Related papers: Implications of Kunita-It\^o-Wentzell formula for …
We extend the It\^o-Wentzell formula for the evolution along a continuous semimartingale of a time-dependent stochastic field driven by a continuous semimartingale to tensor field-valued stochastic processes on manifolds. More concretely,…
We provide a general It\=o\,-Wentzell formula for a random field of maps on the Wasserstein space of probability measures, defined by continuous semimartingales, and evaluated along the flow of conditional distributions of another…
This paper derives stochastic partial differential equations (SPDEs) for fluid dynamics from a stochastic variational principle (SVP). The Legendre transform of the Lagrangian formulation of these SPDEs yields their Lie-Poisson Hamiltonian…
Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated It\^o stochastic process (with zero mean) obtained from data which is taken in…
We establish It\^o's formula along flows of probability measures associated with general semimartingales; this generalizes existing results for flows of measures on It\^o processes. Our approach is to first establish It\^o's formula for…
We discuss stochastic derivations, stochastic Hamiltonians and the flows that they generate, algebraic fluctuaion-dissipation theorems, etc., in a language common to both classical and quantum algebras. It is convenient to define distinct…
The present paper is an extension of Fadle-Touzi (2024). Following the same methodology, merely based on Taylor expansions, we establish the It\^o and It\^o-Wentzell formulae for flows of conditional distributions of general…
Stochastic processes of evolving shapes are used in applications including evolutionary biology, where morphology changes stochastically as a function of evolutionary processes. Due to the non-linear and often infinite-dimensional nature of…
In this paper, we establish the It\^o-Wentzell-Lions formulae for flows of both full and conditional measures on general semimartingales. This generalizes the existing works on flows of measures on It\^o processes. The key technical…
In this work, we introduce an effective model for both ideal and viscous fluid dynamics within the framework of kinetic field theory (KFT). The main application we have in mind is cosmic structure formation where gaseous components need to…
In this paper, we study the stochastic Hamiltonian flow in Wasserstein manifold, the probability density space equipped with $L^2$-Wasserstein metric tensor, via the Wong--Zakai approximation. We begin our investigation by showing that the…
Stochastic geometric mechanics (SGM) is known for its potential utility in quantifying uncertainty in global climate modelling of the Earth's ocean and atmosphere while also preserving the fundamental advective transport properties of ideal…
In this paper we investigate classical solution of a semi-linear system of backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. By proving an It\^{o}-Wentzell formula for jump…
In this manuscript, we extend Constantin-Iyer's Lagrangian formulation of Navier-Stokes Equation to a wider class of hydrodynamic models. Moreover, we prove that such Lagrangian formulation is naturally derived from a stochastic…
A semi-classical non-Hamiltonian model of a spontaneous collapse of unstable quantum system is given. The time evolution of the system becomes non-Hamiltonian at random instants of transition of pure states to reduced ones, given by a…
We present a novel backward It{\^o}-Ventzell formula and an extension of the Aleeksev-Gr\"obner interpolating formula to stochastic flows. We also present some natural spectral conditions that yield direct and simple proofs of time uniform…
We introduce Wilson-It\^o diffusions, a class of random fields on $\mathbb{R}^d$ that change continuously along a scale parameter via a Markovian dynamics with local coefficients. Described via forward-backward stochastic differential…
The general idea of a stochastic gauge representation is introduced and compared with more traditional phase-space expansions, like the Wigner expansion. Stochastic gauges can be used to obtain an infinite class of positive-definite…
We suggest that the tools of contraction analysis for deterministic systems can be applied towards studying the convergence behavior of stochastic dynamical systems in the Wasserstein metric. In particular, we consider the case of Ito…
We prove an It\^o-Wentzell formula for the fractional Brownian motion. As an application we derive an existence and uniqueness result for a class of stochastic differential equations driven by this stochastic process.