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Asymptotic expansion of the density of a degenerated diffusion. (Développement asymptotique de la densité d’une diffusion dégénérée.) (French) Zbl 0749.60054

The author proves the following main result. Suppose \(X_ 1,\dots,X_ m\) are \(C^ \infty\)-vectors fields whose derivatives of each order are bounded. The Lie algebra generated by the field \(X_ i\) is equal to \(\mathbb{R}^ d\) in every point \(x_ 0\). Consider the semigroup \(P_ t\) generated by \({1\over 2}\sum_{i=1}^ m X^ 2_ i\). By the Hörmander theorem, for \(t>0\) it exists a \(C^ \infty\)-function \(p_ t'(x_ 0,y)\) in the variable \(y\) such that \[ P_ t f(x_ 0)=\int_{R^ d} f(y)p_ t'(x_ 0,y)dy. \] Let be given the Stratonovitch differential equation in the form \[ dx_ t(\varepsilon,x_ 0)=\varepsilon \sum_{i=1}^ m X_ i(x_ t(\varepsilon,x_ 0))dw_ i, \qquad x_ 0(\varepsilon,x_ 0)=x_ 0. \] For some \(\varepsilon\), \(x_ 1(\varepsilon,x_ 0)\) has density \(p_ \varepsilon(x_ 0,y)\neq 0\) [see N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes (1981; Zbl 0495.60005)]; moreover, \(p_{\varepsilon^ 2}'(x_ 0,y)=p_ \varepsilon(x_ 0,y)\). Let \(E_ j(X,x_ 0)\) be the space in \(x_ 0\) generated by the Lie bracket and fields \(X_ i\), \(i=1,\dots,d\); moreover, let \(n_ j(X,x_ 0)\) be the dimension of this space. Put \[ N(X,x_ 0)=\sum_ j j(n_ j(X,x_ 0)-n_{j-1}(X,x_ 0))\text{ and } n_ 0(X,x_ 0)=0. \] If the Lie algebra generated by the field \(X_ i\) is equal to \(R^ d\) in \(x_ 0\), there exist constants \(a_ j(x_ 0)\), such that for \(\varepsilon\) sufficiently small we have \[ p_ \varepsilon(x_ 0,x_ 0)=\varepsilon^{-N(X,x_ 0)}\left(\sum_{j=0}^ N a_ j(x_ 0)^ j+o(\varepsilon^ N)\right),\tag{1} \] and \(a_ 0(x_ 0)\) is positive. If \(n_ j(X,x_ 0)\) is independent of \(x_ 0\) in compact \(K\subset\mathbb{R}^ d\), then we have (1) uniformly in \(K\) and \(a_ j(x_ 0)\) belongs to \(C^ \infty\) for \(x_ 0\).

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus

Citations:

Zbl 0495.60005
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