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This realization lets us continue along the path to develop thermodynamic theory for general stochastic processes and confirm the universal ideal behavior in Orntein-Uhlenbeck processes.

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The realization also prompts us to continue our excursion further into applications. On the modeling side, we discover a way to analyze noise induced phenomena in reaction diffusion equations.

Stochastic Dynamics and Irreversibility

We are able to quantitatively explain the onset of pattern formations introduced by chemical reaction noise. Looking over to the Bayesian inference side for the learning of model parameters from data , we find ourselves in the position of digging into a critical problem: computation with stochasticity.

As the defacto approaches for Bayesian inference, Markov chain Monte Carlo MCMC methods have always been criticized for their slow convergence mixing rates and huge amount of computation required for large data sets scalability. It has been discovered that introduction of irreversibility increases the mixing of Markov processes. Using the decomposition of general Markov processes, we reparametrize the space of viable Markov processes for sampling purpose, so that the search for the correct MCMC algorithm turns into a game of plug and play with two matrices or transition probabilities to choose from.

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  5. Irreversibility is automatically incorporated as one of the components to specify. Digging even deeper into a new world of scalable Bayesian inference, we start to make use of stochastic gradient techniques for excessively large data sets.

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    With independent and identically distributed data, our previous results with continuous Markov process can be revised and provide a complete recipe to construct new stochastic gradient MCMC algorithms. Within our recipe, we pick some of the nice attributes of the previous methods and combine them to form an algorithm that excels at learning topics in Wikipedia entries in a streaming manner.

    With correlated data, we find a huge void space to explore.

    • Irreversibility in Stochastic Dynamic Models and Efficient Bayesian Inference?
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    As the first step, we visit time dependent data and harness the memory decay to generalize the stochastic gradient MCMC methods to hidden Markov models. A standard approach to bridge dynamics with phenomenology is to consider the Markovian approximation of the former. In this paper, we present a formulation in terms of dressed particles, which gives exact Markovian equations.

    Time reversibility

    Lambda is obtained by an extension of the canonical unitary transformation operator U that eliminates interactions for integrable systems. The unitarity of U is extended to "star unitarity" for Lambda. We show that Lambda-transformed variables have the same time evolution as stochastic variables obeying Langevin equations, and that Lambda-transformed distribution functions satisfy exact Fokker-Planck equations.