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Let $f\geq 0$ be a Lipschitz function and let $(L_t)_{t\geq 0}$ be an $\alpha$-stable Lévy process ($0<\alpha<2$, possibly multivariate). Consider the process given by $$dX_t=-\nabla f(X_t)dt+\sigma dL_t$$ where $\sigma>0$ is a constant. This SDE is reminiscient of Langevin Dynamics, where we usually let the process be driven by Brownian Motion instead of a Lévy process.

I am interested in seeing which results from the "Brownian setting" extend to the setting with $\alpha$-stable Lévy processes as described above.

Most importantly, my main question is: Does $X_t$ admit a stationary distribution? When the process is driven by Brownian Motion, it is known that there exists a stationary distribution, for which we can also find a closed form expression for the density.

Example: Ornstein-Uhlenbeck Process

If we consider the case $f(x)=\frac12 |x|^2$ we get the Ornstein-Uhlenbeck process driven by the Lévy process $L$. There it is known, see e.g. Topics in Infinitely Divisible Distributions and Lévy Processes Theorem 2.17, that the process admits a stationary distribution.

Let $f\geq 0$ be a Lipschitz function and let $(L_t)_{t\geq 0}$ be an $\alpha$-stable Lévy process ($0<\alpha<2$). Consider the process given by $$dX_t=-\nabla f(X_t)dt+\sigma dL_t$$ where $\sigma>0$ is a constant. This SDE is reminiscient of Langevin Dynamics, where we usually let the process be driven by Brownian Motion instead of a Lévy process.

I am interested in seeing which results from the "Brownian setting" extend to the setting with $\alpha$-stable Lévy processes as described above.

Most importantly, my main question is: Does $X_t$ admit a stationary distribution? When the process is driven by Brownian Motion, it is known that there exists a stationary distribution, for which we can also find a closed form expression for the density.

Example: Ornstein-Uhlenbeck Process

If we consider the case $f(x)=\frac12 |x|^2$ we get the Ornstein-Uhlenbeck process driven by the Lévy process $L$. There it is known, see e.g. Topics in Infinitely Divisible Distributions and Lévy Processes Theorem 2.17, that the process admits a stationary distribution.

Let $f\geq 0$ be a Lipschitz function and let $(L_t)_{t\geq 0}$ be an $\alpha$-stable Lévy process ($0<\alpha<2$, possibly multivariate). Consider the process given by $$dX_t=-\nabla f(X_t)dt+\sigma dL_t$$ where $\sigma>0$ is a constant. This SDE is reminiscient of Langevin Dynamics, where we usually let the process be driven by Brownian Motion instead of a Lévy process.

I am interested in seeing which results from the "Brownian setting" extend to the setting with $\alpha$-stable Lévy processes as described above.

Most importantly, my main question is: Does $X_t$ admit a stationary distribution? When the process is driven by Brownian Motion, it is known that there exists a stationary distribution, for which we can also find a closed form expression for the density.

Example: Ornstein-Uhlenbeck Process

If we consider the case $f(x)=\frac12 |x|^2$ we get the Ornstein-Uhlenbeck process driven by the Lévy process $L$. There it is known, see e.g. Topics in Infinitely Divisible Distributions and Lévy Processes Theorem 2.17, that the process admits a stationary distribution.

Source Link
Small Deviation
Small Deviation
  • 123
  • 1
  • 6

Stationary Distribution of Langevin Dynamics driven by Lévy Process

Let $f\geq 0$ be a Lipschitz function and let $(L_t)_{t\geq 0}$ be an $\alpha$-stable Lévy process ($0<\alpha<2$). Consider the process given by $$dX_t=-\nabla f(X_t)dt+\sigma dL_t$$ where $\sigma>0$ is a constant. This SDE is reminiscient of Langevin Dynamics, where we usually let the process be driven by Brownian Motion instead of a Lévy process.

I am interested in seeing which results from the "Brownian setting" extend to the setting with $\alpha$-stable Lévy processes as described above.

Most importantly, my main question is: Does $X_t$ admit a stationary distribution? When the process is driven by Brownian Motion, it is known that there exists a stationary distribution, for which we can also find a closed form expression for the density.

Example: Ornstein-Uhlenbeck Process

If we consider the case $f(x)=\frac12 |x|^2$ we get the Ornstein-Uhlenbeck process driven by the Lévy process $L$. There it is known, see e.g. Topics in Infinitely Divisible Distributions and Lévy Processes Theorem 2.17, that the process admits a stationary distribution.