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In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a stochastic process is a partial differential operator that encodes a great deal of information about the process. The generator is used in evolution equations such as the Kolmogorov backward equation (which describes the evolution of statistics of the process); its ''L''2 Hermitian adjoint is used in evolution equations such as the Fokker–Planck equation (which describes the evolution of the probability density functions of the process). ==Definition== Let ''X'' : [0, +∞) × Ω → R''n'' defined on a probability space (Ω, Σ, P) be an Itô diffusion satisfying a stochastic differential equation of the form : where ''B'' is an ''m''-dimensional Brownian motion and ''b'' : R''n'' → R''n'' and ''σ'' : R''n'' → R''n''×''m'' are the drift and diffusion fields respectively. For a point ''x'' ∈ R''n'', let P''x'' denote the law of ''X'' given initial datum ''X''0 = ''x'', and let E''x'' denote expectation with respect to P''x''. The infinitesimal generator of ''X'' is the operator ''A'', which is defined to act on suitable functions ''f'' : R''n'' → R by : The set of all functions ''f'' for which this limit exists at a point ''x'' is denoted ''D''''A''(''x''), while ''D''''A'' denotes the set of all ''f'' for which the limit exists for all ''x'' ∈ R''n''. One can show that any compactly-supported ''C''2 (twice differentiable with continuous second derivative) function ''f'' lies in ''D''''A'' and that : or, in terms of the gradient and scalar and Frobenius inner products, : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Infinitesimal generator (stochastic processes)」の詳細全文を読む スポンサード リンク
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