comparing diffusions

Question

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 ?

Answer 1

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.

Answer 2

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.

Answer 3

Hi Alekk

You might take a look at this paper :

Debussche,Faou - Weak Backward Error Analysis for SDEs

Regards