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We investigate conditional McKean-Vlasov equations driven by time-space white noise, motivated by the propagation of chaos in an N-particle system with space-time Ornstein-Uhlenbeck dynamics. The framework builds on the stochastic calculus…
Stochastic evolution underpins several approaches to the dynamics of open quantum systems, such as random modulation of Hamiltonian parameters, the stochastic Schrodinger equation (SSE), and the stochastic Liouville equation (SLE). These…
The derivation of the equations of motion for nonholonomic systems remains a central issue in analytical mechanics, primarily due to the tension between the d'Alembert-Lagrange differential principle and integral variational approaches.…
Motivated by studies of indirect measurements in quantum mechanics, we investigate stochastic differential equations with a fixed point subject to an additional infinitesimal repulsive perturbation. We conjecture, and prove for an important…
In this work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure…
Timesteppers constitute a powerful tool in modern computational science and engineering. Although they are typically used to advance the system forward in time, they can also be viewed as nonlinear mappings that implicitly encode steady…
In this note, we introduce a suitable generalization of the momentum map for nonholonomic field theories and prove a covariant form of the nonholonomic momentum equation. We show that these covariant objects coincide with their counterparts…
Recently, extracting data-driven governing laws of dynamical systems through deep learning frameworks has gained a lot of attention in various fields. Moreover, a growing amount of research work tends to transfer deterministic dynamical…
We introduce a stochastic perturbation of the Camassa-Holm equation such that, unlike previous formulations, energy is conserved by the stochastic flow. We compare this to a complementary approach which preserves Casimirs of the Poisson…
L\'evy's stochastic area for planar Brownian motion is the difference of two iterated integrals of second rank against its component one-dimen\-sional Brownian motions. Such iterated integrals can be multiplied using the sticky shuffle…
The famous It\^o-Stratonovich dilemma arises when one examines a dynamical system with a multiplicative white noise. In physics literature, this dilemma is often resolved in favour of the Stratonovich prescription because of its two…
We prove that a solution, in a variational framework, to the Stratonovich stochastic partial differential equation with noise $G\left(t, \Psi_t\right) \circ dW_t$ is given by a solution to the It\^{o} equation with It\^{o}-Stratonovich…
We develop a geometric formulation of stochastic dynamics in which noise, diffusion, path probabilities, fluctuation theorems, and entropy production arise from the intrinsic geometry of an evolving manifold rather than from externally…
We introduce natural differential geometric structures underlying the Poisson-Vlasov equations in momentum variables. We decompose the space of all vector fields over particle phase space into a semi-direct product algebra of Hamiltonian…
Many real-world systems exhibit ``noisy'' evolution in time; interpreting their finitely-sampled behavior as arising from continuous-time processes (in the It\^o or Stratonovich sense) has led to significant success in modeling and analysis…
Interpreting the noise in a stochastic differential equation, in particular the It\^o versus Stratonovich dilemma, is a problem that has generated a lot of debate in the physical literature. In the last decades, a third interpretation of…
This paper is concerned with stochastic Hamiltonian systems which model a class of open dynamical systems subject to random external forces. Their dynamics are governed by Ito stochastic differential equations whose structure is specified…
An effective characterization of chaotic conservative Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor derived from the structure of the Hamiltonian has been extended to a wide class of potential…
Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a…
The purpose of this paper is to establish the well-posedness of the stochastic Stefan problem on moving hypersurfaces. Through a specially designed transformation, it turns out we need to solve stochastic partial differential equations on a…