The Fractional Derivative and Stochastic Processes/ Fokker Planck equation

Question

In this question I am interested in Fractional Derivatives and the Fractional Fokker Planck Equation. Note I only mention fractional in time, however if someone wants to reference to fractional in space too then that is also helpful.

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$\textbf{Notation :}$

The LHS Caputo fractional derivative is defined as (say for $\alpha \in \mathbb{R}^+ $)

$$ _aD_t^\alpha \rho_t := \frac{1}{\Gamma(1-\alpha)} \int_a^t (t-s)^{-\alpha} \rho_s ds, $$ with $\Gamma$ the usual Gamma function.

Consider the following (time) fractional PDE

\begin{equation}\tag{*} _0D_t^\alpha \rho=\text{div}(\rho\nabla \Psi)+\Delta \rho, \end{equation}

for some potential $\Psi : \mathbb{R}^d \to \mathbb{R}$, and $\rho : [0,T] \times \mathbb{R}^d \to \mathbb{R}$, and initial condition $\rho_0$ given.

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$\textbf{Comments :}$

I am not well versed in this subject so definitely read these with caution.

I have been told that the PDE $(^*)$ can be the associated Fokker Planck equation of a diffusion process, that is $\rho$ is the density which describes the evolution of the associated stochastic process. However the occurrence of the fractional derivative in ($^*$) means that the associated stochastic process will not be the usual diffusion

$$ dX_t=-\nabla \Psi(X_t) dt+\sqrt{2}dW_t, $$ where $W_t$ is distributed as $N(0,t)$. Instead I have heard that the diffusion associated to $(^*)$ (sometimes called an anomalous diffusion) will driven by some other process, lets call it $B_t$ where $B_t$ can be :

• A time-changed Brownian Motion.

• A time-changed fractional Brownian Motion (see https://arxiv.org/abs/1002.1494).

• A process constructed from heavy tailed distributions?

Lasty its worth mentioning that the fractional derivative is non-local, hence $(^*)$ is suited to describing stochastic processes which have some memory, i.e are not Markovian.

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$\textbf{Questions : }$

Any insight, or reference regarding any of the following questions would be much appreciated. Especially introductory materials (I have only done stochastic calculus using the classical Brownian Motion).

1. What are the most common examples of $B_t$ which give rise to $(^*)$ as an associated Fokker Planck?

2. For each choice of $B_t$ do we have to construct a stochastic calculus (like was done for $W_t$).

3. What are the relation between $(^*)$ and stochastic processes with memory?

4. What are the relation between $(^*)$ and heavy tailed distributions?