Probabilistic Deformation of Contact Geometry, Diffusion Processes and Their Quadratures

Reduction and reconstruction of stochastic differential equations via symmetries

“…Uα(Xs)dW α s (see, e.g., [7,27]). There is a growing interest for this kind of stochastic perturbations of Lagrangian and Hamiltonian systems due both to their special mathematical properties and to their applications in mathematical physics (see, e.g., [2,3,24,25,30,31,39]). In the following we propose a method to obtain a SDE of the form (21) which can be interpreted as a symmetric stochastic perturbation of a symmetric ODE of the form (22).…”

“…The underlying idea is to provide an algorithmic procedure in order to identify symmetric ODEs and to exploit their symmetries for simplifying them. Despite the well acknowledged and rich literature in the deterministic setting, the concept of infinitesimal symmetries for SDEs is quite recent [14,18,28,30,35,45] and their use for reduction purposes is not yet completely developed. The principal aim of this paper is to investigate the possible applications of a symmetry approach to SDEs taking the cue from the deterministic case.…”

The Research Program of Stochastic Deformation (with a View Toward Geometric Mechanics)

“…The definition of Symmetries for HJB is the same as classically, i.e Eq (5.11), for the deformed ideal (5.12), and the calculation of the coefficients N • of Eq (5.10) is a rather tiring exercise (cf. [40]). But it is quite rewarding: The proof of the Theorem shows that it is, in fact, sufficient to consider symmetry contact Hamiltonians of the form N (τ, x, S, E, P ) = N x (x, τ )P + N τ (x, τ )E + N S (x, τ ), so that N x = Q, N τ = T and N S = −φ in the notations of the stochastic Noether Theorem, where T, Q and φ solve its Determining Equations.…”

“…[40]) Along any N -variation as before, I HJB and the Lagrangian L satisfy the following invariance conditions(1) L N (ω P C ) = −dN S (2) L N (Ω) = 0 (5.14) L N (L) + L dN τ dτ = −D τ N S .This Theorem seems purely algebraic but encodes a lot of informations about our stochastic deformation, resulting from the substitution of smooth classical paths τ → ω(τ ) by Bernstein diffusion sample paths τ → X(τ ). Eq (1) means that Poincaré-Cartan 1-form is invariant up to a phase coefficient N S .…”

A symmetry-adapted numerical scheme for SDEs

“…[4,8,27,26] and references therein), the application of the same techniques in the stochastic setting to the best of our knowledge is not yet pursued, probably because the concept of symmetry of a SDE has been quite recently developed (see e.g. [7,6,12,23,25,29]). In this paper we introduce two different numerical methods taking advantage of the presence of Lie symmetries in a given SDE in order to provide a more efficient numerical integration of it.…”