Differential Analysis

Special Notation

Relatively open: no one in V is infinitely close to any member of U

U \subset^{\circ} V

Compactly Contained: any possible limit points of U lie within V

U \subset\subset V

Definition

We introduce the following important notations to be used for the preceding discussions
• for $f \in L_{loc}^{1} (U)$ we identify the function $f$ with the distribution $\tilde{f}$ from 28
• for $f \in L_{loc}^{1} (U)$ and $φ \in C_{c}^{\infty} (U)$ we define the pairing

(f, φ) = \int_{U} f (x) φ (x) d x

• for $u \in D^{'} (U)$ and $φ \in C_{c}^{\infty} (U)$ we define the pairing

$(u, φ) = u (φ)$

Remark

It is common to write

(u, φ) = \int_{U} u (x) φ (x) d x

as well but take caution that this only makes sense when you write it like this. If you try to isolate $u (x)$ and it as a classical function the result will not make sense

Example

Consider isolating the dirac delta function $δ (x - x^{'}) = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{i k (x - x^{'})} d k$ from the fourier decomposition of the wave function as we did in Quantum Mechanics 1

Ψ (x) = \frac{1}{\sqrt{2 π}} \int [\frac{1}{\sqrt{2 π}} \int Ψ (x^{'}) e^{- i k x^{'}} d x^{'}] e^{i k x} d k,

Ψ (x) = \int_{- \infty}^{\infty} Ψ (x^{'}) \frac{1}{2 π} \int e^{i k (x - x^{'})} d k d x^{'} .

if you took this in a classical sense after attempting to isolate the dirac delta "function" you get for $x = x^{'}$ (sifting property)

Ψ (x) = Ψ (x) ∙ \infty

which is nonsense

Lemma

Suppose that $(X_{t})_{t \in [0, 1]}$ is any stochastic process satisfying $E [| X_{s} - X_{t} |^{q}] \leq C | s - t |^{1 + ϵ}$ for all $s, t \in [0, 1]$ then for all $α \in (0, \frac{ϵ}{q})$ there exist $K_{α} (ω) < \infty$ such that

| X_{i / 2^{n}} (ω) - X_{(i - 1) / 2^{α} (ω)} | < \frac{K_{α} (ω)}{2^{n α}}

for all $n \geq 1$ and $1 \leq i \leq 2^{n} - 1$