# Short-Time Geometry of Random Heat Kernels

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*Richard B. Sowers*

This volume studies the behavior of the random heat kernel associated with the stochastic partial differential equation \(du=\tfrac {1}{2} {\Delta}udt = (\sigma, \nabla u) \circ dW_t\), on some Riemannian manifold \(M\). Here \(\Delta\) is the Laplace-Beltrami operator, \(\sigma\) is some vector field on \(M\), and \(\nabla\) is the gradient operator. Also, \(W\) is a standard Wiener process and \(\circ\) denotes Stratonovich integration. The author gives short-time expansion of this heat kernel. He finds that the dominant exponential term is classical and depends only on the Riemannian distance function. The second exponential term is a work term and also has classical meaning. There is also a third non-negligible exponential term which blows up. The author finds an expression for this third exponential term which involves a random translation of the index form and the equations of Jacobi fields. In the process, he develops a method to approximate the heat kernel to any arbitrary degree of precision.

#### Table of Contents

# Table of Contents

## Short-Time Geometry of Random Heat Kernels

- Contents vii8 free
- Chapter 1. Introduction 110 free
- Chapter 2. Guessing the Dominant Asymptotics 2635
- Chapter 3. Initial Condition and Evolution of the Approximate Kernel 4857
- Chapter 4. The Minakshisundaram-Pleijel Coefficients 6271
- Chapter 5. Error Estimates, Proof of the Main Theorem, and Extensions 8594
- Appendices 111120
- Bibliography 128137