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Firstly, I'm not good at English, sorry. Please tell me this stochastic process's name $x_t=x_{t−1}−λ\operatorname{sgn}(x_{t−1})+ε$, where $ε$ is Gaussian noise and $λ>0.$

Michael Hardy
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tn1021
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  • Why should it have a particular name in the first place? Where does it come from or where have you seen it? Looks to me like some sort of ARMA model. – Tobsn Oct 22 '20 at 07:30
  • On deep learning, I saw this stochastic process. We use lasso model, so loss function is added |ω|, that differentiated sgn(ω). It looks to me like AR model,too. – tn1021 Oct 22 '20 at 09:17
  • I want to proove this model($x_t$) converges 0, so I want to know this process's name. – tn1021 Oct 22 '20 at 10:08
  • You don't need the name of something in order to study its convergence properties. And due to the answer user619894 gave below, I highly doubt it's converging to $0$ btw. – Tobsn Oct 22 '20 at 12:35
  • This process seems for me that is simple. So I think there is this process's name. Thanks. – tn1021 Oct 23 '20 at 07:54

1 Answers1

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The continuum limit (the limit of short time steps) is $$ \dot x =-\lambda \operatorname{sgn}(x)+\epsilon$$ which is a Smolochowski diffusion equation.

In fact, the force $F(x)=\operatorname{sgn}(x)$ comes from a potential field $F(x)=\lambda \nabla|x|$ So it admits a stationary Boltzmann solution $$P(x)=e^{-|x|\beta}$$ where $\beta$ is a function of $\lambda $ and $\sigma$.

user619894
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