Applications of the Malliavin calculus. II.

*(English)*Zbl 0568.60059In part I [Stochastic analysis, Proc. Taniguchi Int. Symp., Katata & Kyoto/Jap. 1982, North-Holland Math. Libr. 32, 271-306 (1984; Zbl 0546.60056)] the authors established general regularity properties of Ito processes which are not necessarily Markov processes by means of the Malliavin calculus. The present article centers on the Markovian setting. In this situation the Malliavin covariance matrix A(t,x) can be expressed in a much more tractable way. From this presentation the authors obtain in section 2 estimates on which the analysis is resting. The mentioned estimate gives an exponential upper bound for
\[
{\mathcal W}(\lambda (T/K^{1/(L+1)},x,F)/T^ L\leq 1/K)
\]
where \({\mathcal W}\) is the Wiener measure, \(F\subseteq S^{N-1}\) is a closed set, K,L\(\in [1,\infty)\), \(T\in (0,1]\) and \(\lambda (t,x,F)=\inf_{\eta \in F}(\eta,A(t,x)\eta).\) The exact formulation of this and all other results of the paper needs too much notation and technical details in order to reproduce it here. For this reason we describe further content in the verbal way similar to the authors’ introduction.

In the following sections 3 to 5 regularity properties of the transition probability function of the considered diffusion process are obtained. Section 3 is concerned with global properties under global assumptions on the coefficients in the Ito equation. In section 4 the authors impose local regularity assumptions on the coefficients and obtain local regularity properties, i.e. regularity properties on an open subset of \({\mathbb{R}}^ N\). In section 5 they ”microlocalize” their results which means they study the regularity of a marginal distribution of the diffusion on a submanifold. In the strictly elliptic context this was also done in the second author’s paper, Ecole d’été de probabilités de Saint-Flour XI-1981, Lect. Notes Math. 976, 267-382 (1983; Zbl 0494.60060).

Section 6 is devoted to transfering the regularity properties from the transition density to the resolvent kernel and its relation to the rank of the associated Lie algebra. In section 7 the authors provide criteria which guarantee that the fundamental solution is regular even when Hörmander’s condition on the rank of the associated Lie algebra fails to hold.

The last section contains applications to the study of hypoellipticity. Here again hypoellipticity is proved for operators which do not satisfy Hörmander’s condition. The authors discuss the connection between hypoellipticity and subellipticity [cf. C. Fefferman and D. H. Phong, Harmonic analysis, Conf. in Honor A. Zygmund, Chicago 1981, Vol. 2, 590-606 (1983; Zbl 0503.35071)]. The distinction lies in the way the transition density explodes for \(t\downarrow 0:\) if it explodes polynomially the operator is subelliptic, if the explosion is faster the operator may be hypoelliptic but will not be subelliptic.

Finally the authors consider a special example on \({\mathbb{R}}^ 3\). Suppose \(\sigma \in C_ b^{\infty}({\mathbb{R}}^ 1)\) has the properties \(\sigma (x)=0\) if and only if \(x=0\), \((\sigma (\cdot))^ 2\) is nondecreasing on [0,\(\infty)\) and even on \({\mathbb{R}}^ 1\). With this \(\sigma\) define \[ L_{\sigma}:=\frac{1}{2}(\frac{\partial^ 2}{\partial x^ 2}+(\sigma (x)\frac{\partial}{\partial y})^ 2+\frac{\partial^ 2}{\partial z^ 2}). \] It is proven that \(L_{\sigma}\) is hypoelliptic if and only if \(\lim_{x\to 0}x \log (1/| \sigma (x)|)=0\).

In the following sections 3 to 5 regularity properties of the transition probability function of the considered diffusion process are obtained. Section 3 is concerned with global properties under global assumptions on the coefficients in the Ito equation. In section 4 the authors impose local regularity assumptions on the coefficients and obtain local regularity properties, i.e. regularity properties on an open subset of \({\mathbb{R}}^ N\). In section 5 they ”microlocalize” their results which means they study the regularity of a marginal distribution of the diffusion on a submanifold. In the strictly elliptic context this was also done in the second author’s paper, Ecole d’été de probabilités de Saint-Flour XI-1981, Lect. Notes Math. 976, 267-382 (1983; Zbl 0494.60060).

Section 6 is devoted to transfering the regularity properties from the transition density to the resolvent kernel and its relation to the rank of the associated Lie algebra. In section 7 the authors provide criteria which guarantee that the fundamental solution is regular even when Hörmander’s condition on the rank of the associated Lie algebra fails to hold.

The last section contains applications to the study of hypoellipticity. Here again hypoellipticity is proved for operators which do not satisfy Hörmander’s condition. The authors discuss the connection between hypoellipticity and subellipticity [cf. C. Fefferman and D. H. Phong, Harmonic analysis, Conf. in Honor A. Zygmund, Chicago 1981, Vol. 2, 590-606 (1983; Zbl 0503.35071)]. The distinction lies in the way the transition density explodes for \(t\downarrow 0:\) if it explodes polynomially the operator is subelliptic, if the explosion is faster the operator may be hypoelliptic but will not be subelliptic.

Finally the authors consider a special example on \({\mathbb{R}}^ 3\). Suppose \(\sigma \in C_ b^{\infty}({\mathbb{R}}^ 1)\) has the properties \(\sigma (x)=0\) if and only if \(x=0\), \((\sigma (\cdot))^ 2\) is nondecreasing on [0,\(\infty)\) and even on \({\mathbb{R}}^ 1\). With this \(\sigma\) define \[ L_{\sigma}:=\frac{1}{2}(\frac{\partial^ 2}{\partial x^ 2}+(\sigma (x)\frac{\partial}{\partial y})^ 2+\frac{\partial^ 2}{\partial z^ 2}). \] It is proven that \(L_{\sigma}\) is hypoelliptic if and only if \(\lim_{x\to 0}x \log (1/| \sigma (x)|)=0\).

Reviewer: M.Breger

##### MSC:

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60J60 | Diffusion processes |

65H10 | Numerical computation of solutions to systems of equations |

58J65 | Diffusion processes and stochastic analysis on manifolds |