# Lagrangian Descriptors for Stochastic Differential Equations: A Tool for Revealing the Phase Portrait of Stochastic Dynamical Systems

## TL;DR

This paper introduces a novel stochastic extension of Lagrangian descriptors to visualize phase portraits of stochastic differential equations, revealing key structures like fixed points and manifolds.

## Contribution

It generalizes the deterministic Lagrangian descriptor method to stochastic systems, enabling graphical identification of stochastic phase space structures.

## Key findings

- Successfully visualizes stochastic fixed points and manifolds.
- Identifies barriers to transport in stochastic systems.
- Demonstrates effectiveness on benchmark stochastic models.

## Abstract

In this paper we introduce a new technique for depicting the phase portrait of stochastic differential equations. Following previous work for deterministic systems, we represent the phase space by means of a generalization of the method of Lagrangian descriptors to stochastic differential equations. Analogously to the deterministic differential equations setting, the Lagrangian descriptors graphically provide the distinguished trajectories and hyperbolic structures arising within the stochastic dynamics, such as random fixed points and their stable and unstable manifolds. We analyze the sense in which structures form barriers to transport in stochastic systems. We apply the method to several benchmark examples where the deterministic phase space structures are well-understood. In particular, we apply our method to the noisy saddle, the stochastically forced Duffing equation, and the stochastic double gyre model that is a benchmark for analyzing fluid transport.