Symmetry of stochastic non-variational differential equations

Abstract

I will sketchily illustrate how the theory of symmetry helps in determining solutions of (deterministic) differential equations, both ODEs and PDEs, staying within the classical theory. I will then present a quick discussion of some more and less recent attempts to extend this theory to the study of stochastic differential equations, and briefly mention some perspective in this direction.

Review paper (119 pages) here in bookstyle. The published version may differ from the present (preprint) one. Version 2 (November 2017): due to a mistake in a basic formula (in a previous paper of mine) the content of Section 5.5 gives wrong statement; an ERRATUM has been added at the end of the file; see also arXiv:1711.01999 for correct results on that topic

References in corpus (8)

• State-dependent diffusion: thermodynamic consistency and its path integral formulation
• Superposition rules, Lie theorem and partial differential equations
• Nonlocal aspects of $\lambda$-symmetries and ODEs reduction
• Local and nonlocal solvable structures in ODEs reduction
• Symmetry of stochastic non-variational differential equations
• Symmetry of stochastic equations
• A symmetry breaking mechanism for selecting the speed of relativistic solitons
• Solving stochastic differential equations with Cartan's exterior differential systems

Cited by in corpus (10)

• Symmetry of stochastic non-variational differential equations
• W-Symmetries of Ito stochastic differential equations
• Integration of the stochastic logistic equation via symmetry analysis
• Symmetries and invariance properties of stochastic differential equations driven by semimartingales with jumps
• Integrable Ito equations and properties of the associated Fokker-Planck equations
• Symmetry classification of scalar autonomous Ito stochastic differential equations with simple noise
• Recent advances in symmetry of stochastic differential equations
• On the integration of Ito equations with a random or a W-symmetry
• Integrable Ito equations with multiple noises
• Asymptotic symmetry and asymptotic solutions to Ito stochastic differential equations