Welcome to MathOverflow

Question

5

I would like to know if there exists a formula to compute the $L^2$-Wasserstein distance between the laws $P_1$ and $P_2$ in path space of two diffusion processes: $$dx_t=f_1(x_t)dt + \sigma_1(x_t)dB^1_t$$ $$dy_t=f_2(y_t)dt + \sigma_2(y_t)dB^2_t$$ that is:

$$W(P_1,P_2):=\inf_{couplings} \mathbb{E}\left[\int_0^T |x_t - y_t|^2dt\right]^{1/2}$$ if my understanding of the Wasserstein metric is correct. By formula, I mean something that would depend only on functions $f_1,f_2,\sigma_1,\sigma_2$ and not on the processes $x_t$ and $y_t$.

As an example it is easy to compute it in the case $f_1=f_2=0$, $\sigma_1=1,\sigma_2=2$ for instance (write two correlated Brownian motions with correlation $\rho$ and minimize over $\rho$). In the general case I am lost. Any idea?

edited Mar 1 at 9:03

asked Jun 5 2012 at 17:06

unknown (google)
249●5

	Are you familiar with the Brenier theorem for existence of optimal transport plans for the $L^2$ Wasserstein metric? This may be useful. – Dorian Jun 5 2012 at 19:00
	@Dorian: No, I am not. How could it be useful? – unknown (google) Jun 8 2012 at 9:54
	Explicit formulas may be too much to hope for in general, but you can certainly bound the distance, at least if the $f_i,\sigma_i$ are Lipschitz. Solve the SDEs strongly, with respect to the same Brownian motion, and then use the inequality $W(P_1,P_2)^2 \le \mathbb{E}[\int_0^T(X_t - Y_t)^2dt]$ along with standard stability estimates. – Dan Mar 1 at 14:06
	yes Dan, that's a good idea, thanks. – unknown (google) Mar 5 at 17:26

Wasserstein distance between two diffusion processes.

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.