## 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

Zbl 0495.60005
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