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Exactly solvable diffusions from space-time transformations

Costantino Di Bello,Édgar Roldán,R. Metzler

Published 2025 in New Journal of Physics

ABSTRACT

We consider a general one-dimensional overdamped diffusion model described by the Itô stochastic differential equation (SDE) dXt=μ(Xt,t)dt+σ(Xt,t)dWt, where Wt 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.

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