Celebratio Mathematica — Varadhan

Diffusion theory

Works connected to Jeremy Quastel

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J. Quastel and S. R. S. Varadhan: “Diffusion semigroups and diffusion processes corresponding to degenerate divergence form operators,” Comm. Pure Appl. Math. 50 : 7 (July 1997), pp. 667–706. MR 1447057 Zbl 0907.47040 article

In this article we study existence and uniqueness problems for semigroups and processes determined by time-dependent, second-order linear partial differential operators of the form \[ L_tu = \nabla\cdot a\nabla u + \sigma c\cdot\nabla u .\] For simplicity we will take the space to be the \( d \)-dimensional torus \( \mathbb{T}^d \). The matrix \[ a:[0,T]\times\mathbb{T}^d\to\mathbb{R}^d\otimes\mathbb{R}^d \] is assumed to be symmetric and nonnegative definite. We define \( \sigma \) to be the symmetric, nonnegative definite square root of \( a \), and \( c:[0,T]\times\mathbb{T}^d\to\mathbb{R}^d \) to be a function about which assumptions will be made later. The focus of our study is on cases where the coefficients \( a \) and \( c \) have minimal regularity, and \( a \) is allowed to degenerate.