**Title:** How to discretize path integrals and related dynamics

**Speaker:** Zhehua Li

**Speaker Info:** Northwestern University

**Brief Description:**

**Special Note**:

**Abstract:**

Let $(M,g,o)$ be a pointed Riemannian manifold, and the sets of continuous paths or loops are referred to as curved spaces in this talk. The energy of a sample path or loop $\sigma(\cdot)$ in curved spaces is formally defined as \[ E(\sigma) := \int_0^T \langle \sigma'(s), \sigma'(s) \rangle_g ds\,. \] In this talk, we will report on some recent work centered around this energy from both the static and dynamic points of view.From the static point of view, the corresponding Gibbs measure \[ \frac{1}{Z} E(\sigma) \mathcal{D} \sigma \] is a mathematically ill-defined path integral, we make sense of it using finite-dimensional approximations. In more detail, we will interpret the above path integral expression as a limit of measures, $\nu_{\mathcal{P}}$, indexed by partitions, $\mathcal{P}$ of $[0,T]$. The measures $\nu_{\mathcal{P}}$ are constructed by restricting the above path integral expression to the finite dimensional manifolds of piecewise geodesics. The beauty of this kind of approximation lies in its nonuniqueness due to the underlying geometric structure, a phenomenon similar to operator ordering in geometric quantization.

Dynamically, we will consider a stochastic Langevin process associated with $E(\sigma)$. These formal dynamics give rise to an ill-defined stochastic PDE known as a coupled KPZ equation. Although its solution theory can be characterized by Martin Hairer's regularity structure theory, the characterization of its invariant measure is an open question. We will mention an incomplete attempt using the static theory developed above.

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