This post is about the deformation of a Markov diffusion by a diffeomorphism.
A Markov diffusion and a diffeomorphism. The class of Markov diffusion processes is globally stable under diffeomorphisms. Indeed, let ${(X_t)}_{t\geq0}$ be a Markov diffusion process on $\mathbb{R}^d$, solving the stochastic differential equation \[ \mathrm{d}X_t=\sigma(X_t)\mathrm{d}B_t+b(X_t)\mathrm{d}t \] where ${(B_t)}_{t\geq0}$ is a standard Brownian motion on $\mathbb{R}^n$, $\sigma$ is a $\mathcal{C}^2$ matrix field of dimension $d\times n$, and $b$ is a $\mathcal{C}^2$ vector field of dimension $d$, both Lipschitz. We say that $\sigma$ is the diffusion matrix and that $b$ is the drift of the stochastic process $X$. Now we pick a diffeomorphism $\varphi:\mathbb{R}^d\to\mathbb{R}^d$. By Itô's formula, the deformed diffusion process \[ {(Y_t)}_{t\geq0}={(\varphi(X_t))}_{t\geq0} \] solves the stochastic differential equation \[ \mathrm{d}Y_t=\widetilde\sigma(Y_t)\mathrm{d}B_t+\widetilde b(Y_t)\mathrm{d}t \] where \[ \widetilde\sigma(y)=\varphi'(x)\sigma(x) \quad\text{and}\quad \widetilde b(y)=\varphi'(x)b(x)+\tfrac{1}{2}\sigma^2(x)\varphi''(x) \] when $n=d=1$, with $x=\varphi^{-1}(y)$, and more generally \[ \widetilde\sigma(y)=\varphi'(x)\sigma(x) \quad\text{and}\quad \widetilde b(y)=\varphi'(x)b(x)+[\tfrac{1}{2}\sigma(x)\sigma^\top(x):\varphi''(x)] \] where \[ [A(x):\varphi''(x)]_i=\sum_{jk}A_{jk}(x)((\varphi_i)'')_{jk}(x). \] Since $\varphi''(x)$ is $3$-tensor, what we do here is the contraction of two indices. This term involving $\varphi''(x)$ is the contribution of the non-affine part of $\varphi$, the effect of $\sigma$ on $\widetilde b$.
We use here the lightweight notation $\varphi'=D\varphi$ and $(\varphi_i)''=D^2(\varphi_i)=\mathrm{Hess}(\varphi_i)$.
In the isotropic case $d=n$ and $\sigma(x)=\sigma(x)\mathrm{Id}$, $\sigma(x) > 0$, we find \[ [\tfrac{1}{2}\sigma(x)\sigma(x)^\top:\varphi''(x)]_i =\tfrac{1}{2}\sigma^2(x)\mathrm{Tr}((\varphi_i)''(x)) =\tfrac{1}{2}\sigma^2(x)\Delta(\varphi_i)(x). \] The coefficients $\widetilde\sigma$ and $\widetilde b$ depend on the diffeomorphism $\varphi$ only via its first and second derivatives, but the initial condition of the stochastic differential equation is $Y_0=\varphi(X_0)$.
We recover $Y=X$ when $\varphi(x)=x$ for all $x$. Indeed, we have then $\varphi'(x)=\mathrm{Id}$ for all $x$, thus $\varphi''\equiv0$, giving $\widetilde\sigma=\sigma$ and $\widetilde b=b$. More generally, if $\varphi$ is affine, say $\varphi(x)=Bx+C$ for fixed matrices $B$ and $C$, then $\varphi'\equiv B$ and $\varphi''\equiv0$, giving $\widetilde\sigma=B\sigma$ and $\widetilde b=Bb$.
Semigroup and conjugacy. The Markov semigroup ${(Q_t)}_{t\geq0}$ of $Y$ is related to the Markov semigroup ${(P_t)}_{t\geq0}$ of $X$. Indeed, for every bounded measurable $f:\mathbb{R}^d\to\mathbb{R}$, \begin{align*} Q_t(f)(y) &=\mathbb{E}(f(Y_t)\mid Y_0=y)\\ &=\mathbb{E}(f(\varphi(X_t))\mid\varphi(X_0)=y)\\ &=\mathbb{E}((f\circ\varphi)(X_t)\mid X_0=\varphi^{-1}(y))\\ &=P_t(f\circ\varphi)(\varphi^{-1}(y)) \end{align*} In other words $Q_t$ is obtained from $P_t$ by conjugacy in the sense that \[ Q_t=\Phi^{-1}\circ P_t\circ\Phi \] where $\Phi$ is the linear map $f\mapsto f\circ\varphi$, with inverse $f\mapsto f\circ\phi^{-1}$.
