Supersymmetric properties of one-dimensional Markov generators with the links to Markov-dualities and to shape-invariance-exact-solvability

Abstract

For diffusion process involving the force $F(x)$ and the diffusion coefficient $D(x)$, the continuity equation ${\partial }_t{P}_t(x)=-{\partial }_x{J}_t(x)$ gives the dynamics of the probability $P_t(x)$ in terms of the current $J_t(x)=F(x)P_t(x)-D(x){\partial }_x{P}_t(x)=\mathcal{J}{P}_t(x)$ obtained from $P_t(x)$ via the application of the first-order differential current-operator $\mathcal{J}$. So the dynamics of the probability $P_t(x)$ is governed by the factorized Fokker-Planck generator $\mathcal{F}=-{\partial }_x\mathcal{J}$, while the dynamics of the current $J_t(x)$ is governed by its supersymmetric partner $\hat{\mathcal{F}}=-\mathcal{J}{\partial }_x$, so that their right and left eigenvectors are directly related using the two intertwining relations $\mathcal{J}\mathcal{F}=-\mathcal{J}{\partial }_x\mathcal{J}=\hat{\mathcal{F}}\mathcal{J}$ and $\mathcal{F}{\partial }_x=-{\partial }_x\mathcal{J}{\partial }_x={\partial }_x\hat{\mathcal{F}}$. We also describe the link with the factorization of the adjoint ${\mathcal{F}}^{†}=\frac{d}{dm(x)}\frac{d}{ds(x)}$ in terms of the scale function $s(x)$ and speed measure $m(x)$. We then analyze how the supersymmetric partner $\hat{\mathcal{F}}=-\mathcal{J}{\partial }_x$ can be re-interpreted in two ways: (1) as the adjoint ${\mathring{\mathcal{F}}}^{†}={\mathring{\mathcal{J}}}^{†}{\partial }_x$ of the Fokker-Planck generator $\mathring{\mathcal{F}}=-{\partial }_x\mathring{\mathcal{J}}$ associated to the dual force $\mathring{F}(x)=-F(x)-D^{\prime }(x)$, that unifies various known Markov dualities; (2) as the non-conserved Fokker-Planck generator ${\tilde{\mathcal{F}}}_{nc}=-{\partial }_x\tilde{\mathcal{J}}-\tilde{K}(x)$ involving the force $\tilde{F}(x)=F(x)+D^{\prime }(x)$ and the killing rate $\tilde{K}(x)=-F^{\prime }(x)-D^{\prime \prime }(x)$, with application to shape-invariance-solvability. Finally, we describe how all these ideas can be also applied to Markov jump processes with nearest-neighbors transition rates $w(x\pm 1,x)$.

v2=final version (30 pages)