Feynman formulas and functional integrals for diffusion with drift in a domain on a manifold

We obtain representations for the solution of the Cauchy-Dirichlet problem for the diffusion equation with drift in a domain on a compact Riemannian manifold as limits of integrals over the Cartesian powers of the domain; the integrands are elementary functions depending on the geometric characteristics of the manifold, the coefficients of the equation, and the initial data. It is natural to call such representations Feynman formulas. Besides, we obtain representations for the solution of the Cauchy-Dirichlet problem for the diffusion equation with drift in a domain on a compact Riemannian manifold as functional integrals with respect to Weizsäcker-Smolyanov surface measures and the restriction of the Wiener measure to the set of trajectories in the domain; such a restriction of the measure corresponds to Brownian motion in a domain with absorbing boundary. In the proof, we use Chernoff’s theorem and asymptotic estimates obtained in the papers of Smolyanov, Weizsäcker, and their coauthors.     

Function integrals corresponding to a solution of the Cauchy-Dirichlet problem for the heat equation in a domain of a Riemannian manifold

A solution of the Cauchy-Dirichlet problem is represented as the limit of a sequence of integrals over finite Cartesian powers of the domain of the manifold considered. It is shown that these limits coincide with the integrals with respect to surface measures of Gauss type on the set of trajectories in the manifold. Moreover, the integrands are a combination of elementary functions of the coefficients of the equation considered and geometric characteristics of the manifold. Also, a solution of the Cauchy-Dirichlet problem in the domain of the manifold is represented as the limit of a solution of the Cauchy problem for the heat equation on the whole manifold under an infinite growth of the absolute value of the potential outside the domain. The proof uses some asymptotic estimates for Gaussian integrals over Riemannian manifolds and the Chernoff theorem. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 3–15, 2006.     

Two Dimensional Landau-Lifshitz Equation

In this paper, we consider the Cauchy problem for two dimensional Landau-Lifshitz equation on 2-dimenslonal Riemannian manifold M without boundary. We proved that if u: M × R_+ → S², is a weak solution, then u is unique and smooth on M × R_+ with the exception of finitely many points.     

紧致黎曼流形上一类非线性热方程的梯度估计

In this paper,we study gradient estimates for the nonlinear heat equation ut-△u =au log u,on compact Riemannian manifold with or without boundary.We get a Hamilton type gradient estimate for the positi...     

Local Schrodinger flow into Kahler manifolds

In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the Schrodinger flow for maps from a compact Riemannian manifold into a complete Kahler manifold, or from a Euclidean space Rm into a compact Kahler manifold. As a consequence, we prove that Heisenberg spin system is locally well-posed in the appropriate Sobolev spaces.     

Local Schr?dinger flow into K?hler manifolds

In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the Schr?dinger flow for maps from a compact Riemannian manifold into a complete K?hler manifold, or from a Euclidean space R^m into a compact K?hler manifold. As a consequence, we prove that Heisenberg spin system is locally well-posed in the appropriate Sobolev spaces.     

Local Existence for Inhomogeneous Schrödinger Flow into Kähler Manifolds

In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the inhomogeneous Schr?dinger flow for maps from a compact Riemannian manifold M with dim(M) ≤ 3 into a compact K?hler manifold (N, J) with nonpositive Riemannian sectional curvature Received November 1, 1999, Revised January 14, 2000, Accepted March 29, 2000     

Quasi-invariant stochastic flows of SDEs with non-smooth drifts on compact manifolds

In this article we prove that stochastic differential equation (SDE) with Sobolev drift on a compact Riemannian manifold admits a unique ν-almost everywhere stochastic invertible flow, where ν is the Riemannian measure, which is quasi-invariant with respect to ν. In particular, we extend the well-known DiPerna-Lions flows of ODEs to SDEs on a Riemannian manifold.     

A remark on the Harnack inequality for non-self-adjoint evolution equations

In this paper we consider a non-self-adjoint evolution equation on a compact Riemannian manifold with boundary. We prove a Harnack inequality for a positive solution satisfying the Neumann boundary condition. In particular, the boundary of the manifold may be nonconvex and this gives a generalization to a theorem of Yau.

     

Energy Decay for the Cauchy Problem of the Linear Wave Equation of Variable Coefficients with Dissipation

Decay of the energy for the Cauchy problem of the wave equation of variable coefficients with a dissipation is considered. It is shown that whether a dissipation can be localized near infinity depends on the curvature properties of a Riemannian metric given by the variable coefficients. In particular, some criteria on curvature of the Riemannian manifold for a dissipation to be localized are given.     

Energy Decay for the Cauchy Problem of the Linear Wave Equation of Variable Coeffcients with Dissipation

Decay of the energy for the Cauchy problem of the wave equation of variable coeffcients with a dissipation is considered.It is shown that whether a dissipation can be localized near infinity depends on the curvature properties of a Riemannian metric given by the variable coeffcients.In particular,some criteria on curvature of the Riemannian manifold for a dissipation to be localized are given.     

