Malliavin calculus for stochastic flows

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Certain infinite particle systems with interactions are defined by stochastic differential equations (SDE's) on infinite dimensional spaces. It would be an interesting problem to study on the partial hypoellipticity of infinitesimal generators on infinite dimensional spaces associated with those SDE's. The main work of the present research is the study on the partial hypoellipticity vi the Malliavin calculus for stochastic flows on infinite dimensional spaces defined by SDE's.Let a_0(x, x^^-) be an R^N-valued smooth function on R^NxR^N, ν(dθ) be a finite measure on a discrete space θ, β_t=(β^θ_t) be a Wiener process, and let μ(dυ) be a measure satisfying a strong integrability condition. Consider a system of SDE's of the specific type : for u∈R^d,dx^u(t)=(∫a_0(x^u(t), x^υ(t))μ(dυ))dt+∫ν(dθ)a_θ(x^u(t))οdβ^θ_t.Assume that x^u(0) is smooth in u and the process x(t)=(x^u(t)) takes its values in the Hilbert space H=(L^2(R^d, β(R^d),μ))^N. Let π: H→R^M be a bounded linear mapping. The existence of the smooth density of the law of the random variable π(x(T)) is called the partial hypoellipticity of the SDE. Introduce the partial Hormander condition for vector fields on H:A_0=∫∫μ(du)μ(dυ)a_0(x^u, x^υ)・∂/(∂x^u), A_θ=∫μ(du)a_θ(x^u)・∂/(∂x^u)・The partial Hormander theorem that the partial hypoellipticity holds under the partial Hormander condition is proved proceeding the Malliavin calculus for SDE's on the Hilbert space H. The partial Hormander theorem can be applied to the problem about the propagation of absolute continuity of measures induced by stochastic flows defined by a certain system of SDE's.