# Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Stochastic Mechanics (SM)

#### 1.2. Scale Relativity Theory (SR)

## 2. Stochastic Representation of Trajectories

- divergence of the velocity, that is divergence of the difference quotient,
- oscillatory singularity or
- difference between forward and backward velocities.

**equivalence class of stochastic paths**having the same expectation as the given deterministic signal. Mathematical notation and preliminaries for the subsequent treatment are presented in Appendix A. A possibly non-differentiable continuous trajectory is represented by a continuous Markov stochastic process evaluated in the virtual space of paths as follows:

**Definition**

**1**(Markov Stochastic Embedding)

**.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**1**(Gaussian stochastic embedding)

**.**

- 1.
- $\Omega \subseteq {\chi}_{\beta}$,
- 2.
- ${X}_{t}$ has i. i. d. Gaussian increments,
- 3.
- $\mathbb{E}{X}_{t}=x\left(t\right)$ and
- 4.
- ${\upsilon}_{+}^{\beta}\mathbb{E}\left(\right)open="("\; close=")">|{X}_{t}\left|\right|{X}_{t}=x$ hold almost sure.

**consistent stochastic embedding**. This theorem allows for Nelson’s characterization of the Langevin diffusion process.

## 3. Nelson’s Characterization

## 4. The Complex Velocity Operator in SR and SM Theories

**absolutely continuous**part of the trajectory, while the fractal part corresponds to the

**singular**and possibly

**oscillatory**parts. Since, at the level of physical description, there is no way to favor the forward rather than the backward velocity, the description should incorporate them on equal grounds, i.e., forming a bivariate vector field $\mathbb{R}\otimes \mathbb{R}$

**Remark**

**1.**

## 5. The Real Stochastic Geodesic Equations

#### 5.1. Path-Wise Separable Solutions

#### 5.2. Stationary Drift Fields

#### 5.3. Stationary Density Solutions

#### 5.4. Transient Drift Fields

#### 5.5. Asymptotic Density Solutions

## 6. The Complex Stochastic Geodesic Equations

## 7. Real-Valued and Complex Cole–Hopf Transformations

## 8. Numerical Results

#### 8.1. Exact Simulations

#### 8.2. Free Diffusion

#### 8.3. Particle in a Box

## 9. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Notations, General Definitions and Properties of Fractional Velocity

**Definition**

**A1**

**(Asymptotic**

**O**

**notation)**

**Definition**

**A2.**

**Definition**

**A3.**

**Definition**

**A4.**

**Definition**

**A5**

**(Fractional**

**order**

**velocity)**

**Definition**

**A6.**

**set of change**${\chi}_{\pm}^{\beta}\left(f\right):=\left(\right)open="\{"\; close="\}">x:{\upsilon}_{\pm}^{\beta}f\left(x\right)\ne 0$.

**Definition**

**A7.**

- Product rule$$\begin{array}{cc}\hfill {\upsilon}_{+}^{\beta}\left[f\phantom{\rule{0.166667em}{0ex}}g\right]\left(x\right)& ={\upsilon}_{+}^{\beta}f\left(x\right)g\left(x\right)+{\upsilon}_{+}^{\beta}g\left(x\right)f\left(x\right)+{[f,g]}_{\beta}^{+}\left(x\right),\hfill \\ \hfill {\upsilon}_{-}^{\beta}\left[f\phantom{\rule{0.166667em}{0ex}}g\right]\left(x\right)& ={\upsilon}_{-}^{\beta}f\left(x\right)g\left(x\right)+{\upsilon}_{-}^{\beta}g\left(x\right)f\left(x\right)-{[f,g]}_{\beta}^{-}\left(x\right),\hfill \end{array}$$
- Quotient rule$$\begin{array}{cc}\hfill {\upsilon}_{+}^{\beta}[f/g]\left(x\right)& =\frac{{\upsilon}_{+}^{\beta}f\left(x\right)g\left(x\right)-{\upsilon}_{+}^{\beta}g\left(x\right)f\left(x\right)-{[f,g]}_{\beta}^{+}}{{g}^{2}\left(x\right)},\hfill \\ \hfill {\upsilon}_{-}^{\beta}[f/g]\left(x\right)& =\frac{{\upsilon}_{-}^{\beta}f\left(x\right)g\left(x\right)-{\upsilon}_{-}^{\beta}g\left(x\right)f\left(x\right)+{[f,g]}_{\beta}^{-}}{{g}^{2}\left(x\right)},\hfill \end{array}$$

- $f\in {\mathbb{H}}^{\phantom{\rule{0.166667em}{0ex}}\beta}$ and $g\in {\mathbb{C}}^{\phantom{\rule{0.166667em}{0ex}}1}$$$\begin{array}{cc}\hfill {\upsilon}_{+}^{\beta}f\circ g\left(x\right)& ={\upsilon}_{+}^{\beta}f\left(g\right){\left(\right)}^{{g}^{\prime}}\beta ,\hfill \end{array}$$
- $f\in {\mathbb{C}}^{\phantom{\rule{0.166667em}{0ex}}1}$ and $g\in {\mathbb{H}}^{\phantom{\rule{0.166667em}{0ex}}\beta}$$$\begin{array}{cc}\hfill {\upsilon}_{+}^{\beta}f\circ g\left(x\right)& ={f}^{\prime}\left(g\right)\phantom{\rule{0.166667em}{0ex}}{\upsilon}_{+}^{\beta}g\left(x\right),\hfill \\ \hfill {\upsilon}_{-}^{\beta}f\circ g\left(x\right)& ={f}^{\prime}\left(g\right)\phantom{\rule{0.166667em}{0ex}}{\upsilon}_{-}^{\beta}g\left(x\right).\hfill \end{array}$$

## Appendix B. The Stochastic Variation Problem

**Remark**

**A1.**

## Appendix C. Matlab Simulation Code

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**Figure 1.**Virtual trajectories of the separable process. (

**A**) virtual trajectories; (

**B**) empirical vs. theoretical density. (

**A**) exact simulation of separable process is compared with the Euler–Maruyama algorithm. E—Exact simulation, E–M—Euler–Maruyama simulation; An offset is added to the exact solution for appreciation. Time is given in arbitrary units; (

**B**) the empirical transition density is estimated from $n={log}^{2}\left({N}_{s}\phantom{\rule{0.166667em}{0ex}}N\right)$ bins. Pearson’s correlation is given as an inset—r = 0.9976.

**Figure 3.**Simulations of free particles virtual trajectories. (

**A**) virtual trajectories: free particles; (

**B**) empirical vs. theoretical density. Simulations are based on $N=10,000$ points in ${N}_{s}=1000$ simulations. (

**A**,

**B**) width of potential well is $2L=100$ units. The empirical pdf is estimated from $n={log}^{2}\left({N}_{s}\phantom{\rule{0.166667em}{0ex}}N\right)$ bins. Pearson’s correlations are given as inset—r = 0.9883. Norming of the free particle transient results in Rayleigh density.

**Figure 4.**Simulations of particles in a box for two quantum numbers. (

**A**) virtual trajectories: particle in a box, n = 2; (

**B**) empirical vs. theoretical density; (

**C**) test particles in a box, n = 8; (

**D**) empirical vs. theoretical density. (

**A**,

**B**) width of potential well is $2L=100$ units.

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Prodanov, D.
Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation. *Entropy* **2018**, *20*, 492.
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Prodanov D.
Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation. *Entropy*. 2018; 20(7):492.
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**Chicago/Turabian Style**

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2018. "Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation" *Entropy* 20, no. 7: 492.
https://doi.org/10.3390/e20070492