Beyond Itô versus Stratonovich

Abstract

Recently, a novel framework to handle stochastic processes has emerged from a series of studies in biology, showing situations beyond 'Itô versus Stratonovich'. Its internal consistency can be demonstrated via the zero mass limit of a generalized Klein-Kramers equation. Moreover, the connection to other integrations becomes evident: the obtained Fokker-Planck equation defines a new type of stochastic calculus that in general differs from the α-type interpretation. A unique advantage of this new approach is a natural correspondence between stochastic and deterministic dynamics, which is useful or may even be essential in practice. The core of the framework is a transformation from the usual Langevin equation to a form that contains a potential function with two additional dynamical matrices, which reveals an underlying symplectic structure. The framework has a direct physical meaning and a straightforward experimental realization. A recent experiment has offered a first empirical validation of this new stochastic integration.

20 pages, 1 figure

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• Determine dynamical behaviors by the Lyapunov function in competitive Lotka-Volterra systems
• Summing over trajectories of stochastic dynamics with multiplicative noise
• Generally covariant state-dependent diffusion
• Boltzmann-Gibbs entropy is sufficient but not necessary for the likelihood factorization required by Einstein
• Work Relations Connecting Nonequilibrium Steady States Without Detailed Balance
• Non-equilibrium work relation beyond Boltzmann-Gibbs distribution
• A Stochastic Model of Inward Diffusion in Magnetospheric Plasmas
• Golden ratio autocorrelation function and the exponential decay
• Comment on "On Uniqueness of SDE Decomposition in A-type Stochastic Integration" [arXiv:1603.07927v1]
• Stochastic cloaking: concealing a region from diffusive particles
• Broken Detailed Balance and Non-Equilibrium Dynamics in Noisy Social Learning Models