Exactly solvable diffusions from space-time transformations - Costantino Di Bello; Édgar Roldán; Ralf Metzler (New Journal of Physics)
We consider a general one-dimensional overdamped diffusion model described by the Itô stochastic differential equation (SDE) ${\mathrm dX_t = μ(X_t,t)\mathrm dt+σ (X_t,t)\mathrm dW_t}$ , where W _t is the standard Wiener process. We obtain a specific condition that µ and σ must fulfil in order to be able to solve the SDE via mapping the generic process, using a suitable space-time transformation, onto the simpler Wiener process. By taking advantage of this transformation, we obtain the propagator in the case of open, reflecting, and absorbing time-dependent boundary conditions for a large class of diffusion processes. In particular, this allows us to derive the first-passage time statistics of such a large class of models, some of which were so far unknown. While our results are valid for a wide range of non-autonomous, non-linear and non-homogeneous processes, we illustrate applications in stochastic thermodynamics by focusing on the propagator and the first-passage-time statistics of isoentropic processes that were previously realised in the laboratory by Brownian particles trapped with optical tweezers.