameron–Martin space, and not by arbitrary elements of E. If the Cameron–Martin formula did hold for arbitrary translations, it would contradict the following result: If E is a separable Banach space and μ is a locally finite Borel measure on E that is equivalent to its own push forward under any translation, then either E has finite dimension or μ is the trivial (zero) measure. (See quasi-invariant measure.) In fact, γ is quasi-invariant under translation by an element v if and only if v ∈ i(H). Vectors in i(H) are sometimes known as Cameron–Martin directions.

Integration by parts The Cameron–Martin formula gives rise to an integration by parts formula on E: if F : E → R has bounded Fréchet derivative DF : E → Lin(E; R) = E∗, integrating the Cameron–Martin formula with respect to Wiener measure on both sides gives {displaystyle int _{E}F(x+ti(h)),mathrm {d} gamma (x)=int _{E}F(x)exp left(tlangle h,xrangle ^{sim }-{tfrac {1}{2}}t^{2}|h|_{H}^{2}right),mathrm {d} gamma (x)} for any t ∈ R. Formally differentiating with respect to t and evaluating at t = 0 gives the integration by parts formula {displaystyle int _{E}mathrm {D} F(x)(i(h)),mathrm {d} gamma (x)=int _{E}F(x)langle h,xrangle ^{sim },mathrm {d} gamma (x).} Comparison with the divergence theorem of vector calculus suggests {displaystyle mathop {mathrm {div} } [V_{h}](x)=-langle h,xrangle ^{sim },} where Vh : E → E is the constant "vector field" Vh(x) = i(h) for all x ∈ E. The wish to consider more general vector fields and to think of stochastic integrals as "divergences" leads to the study of stochastic processes and the Malliavin calculus, and, in particular, the Clark–Ocone theorem and its associated integration by parts formula.

An application Using Cameron–Martin theorem one may establish (See Liptser and Shiryayev 1977, p. 280) that for a q × q symmetric non-negative definite matrix H(t) whose elements Hj,k(t) are continuous and satisfy the condition {displaystyle int _{0}^{1}sum _{j,k=1}^{q}|H_{j,k}(t)|,dt

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