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

Lyons, Terry; Stoica, Lucreţiu

Ann. Probab. 27, No. 1, 1-49 (1999).

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

Keywords:

Lyons-Zheng decomposition; martingale decomposition; stochastic integral; singular integral; path integral

Citations:

Zbl 0654.60059; Zbl 0804.60044

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