## 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)

### MSC:

 60J60 Diffusion processes 60H05 Stochastic integrals 31C25 Dirichlet forms

### Citations:

Zbl 0654.60059; Zbl 0804.60044
Full Text:

### References:

 [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
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