By Fabrice Baudoin

This booklet goals to supply a self-contained creation to the neighborhood geometry of the stochastic flows. It stories the hypoelliptic operators, that are written in Hörmander’s shape, through the use of the relationship among stochastic flows and partial differential equations.

The booklet stresses the author’s view that the neighborhood geometry of any stochastic move is decided very accurately and explicitly via a common formulation known as the Chen-Strichartz formulation. The usual geometry linked to the Chen-Strichartz formulation is the sub-Riemannian geometry, and its major instruments are brought during the textual content.

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Extra resources for An Introduction to the Geometry of Stochastic Flows

Example text

Every joint (probability) density has properties f X,Y (x, y) ≥ 0, +∞ +∞ ∫ −∞ ∫ −∞ f X,Y (x, y) dx dy = 1. Conversely, any function of two variables x and y satisfying these two conditions can be considered to be the joint density of a random vector (X, Y). 57), one obtains the marginal densities of (X, Y) : +∞ f X (x) = ∫ −∞ f X,Y (x, y) dy, © 2006 by Taylor & Francis Group, LLC +∞ f Y (y) = ∫ −∞ f X,Y (x, y) dx . 58) 1 PROBABILITY THEORY 49 Thus, the respective marginal densities of (X, Y) are simply the densities of X and Y.

Notation: X ≤ Y hr Properties of the hazard rate order: 1) If X and Y have continuous densities so that the respective failure rates λ X (t) and λ Y (t) exist, then X ≤ Y if and only if λ X (t) ≥ λ Y (t) for t ≥ 0. hr 2) Let X ≤ Y and h(⋅) be an increasing real function. Then, h(X) ≤ h(Y). hr hr 3) If X ≤ Y , then X ≤ Y . st hr Convex Orders The usual stochastic order and the hazard rate order refer to the absolute sizes of the random variables to be compared. However, for many applications it is useful to include the variability aspect.

11 Let X have an exponential distribution with parameter λ : f (x) = λ e −λ x , x ≥ 0. The Laplace transform of f (x) is f (s) = ∫ 0 e −s x λ e −λ x dx = λ ∫ 0 e −(s+λ) x dx = λ . s+λ ∞ ∞ It exists for s > −λ. The n th derivative of f (s) is d n f (s) λ n! n ; λ n = 0, 1, ... 12 The definition of the Laplace transform can be extended to functions defined on the whole real axis (−∞, +∞). For instance, consider the density of an N(μ, σ 2 ) -distribution: f (x) = 1 e− 2π σ (x−μ) 2 2σ 2 ; x ∈ (−∞, +∞).

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