# Questions tagged [stochastic-processes]

A stochastic process is a random process evolving with time , i.e., a time sequence representing the evolution of some system represented by a variable whose change is subject to a random variation.

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### What is the mean of the stochastic differential equation $dX=K dt + sigma X dW$ and how to find it? [migrated]

I have the stochastic differential equation, $$dX=k dt + \sigma XdW,$$ which I expect to have just the mean $kt$, since taking the expectation of the SDE gives E[dX]=E[k dt] due to the brownian term ...
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1 vote
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### What is the correlation between Brownian noise's low frequency components and the actual movement of particles?

I do have some crude training in mathematics, but I'm not a physicist or engineer. So I'd appreciate a simple not too technical explanation. I conceptually understand how hitting a piece of wood will ...
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### Helmholtz decomposition of flow at non-equilibrium steady state

I'm trying to work through Karl Friston's mathematical derivation of the Free Energy Principle from Langevin Dynamics (see this paper). I'm confused about the part at the end of page 8 where he uses ...
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### Using the functional derivative and delta function for proving The Fokker-Planck Equation

I am reading "Lectures on Phase Transitions" by Nigel Goldenfeld, specifically Chapter 8, where the Fokker-Planck equation is derived. I found the following part of the proof, but there are ...
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1 vote
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### Discrete simulation of a Levy flight

I am trying to construct a discrete simulation of Levy flight in 1D and am wondering what is the best way to do so. For example, for pure diffusive random walk, one may assign probability of $1/2$ to ...
293 views

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### Energy exchanges between a Brownian fluid and particles

In the context of the dynamics of polymeric models, and specifically the dumbbell model, one of the forces acting on a dumbbell spring is said to result from "a time smoothed Brownian force" ...
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### Alternatives to the Fokker-Planck equation for deriving the probability distribution associated with Langevin dynamics

I was wondering if there are any other means of obtaining exact (or analytical approximations) of the phase space probability density for a system evolving according to Langevin dynamics. The typical ...
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### Reduce multiplicative noise to additive noise with singular matrices

I have a stochastic differential equation as $$\dot{\textbf{X}}=\textbf{A}\textbf{X}+\alpha(t)\textbf{B}\textbf{X}-\alpha^*(t)\textbf{B}^T\textbf{X}$$ where $T$ ...
• 87
1 vote
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### Does the master equation break down for negative times?

I'm studying stochastic dynamics and have encountered the framework of the master equation for the study of continuous time Markov processes. First, I'll state some general definitions and then say ...
1 vote
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### What is a "necessary condition" to observe stationary probabilities? [closed]

In our Stat. Mech class the professor said that the detailed balance condition, \begin{align} K_{ij}P_{j}=K_{ji}P_{i} \end{align} is a sufficient but not necessary condition to observe stationary ...
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### Mathematical description of random noise

I am trying to mathematically model the noise affecting a certain physical quantity, say $X(t)$. Then, the noisy quantity would be $X'(t)$ which differs by some small value $\delta$ from the ideal ...
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### Distribution sampling from a trajectory

Consider a sequence of random variables $\{X_1, X_2,.., X_n\}$ corresponding to regular measurements of a single observable over time obtained from a laboratory or computer experiment. Assume that the ...
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### How to show random cluster models with non-integer $q$ have no local description?

It is known that the random cluster model with $q = 1$ corresponds to bond percolation, and $q = 2, 3, ...$ corresponds to the $q$-state Potts model. Both of these have a local description. What ...
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1 vote
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### Mean squared displacement of a particle on a biased random walk [closed]

Given a particle on a 1-D random walk with some drift velocity $\nu_d = \frac{\Delta x_d}{\Delta t}$, the position in at some time step j is given by $$x_j=x_{j-1}+k_j L + \Delta x_d$$ where $L$ is ...
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### Is this a valid alternative definition of the delta function?

The delta function can be defined as: $$\delta(x) = \int_{-\infty}^{\infty} e^{-2\pi i k x} \, dk$$ Loosely speaking, I can understand this because unless $x=0$, the complex exponential oscillates ...
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1 vote