Convergence of the distribution of the Langevin diffusion to its invariant measure
Let $(X_t)_{tge0}$ be a solution of $${rm d}X_t=-h'(X_t){rm d}t+sqrt 2W_t,tag1$$ where $(W_t)_{tge0}$ is a Brownian motion and $h$ is such that $X$ is the unique strong solution of $(1)$. $X$ is a time-homogeneous Markov process whose transition semigroup is stationary with respect to the measure $mu:=e^{-h}lambda$ with density $e^{-h}$ with respect to the Lebesgue measure $lambda$.
Are we able to show that the distribution $mathcal L(X_t)$ converges to $mu$ as $ttoinfty$? If so, for which mode of convergence? Weak convergence? Convergence in total variation distance?
probability-theory stochastic-processes markov-process stochastic-analysis sde
asked Dec 31 '18 at 19:41
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Let $(X_t)_{tge0}$ be a solution of $${rm d}X_t=-h'(X_t){rm d}t+sqrt 2W_t,tag1$$ where $(W_t)_{tge0}$ is a Brownian motion and $h$ is such that $X$ is the unique strong solution of $(1)$. $X$ is a time-homogeneous Markov process whose transition semigroup is stationary with respect to the measure $mu:=e^{-h}lambda$ with density $e^{-h}$ with respect to the Lebesgue measure $lambda$.
Are we able to show that the distribution $mathcal L(X_t)$ converges to $mu$ as $ttoinfty$? If so, for which mode of convergence? Weak convergence? Convergence in total variation distance?
probability-theory stochastic-processes markov-process stochastic-analysis sde
asked Dec 31 '18 at 19:41
0xbadf00d
1,75441430
This question has an open bounty worth +50
reputation from 0xbadf00d ending in 5 days.
This question has not received enough attention.
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Let $(X_t)_{tge0}$ be a solution of $${rm d}X_t=-h'(X_t){rm d}t+sqrt 2W_t,tag1$$ where $(W_t)_{tge0}$ is a Brownian motion and $h$ is such that $X$ is the unique strong solution of $(1)$. $X$ is a time-homogeneous Markov process whose transition semigroup is stationary with respect to the measure $mu:=e^{-h}lambda$ with density $e^{-h}$ with respect to the Lebesgue measure $lambda$.
Are we able to show that the distribution $mathcal L(X_t)$ converges to $mu$ as $ttoinfty$? If so, for which mode of convergence? Weak convergence? Convergence in total variation distance?
probability-theory stochastic-processes markov-process stochastic-analysis sde
asked Dec 31 '18 at 19:41
0xbadf00d
1,75441430
Let $(X_t)_{tge0}$ be a solution of $${rm d}X_t=-h'(X_t){rm d}t+sqrt 2W_t,tag1$$ where $(W_t)_{tge0}$ is a Brownian motion and $h$ is such that $X$ is the unique strong solution of $(1)$. $X$ is a time-homogeneous Markov process whose transition semigroup is stationary with respect to the measure $mu:=e^{-h}lambda$ with density $e^{-h}$ with respect to the Lebesgue measure $lambda$.
Are we able to show that the distribution $mathcal L(X_t)$ converges to $mu$ as $ttoinfty$? If so, for which mode of convergence? Weak convergence? Convergence in total variation distance?
probability-theory stochastic-processes markov-process stochastic-analysis sde
probability-theory stochastic-processes markov-process stochastic-analysis sde
asked Dec 31 '18 at 19:41
0xbadf00d
1,75441430
asked Dec 31 '18 at 19:41
0xbadf00d
1,75441430
asked Dec 31 '18 at 19:41
0xbadf00d
1,75441430
asked Dec 31 '18 at 19:41
0xbadf00d
1,75441430
asked Dec 31 '18 at 19:41
0xbadf00d
1,75441430
1,75441430
This question has an open bounty worth +50
reputation from 0xbadf00d ending in 5 days.
This question has not received enough attention.
This question has an open bounty worth +50
reputation from 0xbadf00d ending in 5 days.
This question has not received enough attention.
add a comment |
add a comment |
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