# Space-Time Inversion of Stochastic Dynamics

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## Abstract

**:**

## 1. Introduction

## 2. Basic Definitions and Statement of the Problem

**Definition**

**1.**

**Definition**

**2.**

## 3. The Prototypical Model

## 4. Mollified Description and Space-Time Inverse Langevin Equation

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proposition**

**3.**

**Proof.**

- for $a\left(y\right)=\mathrm{constant}=1$, Equation (20) provides the classical subordination result [47], corresponding to the dynamics of the space-inverted process given by $dy\left(x\right)=\sqrt{2\phantom{\rule{0.166667em}{0ex}}{K}_{0}/v\left(y\left(x\right)\right)}\phantom{\rule{0.166667em}{0ex}}d\overline{w}\left(x\right)$, where $\overline{W}\left(x\right)$ is a Wiener process and this stochastic equation should be interpreted using the Ito prescription;
- if the original Langevin Equation (8) is nonlinear, that is, $a\left(y\right)$ depends explicitly on y, the Fokker-Planck Equation (20) does not correspond to any simple Langevin equation of the form$$dy\left(x\right)=\sqrt{\frac{2\phantom{\rule{0.166667em}{0ex}}{K}_{0}a\left(y\left(x\right)\right)}{v\left(y\right(x\left)\right)}}{\ast}_{\lambda}d\overline{w}\left(x\right),$$

## 5. Mixed-Order Stochastic Integrals

**Proposition**

**4.**

**Proof.**

## 6. Applications

**Transit-time statistics**—The inverse space-dynamics and the associated Fokker-Planck equation provide a direct way to estimate transit-time statistics and transversal hitting distributions as a direct problem for the space-time inverse dynamics. Consider the Poiseuille flow problem analyzed in the previous Section, and set $a\left(y\right)=1$ for simplicity, leading to a linear Langevin equation. In this case, the inverse space-dynamics is given by

**Stochastic thermodynamics**—Consider a classical Ornstein-Uhlenbeck model for a particle of mass m in a conservative potential $\varphi \left(x\right)$ in the presence of stochastic fluctuations

**Fractal time models**—There is a straightforward application of the space-time inversion related to the idea by R. Hilfer of introducing transport models defined with respect to “a fractal time” [43,44]. To this purpose, consider a pure diffusion process defined by the Langevin model

## 7. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Stationary transversal hitting distribution ${p}_{y}^{*}\left(y\right)={\overline{p}}^{*}\left(y\right)$ vs y for model I and model II described in the main text. Symbols correspond to stochastic simulations using the direct Langevin Equation (8), solid lines to Equation (23). Line (a) and symbols (□) refer to model I, line (b) and symbols (○) to model II.

**Figure 3.**Comparison of the stationary transversal hitting distribution ${p}_{y}^{*}\left(y\right)={\overline{p}}^{*}\left(y\right)$ vs y for model I deriving from direct stochastic simulations of Equation (8) (symbols ○, corresponding to curve d), compared with the stationary predictions deriving from Equation (22) for $\lambda =0$ (line a), $\lambda =1/2$ (line b) and $\lambda =1$ (line c).

**Figure 4.**Lower-order transit time moments for the 2d Poiseuille channel flow problem vs the spatial (axial) coordinate x at ${K}_{0}={10}^{-3}$, $\epsilon ={10}^{-2}$. Panel (

**a**) refers to the first-order moments ${\theta}_{1}\left(x\right)$, panel (

**b**) to the second-order central moments (squared variances) ${\sigma}_{\theta}^{2}\left(x\right)$. Lines (a) and (b) are the results of the stochastic simulations of the inverted process, while symbols represent the corresponding quantities estimated from the direct process. Line (a) and (□) correspond to a uniform inlet distribution ${p}_{\mathrm{in},y}\left(y\right)=1$, while line (b) and (○) to an impulsive inlet condition ${p}_{\mathrm{in},y}\left(y\right)=\delta \left(y\right)$. Lines (c) represent the scalings ${\theta}_{1}\left(x\right)\sim {x}^{2/3}$ in panel (a), and ${\sigma}_{\theta}^{2}\left(x\right)\sim {x}^{4/3}$ in panel (b). Lines (d) correspond to the large-distance linear scaling ${\theta}_{1}\left(x\right)\sim {\sigma}_{\theta}^{2}\left(x\right)\sim x$.

**Figure 5.**Transit time and transverse distribution for the same problem as in Figure 4, that is, 2d Poiseuille flow at ${K}_{0}={10}^{-3}$ and $\epsilon ={10}^{-2}$. Panel (

**a**) refers to the stationary transversal hitting distribution ${p}_{y}^{*}\left(y\right)$: symbols (□) corresponds to the results of the direct simulation Equation (8), symbols (○) to those of the inverse simulations Equation (37). The solid line represents ${p}_{y}^{*}\left(y\right)=A\phantom{\rule{0.166667em}{0ex}}v\left(y\right)$, where A is a normalization constant. Panel (

**b**) depicts the transit-time probability density functions ${p}_{\theta}(\theta ;x)$ vs $\theta $ at different values of the axial coordinate x. Solid lines are the results of the stochastic simulations of the inverse process, symbols those of the direct process. From (a) to (e), $x=20,\phantom{\rule{0.166667em}{0ex}}50,\phantom{\rule{0.166667em}{0ex}}100,\phantom{\rule{0.166667em}{0ex}}200,\phantom{\rule{0.166667em}{0ex}}300$, respectively.

**Figure 6.**Mean square displacement ${R}^{2}\left(\tau \right)$ vs $\tau $ for the inverted process Equation (44) associated with Equation (42) and (43), and corresponding to the occurrence of a “stochastic” time parametrization, for different values of $\nu $. The arrow indicates increasing values of $\nu =0.25,\phantom{\rule{0.166667em}{0ex}}0.5,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}2$. Symbols are the results of the stochastic simulations of Equation (44), while lines depict the expected scalings ${R}^{2}\left(\tau \right)\sim {\tau}^{2/(2+\nu )}$.

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Giona, M.; Brasiello, A.; Adrover, A.
Space-Time Inversion of Stochastic Dynamics. *Symmetry* **2020**, *12*, 839.
https://doi.org/10.3390/sym12050839

**AMA Style**

Giona M, Brasiello A, Adrover A.
Space-Time Inversion of Stochastic Dynamics. *Symmetry*. 2020; 12(5):839.
https://doi.org/10.3390/sym12050839

**Chicago/Turabian Style**

Giona, Massimiliano, Antonio Brasiello, and Alessandra Adrover.
2020. "Space-Time Inversion of Stochastic Dynamics" *Symmetry* 12, no. 5: 839.
https://doi.org/10.3390/sym12050839