A class of infinite dimensional stochastic processes with unbounded diffusion

The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron–Martin space. In particular, it is shown that under a class of changes of the reference measure, quasi-regularity of the form is preserved. We also show that under these changes of the reference measure, derivative and divergence are closable with certain closable inverses. We first treat the case of the classical Wiener space and then we transfer the results to the Wiener space over a Riemannian manifold.     

Stochastic systems in Riemannian manifolds

A formulation of stochastic systems in a Riemannian manifold is given by stochastic differential equations in the tangent bundle of the manifold. Brownian motion is constructed in a compact Riemannian manifold as well as the horizontal lift of this process to the bundle of orthonormal frames. The solution of some stochastic differential equations in the tangent bundle of the manifold is defined by the transformation of the measure for the manifold-valued Brownian motion by a suitable Radon-Nikodym derivative. Real-valued stochastic integrals are defined for this Brownian motion using parallelism along the Brownian paths. A stochastic control problem is formulated and solved for these stochastic systems where a suitable convexity condition is assumed.This research was supported by NSF Grants Nos. GK-32136, ENG-75-06562, and MCS-76-01695.The author wishes to thank D. Gromoll, J. Simons, and J. Thorpe for some helpful conversations on differential geometry.     

Geodesic flows on path spaces

On the path space over a compact Riemannian manifold, the global existence and the global uniqueness of the quasi-invariant geodesic flows with respect to a negative Markov connection are obtained in this paper. The results answer affirmatively a left problem of Li.     

Rigidity results for spin manifolds with foliated boundary

In this paper, we consider a compact Riemannian manifold whose boundary is endowed with a Riemannian flow. Under a suitable curvature assumption depending on the O’Neill tensor of the flow, we prove that any solution of the basic Dirac equation is the restriction of a parallel spinor field defined on the whole manifold. As a consequence, we show that the flow is a local product. In particular, in the case where solutions of the basic Dirac equation are given by basic Killing spinors, we characterize the geometry of the manifold and the flow.     

Quasi-Invariance of the Wiener Measure on Path Spaces: Noncompact Case

For a geometrically and stochastically complete, noncompact Riemannian manifold, we show that the flows on the path space generated by the Cameron-Martin vector fields exist as a set of random variables. Furthermore, if the Ricci curvature grows at most linearly, then the Wiener measure (the law of Brownian motion on the manifold) is quasi-invariant under these flows.     

On the Cauchy problem for the transport equation with random noise

We establish the existence and uniqueness of a solution to the Cauchy problem for the transport equation with random noise in Rd. When the vector field is time-periodic and the noise is multiplicative and nondegenerate, we show the existence of a time-periodic measure, which includes an invariant measure as a special case.     

Quasi-invariance of the Wiener measure on the path space over a complete Riemannian manifold

We prove a generalization of the Cameron-Martin theorem for a geometrically and stochastically complete Riemannian manifold; namely, the Wiener measure on the path space over such a manifold is quasi-invariant under the flow generated by a Cameron-Martin vector field.     

Travel Time Tomography

We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary rigidity problem is whether we can determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points. The lens rigidity problem problem is to determine a Riemannian metric of a Riemannian manifold with boundary by measuring for every point and direction of entrance of a geodesic the point of exit and direction of exit and its length. The linearization of these two problems is tensor tomography. The question is whether one can determine a symmetric two-tensor from its integrals along geodesics. We emphasize recent results on boundary and lens rigidity and in tensor tomography in the partial data case, with further applications.     

Anomaly formulas for the complex-valued analytic torsion on compact bordisms

We extend the complex-valued analytic torsion, introduced by Burghelea and Haller on closed manifolds, to compact Riemannian bordisms. We do so by considering a flat complex vector bundle over a compact Riemannian manifold, endowed with a fiberwise nondegenerate symmetric bilinear form. The Riemmanian metric and the bilinear form are used to define non-selfadjoint Laplacians acting on vector-valued smooth forms under absolute and relative boundary conditions. In order to define the complex-valued analytic torsion in this situation, we study spectral properties of these generalized Laplacians. Then, as main results, we obtain so-called anomaly formulas for this torsion. Our reasoning takes into account that the coefficients in the heat trace asymptotic expansion associated to the boundary value problem under consideration, are locally computable. The anomaly formulas for the complex-valued Ray–Singer torsion are derived first by using the corresponding ones for the Ray–Singer metric, obtained by Brüning and Ma on manifolds with boundary, and then an argument of analytic continuation. In odd dimensions, our anomaly formulas are in accord with the corresponding results of Su, without requiring the variations of the Riemannian metric and bilinear structures to be supported in the interior of the manifold.