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Even though the Wiener process (Brownian motion) is continuous, it has no derivative at any point.

Does it at least have weak derivatives?

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Not a full answer, though it would mean that if $X_t$ is a weak derivative then for any smooth $f$ with a support somewhere in $(a,b)$ it holds that $$\int\limits_a^b X_t f(t)\mathrm dt = -\int\limits_a^b B_t f(t)\mathrm dt = \int\limits_a^bf(t)\mathrm d B_t$$ where the last integral is understood as Ito integral. –  Ilya Dec 4 '11 at 11:40
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What exactly do you mean by "weak derivative"? –  Nate Eldredge Dec 5 '11 at 0:59

2 Answers 2

up vote 5 down vote accepted

Even though what Byron says is 100% true there is a topic called "White Noise Analysis", which might be interesting for Luke purpose and might "in a correct sense and context" correspond to what he had in mind.

The story is based on Minlös-Bochner theorem that allows you to construct a unique probability measure $\mu$ over the space $\mathcal{S}'$ of "Tempered Distributions" of linear forms over the Schwartz Space $\mathcal{S}$ of rapidly decaying smooth functions equiped with its Borel sigma-algebra $\mathcal{B}$ associated to its weak-star topology. Moreover $\mu$ is such that :

$$\int_{\mathcal{S}'} e^{i \langle \phi,\omega \rangle}d\mu(\omega)=e^{-\frac{1}{2}\|\phi\|^2} ,$$ with $\|\phi\|^2=\int_{\mathbb R}\phi(x)^2dx$.

The triplet $(\mathcal{S}',\mathcal{B},\mu)$ is called the white Noise probability space.

The white noise process is then defined as :
$$ w: \mathcal{S}\times \mathcal{S}'\to \mathbb R$$
$$(\phi,\omega)\to \langle \omega,\phi \rangle.$$

Then the framework can be extended to $\phi \in L^2(\mathbb R)$ by a density argument.

Now a "pre-Brownian" motion can be defined by using the family $\phi_t=1_{[0,t]}(s)\in L^2(\mathbb R)$ by the following trick :

$$B_t(\omega)=w(\phi_t,\omega) .$$

Then the existence of a modification of $B_t$ with continuous paths can be done, and conclude by claiming that this modification of $B_t$ is a true Brownian motion.

This sketches the construction of Brownian motion from Tempered Distributions space and is coming from Oksendal's electronic document "Introduction to Malliavin Calculus" Section 3.

After this, if you restrict yourself to the White Noise Space it might be possible to define a weak derivative on the pre-Brownian motion defined here, I let you check the necessary calculations but lagged dirac shouldn't be far.

Anyway as pointed out by Byron Schmuland, you don't have any chance to extend this weak-derivative to the Wiener Space otherwise his argument would be turn out to be false which it is not.

Best regards

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A function with a weak derivative is absolutely continuous, and hence of bounded variation. Since Brownian sample paths have infinite variation over every time interval, they cannot have weak derivatives.

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