# Dilation Vector Field on Wiener Space

• Hélène Airault
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 34)

## Abstract

We consider the heat operator Δ H , heat equation, and heat kernel measures (ν t ) t≥0 on Wiener space Ω as explained in Driver (Contemp. Math. 338:101–141, 2003). We define the notion of heat dilation vector field associated to a family of probability measures (μ t ) t≥0 on Ω. Let ω ∈Ω. The vector field V on Ω is expressed for F(ω)=f(ω(t 1),ω(t 2),,ω(t n )) as VF(ω)=(vf)(ω(t 1), ω(t 2), ,ω(t n )) where $$vf =\sum _{ k=1}^{n}x_{k} \frac{\partial } {\partial x_{k}}$$. The vector field V is shown to be a heat vector field for the heat kernel measures (ν t ) t≥0. We project down “through a nondegenerate map Z”, Ornstein–Uhlenbeck operators defined on Ω by $$\mathcal{L}_{t}F = t\Delta _{H}F - V F$$. We obtain a first-order partial differential equation for the density of the random vector Z. We compare this differential equation to the heat equation and to Stein’s equation for the density.

## Keywords

Wiener space Heat kernel measures Dilation vector fields Ornstein–Uhlenbeck operators Malliavin calculus

## Notes

### Acknowledgements

A preliminary version of this note was presented in June 2011 at the “Journée mathématique d’Amiens”. I thank Frederi Viens for friendly and useful discussions in July 2011. I also thank the referee for significant suggestions for the writing.

## References

1. 1.
Airault, H., Malliavin, P., Viens, F.: Stokes formula on the Wiener space and n-dimensional Nourdin-Peccati analysis. J. Funct. Anal. 258(5), 1763–1783 (2010)
2. 2.
Airault, H., Malliavin, P.: Integration by parts formulas and dilatation vector fields on elliptic probability spaces. Probab. Theor. Rel. Fields. 106(4), 447–494 (1996)
3. 3.
Cameron, R.H., Martin, W.T.: Transformations of Wiener integrals under translations. Ann. Math. 45(2), 386–396 (1944)
4. 4.
Driver, B.K.: Heat kernels measures and infinite dimensional analysis. Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002). Contemp. Math. 338, 101–141 (2003) (American Mathematical Society, Providence)Google Scholar
5. 5.
Malliavin, P.: Analyse différentielle sur l’espace de Wiener. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Warsaw, 1983), pp. 1089–1096. Warsaw (1984)Google Scholar
6. 6.
Malliavin, P.: Stochastic analysis. Grundlehren der Mathematischen Wissenschaften, vol. 313. Springer, Berlin (1997)Google Scholar
7. 7.
Malliavin, P.: Calcul des variations, intégrales stochastiques et complexes de De Rham sur l’espace de Wiener. C.R. Acad. Sci. Paris Série I, 299(8), 347–350 (1984)Google Scholar
8. 8.
Malliavin, P.: Calcul des variations stochastiques subordonné au processus de la chaleur. C. R. Acad. Sci. Paris Sér. I Math. 295(2), 167–172 (1982)
9. 9.
Mancino, M.E.: Dilatation vector fields on the loop group. J. Funct. Anal. 166(1), 130–147 (1999)
10. 10.
Nourdin, I., Peccati, G.: Stein’s method on Wiener chaos. Probab. Theor. Rel. Fields 145(1–2), 75–118 (2009)
11. 11.
Nourdin, I., Viens, F.G.: Density formula and concentration inequalities with Malliavin calculus. Electron. J. Probab. 14(78), 2287–2309 (2009)
12. 12.
Nualart, D.: The Malliavin calculus and related topics. In: Probability and its Applications. Springer, Berlin (1995)Google Scholar
13. 13.
Wiener, N.: Differential space. J. Math. Phys. 2, 131–174 (1923)Google Scholar