Invariant measure. The conjugacy of semigroups implies that if $X$ admits an invariant measure $\mu$ then the measure $\mu\circ\varphi^{-1}$ is invariant for $Y$. A famous case for which $X$ admits an invariant measure is given by $d=n$, a constant diffusion matrix $\sigma=\sigma\mathrm{Id}$, $\sigma > 0$, and a drift $b(x)=-\nabla V(x)$, $V:\mathbb{R}^d\to\mathbb{R}$. Then $\mu=\mathrm{exp}(-\frac{2}{\sigma^2}V(x))\mathrm{d}x$ is invariant.
Generator and flattening. The conjugacy of semigroups gives also that the infinitesimal generator $G$ of ${(P_t)}_{t\geq0}$ and $H$ of ${(Q_t)}_{t\geq0}$ are related by conjugacy, in the sense that \[ H=\Phi^{-1}\circ G\circ\Phi \] namely $H(f)(y)= G(f\circ\varphi)(\varphi^{-1}(y))$. In particular, when $d=n=1$, we find \begin{align*} G(g)(x)&=\tfrac{1}{2}\sigma^2(x)g''(x)+b(x)g'(x)\\ H(f)(y)&=\tfrac{1}{2}\varphi'^2(x)\sigma^2(x)f''(y) +\bigr(\varphi'(x)b(x)+\tfrac{1}{2}\sigma^2(x)\varphi''(x)\bigr)f'(y),\quad x=\varphi^{-1}(y). \end{align*} In particular, in the special case where $\varphi'(x)=1/\sigma(x)$ we find \[ Hf=\frac{1}{2}f'' +\Bigr(\frac{b}{\sigma\circ\varphi^{-1}}-\frac{1}{2}\sigma'\circ\varphi^{-1}\Bigr)f'. \] To summarize, taking $\varphi'=\sigma^{-1}$ has the effect of flattening the diffusion coefficient, making $Y$ a Euclidean diffusion, in the sense that the diffusion coefficient of $Y$ is constant, in other words $Y$ is a Brownian motion with drift, which is easier to study.
Isometry and spectrum. Suppose that $X$ has invariant measure $\mu$. It is then customary to see $G$ as an unbounded operator defined on the Hilbert space $L^2(\mu)$. In this case, the linear map $\Phi$ is an isometry from $L^2(\nu)$ to $L^2(\mu)$ where $\nu=\mu\circ\varphi^{-1}$ is the invariant measure of $Y$. Indeed, for all $f\in L^2(\nu)$, \[ \int\Phi(f)^2\mathrm{d}\mu =\int (f\circ\varphi)^2\mathrm{d}\mu =\int f^2\mathrm{d}\nu. \] Thus $G$ and $H$ have same spectral resolution. In particular, they have same eigenvalues and spectral gap, and the eigenfunctions of $H$ are the ones of $G$ composed with $\Phi^{-1}$.
Curvature-dimension. Thanks to the conjugacy, the processes $X$ and $Y$ satisfy the same curvature-dimension inequalities. The carré du champ $\Gamma$ and the $\Gamma_2$ operators of $X$ and $Y$ are related by the change of variable formulas for $\Gamma$ and $\Gamma_2$. It is convenient to interpret $X$ and $Y$ as diffusions on $\mathbb{R}^d$ seen as a Riemannian manifold with respective curvatures given by the diffusion coefficients. From this point of view, the diffeomorphism can be used to flatten the space : by taking $\varphi=\sigma^{-1}$, the diffusion with variable coefficient $X$ can be studied as a deformation by $\varphi^{-1}$ of the diffusion $Y$ with constant coefficient, they have the same long time behavior and trend toward equilibrium. The intrinsic Riemannian distance is then naturally used to define intrinsic Wasserstein distances, intrinsic regularization, intrinsic functional inequalities, etc.
Further reading.
- Dominique Bakry, Ivan Gentil, and Michel Ledoux
Analysis and Geometry of Markov Diffusion Operators
Springer (2014) - Samuel Chan-Ashing
On The Cutoff Phenomenon For Dyson-Laguerre Processes
Preprint arXiv:2509.19798