The limits of stochastic integrals of differential forms. (English) Zbl 0969.60078

This long article is concerned with developing an appropriate stochastic calculus for a diffusion process \(X\) on a manifold \(E\). It is assumed that \(X\) is associated with a second-order elliptic differential operator in divergence form with measurable coefficients. The first main result is an extension of the Lyons-Zheng decomposition to the case of fixed starting point \(o\in E\); if \(f\) is a bounded function of finite energy, the process \(f(X_t)\) can be uniquely decomposed under \(P^o\) into the sum of an additive functional martingale \((M_t^s,\;t\geq s)\), a backward martingale \((\overline{M}_s^t,\;0<s\leq t)\), and two additive functionals \((\alpha_t^s,\beta_t^s,\;t\geq s)\) of finite variation: \[ f(X_t)=f(X_s)+\frac 12M^s_t-\frac 12\overline{M}^t_s- \alpha_t^s+\beta_t^s\qquad 0<s\leq t . \] This extends results of T. J. Lyons and W. Zheng [in: Les processus stochastiques. Astérisque 157-158, 249-271 (1988; Zbl 0654.60059)] and T. J. Lyons and T. S. Zhang [Ann. Probab. 22, No. 1, 494-524 (1994; Zbl 0804.60044)]. This decomposition is then used to define a Stratonovich type stochastic integral \(\int_s^t\omega \circ dX \) of a differential 1-form \(\omega\) along the path \(X\). This integral is a priori only defined for \(s>0\), but it is shown that it converges under certain conditions \(P^o\)-a.s. to a finite limit as \(s\downarrow 0\). Similarly, the limiting behavior for \(t\uparrow\zeta\) (the lifetime of \(X\)) is studied. Particular results are obtained in the case where \(X\) is a Brownian motion on a negatively curved complete Riemannian manifold. Section 8 contains estimates on the heat kernel and its gradient.
Reviewer: A.Schied (Berlin)


60J60 Diffusion processes
60H05 Stochastic integrals
31C25 Dirichlet forms
Full Text: DOI


[1] Bliedtner, J. and Hansen, W. (1986). Potential Theory. Springer, New York. · Zbl 0706.31001
[2] Blumenthal, R. M. and Getoor, R. K. (1986). Markov Processes and Potential Theory. Academic Press, New York. · Zbl 0169.49204
[3] Constantinescu, C. and Cornea, A. (1972). Potential Theory on Harmonic Spaces Springer, New York. · Zbl 0248.31011
[4] Courr ege, Ph. and Priouret, P. (1965). Recollements de processus de Markov. Publ. Inst. Statist. Univ. Paris 14 275-377. · Zbl 0275.30026
[5] Fabes, E. and Stroock, D. (1986). A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Rational Mech. Anal. 96 327-338. · Zbl 0652.35052
[6] Fukushima, M., Oshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin. · Zbl 0838.31001
[7] Gilbarg, D. and Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order. Springer, Berlin. · Zbl 0562.35001
[8] Ladyzenskaya, O. A., Uraltseva, N. N. and Solonikov, V. A. (1867). Linear and Quasilinear Equations of Parabolic Type. Nauka, Moskow. (In Russian.)
[9] Li, P. and Karp, L. (1998). The heat equation on complete Riemannian manifolds. Unpublished manuscript.
[10] Li, P. and Yau, S. T. (1986). On the upper estimate of the heat kernel of a complete Riemannian manifold. Acta Math. 156 153-201. · Zbl 0611.58045
[11] Lyons, T. (1998). Random thoughts on reversible potential theory. Unpublished manuscript. · Zbl 0757.31007
[12] Lyons, T. and Stoica, L. (1996). On the limit of stochastic integrals of differential forms. Stochastics Monogr. 10. Gordon and Breach, Yverdon. · Zbl 0899.60046
[13] Lyons, T. J. and Zhang, T. S. (1994). Decomposition of Dirichlet processes and its application. Ann. Probab. 22 1-26. · Zbl 0804.60044
[14] Lyons, T. J. and Zheng, W. (1988). A crossing estimate for the canonical process on a Dirichlet space and a tightness result. Astérisque 157-158 249-271. · Zbl 0654.60059
[15] Lyons, T. J. and Zheng, W. A. (1990). On conditional diffusion processes. Proc. Roy. Soc. Edinburgh Sec. A 115 243-255. · Zbl 0715.60097
[16] Ma,M. and R öckner, M. (1992). Introduction to the Theory of Dirichlet Forms. Springer, New York. · Zbl 0826.31001
[17] Prat, J.-J. (1971). Étude asymptotique du mouvement brownian sur une variété riemannienne a courbure négative. C.R. Acad. Sci. Paris Sér. A 272 1586-1589. · Zbl 0296.60053
[18] Saloff-Coste, L. (1992). Uniformly elliptic operators on Riemannian manifolds. J. Diff. Geometry 36 417-450. · Zbl 0735.58032
[19] Stoica, L. (1980). Local Operators and Markov Processes. Lecture Notes in Math. 816. Springer, Berlin. · Zbl 0446.60067
[20] Stroock, D. W. (1988). Diffusion semigroups corresponding to uniformly elliptic divergence form operators. Seminaire de Probabilités XXII Lecture Notes in Math. 1321 316-347. Springer, Berlin. · Zbl 0651.47031
[21] Sullivan, D. (1983). The Dirichlet problem at infinity for a negatively curved manifold. J. Diff. Geometry 18 723-732. · Zbl 0541.53037
[22] Takeda, M. (1991). On the conservatineness of the Brownian motion on a Riemannian manifold. Bull. London Math. Soc. 23 86-88. · Zbl 0748.60070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.