Background: I need some help to understand the concept behind measure-driven differential equations. The solution of an ordinary differential equation is continuous. In order to describe discontinuous trajectories we use the concept of distributions (very well described in the book "Functional Analysis" by W. Rudin). In brief, every function $x:\Re\to\Omega\subseteq \Re^n$ that is locally integrable over the open set $\Omega$ is mapped to a functional $T_x:\mathcal{D}(\Re^n)\to\Re^n$ as follows:

$$ T_x(\phi)= \int_\Omega x\phi d\mu $$

where $\mathcal{D}(\Re^n)$ is the set of test functions (infinitely many times differentiable and with compact support). $\mu$ is the Lebesgue measure over $\Omega$ - meaning that the integration is carried out in the Lebesgue sense.

Every distribution (i.e. a functional $T\in\mathcal{D}^\star(\Omega)$) has a derivative given by:

$$ (DT)(\phi)=-T(D\phi) $$

In that sense we can construct generalized differential equations that look like:

$$ DT=g(T) $$

Using this framework we can describe solutions that encounter jumps such as impulsive differential equations. This is accomplished using the Dirac functional $\delta(\phi)=\phi(0)$. (I don't want to go into much detail to keep the question short).

The problem: I recently stumbled on a thing called "Measure-driven differential equations". These have the form:

$$ dx=f(x(t),u(t))dt+g(x(t))d\mu(t) $$

where $\mu:\mathcal{B}([t_0,t_1])\to\Re_+$ is a positive measure with the property $\mu(A)\in K$ for all $A\subseteq [t_0,t_1]$ where $K$ is compact. $u$ and $\mu$ here serve as external "signals". The solution of such an equation is reportedly:

$$ x(t)=x(t_0) + \int_{t_0}^t f(x(s),u(s))ds + \int_{[t_0,t]}g(x(s))\mu(ds) $$

(see this article for example).

The questions: (i) I'm a bit puzzled with the notation $d\mu(t)$ and $\mu(ds)$. Can someone elaborate a bit on that? Since $\mu$ is a measure, what exactly is the meaning of $\mu(ds)$? (ii) Is there any advantage from using measures instead of distributions to describe phenomena with discontinuous trajectories? (iii) I would appreciate some reference (preferably a book) to get started with these things.

  • $\begingroup$ For (i), the Lebesgue integral of the function $f$ w.r.t. measure $\mu$ over the set $A$ is often denoted either by $$ \int\limits_A d\mathrm d\mu $$ or by $$ \int\limits_A f(x)\mu(\mathrm dx). $$ These notations are equivalent and are chosen based on personal preferences/notation convenience. I guess, when the measure-driven ODE is written, the form $\mathrm d\mu(t)$ is chosen to hold the similarity with $\mathrm dt$ (usual increment in ODEs) or $\mathrm dB_t$ (Brownian motion's increment in SDEs) $\endgroup$ – Ilya Aug 27 '12 at 8:13
  • $\begingroup$ The first integral was meant to be $\int_A f\mathrm d\mu$ $\endgroup$ – Ilya Aug 29 '12 at 13:46

Intuitively, if your variable $s$ is take in some $\Omega$ the notation $\mu(ds)$ means that you evaluate the measure on an infinitersimal segment $ds$ (you can think that like an open ball little as you want near $s=s*$) and you multiply this value with the value of the function evaluate in $s*$; so for example: $f(s*)\mu(ds)$, but this is just a particular notation because the true integral is the limsup of simplies function that approximate $f(s)$ with the necessary specification about sigma algebra ecc... For the other question I can't help you now...


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