Diffusions as Integral Curves, or Stratonovich without Itô

Abstract

The purpose of this article is to present a somewhat novel approach to the construction and basic analysis of Stratonovich diffusions. That is, given smooth vector fields V ₀,…, V _d on ℝ^N, let L denote the (possibly degenerate) second order elliptic operator

$$L = \frac{1}{2}\sum\limits_{k = 1}^d {V_k^2 + {V_0},} $$

(0.1)

where (cf. the discussion preceding Lemma 1.2 below) we think of V _k as a first order differential operator (i.e., directional derivative) and use V ² _k to denote the second order operator obtained by applying V _k twice. Then, letting (θ, B _θ, W) denote the standard Wiener space of ℝ^d-valued paths (cf. the discussion preceding (1.10)), our goal is to describe a measurable map $\theta \in \Theta \mapsto X( \cdot , \cdot ,\Theta ) \in {C^{0,\infty }}([0,\infty )x{\mathbb{R}^N};{\mathbb{R}^N})$ with the property that, for each x ∈ ℝ^N, the distribution of $\theta \in \Theta \mapsto X( \cdot ,x,\Theta ) \in C([0,\infty );{\mathbb{R}^N})$ under W solves the martingale problem for L starting at x (cf. [13]).