Consider a probability distribution $\pi$ on the real axis that has a density (w.r.t Lebesgue) proportional to $e^{-V(x)}$, where $V(\cdot)$ is a potential function. For any reasonable volatility function $\sigma:\mathbb{R} \to (0:+\infty)$ the diffusion $$ dX^{\sigma}_t = [ -\frac{1}{2} \sigma(X_t^{\sigma})^2 V'(X_t^{\sigma}) + \sigma(X_t^{\sigma}) \sigma'(X_t^{\sigma}) ] dt + \sigma(X_t^{\sigma}) \, dW_t $$ has $\pi$ as unique invariant distribution.

Question: Given two volatility functions $\sigma_1, \sigma_2$, are there tractable ways of comparing the speed of convergence to equilibrium of the two associated diffusions?

For example, if $\sigma_2(x) = \alpha \cdot \sigma_1(x)$, the diffusion $X^{\sigma_2}$ is just $X^{\sigma_1}$ slowed down by a factor $\alpha$: any ways of comparing the two diffusions should say that if $\alpha > 1$ then $X^{\sigma_2}$ converges 'faster' than $X^{\sigma_1}$. Spectral Gaps work but are not very tractable when comparing two non-proportional diffusions. Is it hopeless ?

Motivations: I consider several MCMC algorithms with target density $\pi$: each one of them, after some time-rescaling, looks like a diffusion $X^{\sigma}$. Which algorithm is the best $i.e.$ what diffusion $X^{\sigma}$ mixes the fastest ?

  • $\begingroup$ Hi Alekk, I was wondering if there exists theorems regarding speed of convergence to $\pi$ when the problem is not about diffusions but about Markov Chains. If there exist such results, maybe you can tackle the problem through Markov Chain discretisations of your diffusions and then try to obtain asymptotic results for the speed of convergence of the diffusions you want to compare. Best Regrads "et bon courage" $\endgroup$ – The Bridge Dec 16 '10 at 8:56

A nice quantitative and very general tool to study the speed to convergence of symmetric Markov processes to equilibrium is the Bakry-Emery criterion. More precisely, let $(X_t)_{t \ge 0}$ be a diffusion Markov process with generator $L$, semigroup $P_t$ and symmetric and invariant probability measure $\pi$. Define the carre du champ by

$\Gamma(f,g)=\frac{1}{2} (L(fg) -fLg-gLf)$

and the iterated carre du champ by

$\Gamma_2(f,f)=\frac{1}{2} (L\Gamma(f,f)- 2\Gamma(f,Lf))$

Assume that $\Gamma_2(f,f) \ge \rho \Gamma(f,f)$ for some positive constant $\rho$, then for every $t \ge 0$ $\int (P_t f -\int f d\pi)^2 d\pi \le e^{-2\rho t} \int ( f -\int f d\pi)^2d\pi$

As a consequence, we get a convenient criterion for exponential speed to equilibrium. In your one -dimensional case, $\Gamma$ and $\Gamma_2$ are easy to explicitly compute so the criterion is easy to check.

Further details on the Bakry-Emery method may be found in these Lecture Notes by Dominique Bakry.

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Depending on the boundary and regularity assumptions, the time evolution of the probability distribution is described by a Fokker-Planck equation (see Wikipedia). For a time-homogenous process with a unique stationary solution, the time evolution is described by an exponential decay of the initial distribution acoording to an eigenvalue expansion of the form $$ p(x, t) = \sum_{k = 0}^{\infty} q_k(x) \exp{(- \lambda_k t)} $$ for eigenvalues $\lambda_k$ with $\lambda_0 = 0$ corresponding to the stationary distribution. The bigger the eigenvalues are, the faster the decay to the stationary distribution will be, so some kind of measure could be the smallest non-zero eigenvalue of the Fokker-Planck operator.

For some concrete examples, have a look at the book Gardiner: "Handbook of Stochastic Methods", chapter 5.2.5 Eigenfunction Methods (Homogeneous Processes).

I don't know though if it is possible to calculate or approximate the eigenvalues for the general equation you stated.

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  • $\begingroup$ thanks Tim: yes, even the case where we are only interested in the most significant eigenvalue i.e. spectral gap, the computations become intractable. I was wondering if there were any other ways of comparing these diffusions, without spectral theory and the non tractable ODEs that are involved. $\endgroup$ – Alekk Dec 13 '10 at 13:07
  • $\begingroup$ Okay, at least I can help to make the question a little bit more precise :-) Do you need quantitative data or would the order itself be enough? $\endgroup$ – Tim van Beek Dec 13 '10 at 13:20
  • $\begingroup$ BTW, is the initial condition supposed to be fixed? $\endgroup$ – Tim van Beek Dec 13 '10 at 13:24
  • $\begingroup$ Tim, thanks for the interest: in fact, the initial distribution is not really what interests me - I would rather like to quantify how fast the diffusion visits the state space. I added more motivations in the question. $\endgroup$ – Alekk Dec 13 '10 at 15:51

Hi Alekk

You might take a look at this paper :

Debussche,Faou - Weak Backward Error Analysis for SDEs


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  • $\begingroup$ Thanks TheBridge. You might want to check the link though: it is pointing to another paper. $\endgroup$ – Alekk May 4 '11 at 12:47
  • $\begingroup$ @ alekk : My mistake correction done $\endgroup$ – The Bridge May 4 '11 at 14:35

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