Markov processes and generalized Schrödinger equations

Abstract

Articles you may be interested inGeneralized Korteweg-de Vries equation induced from position-dependent effective mass quantum models and mass-deformed soliton solution through inverse scattering transformThe Schrödinger equation with friction from the quantum trajectory perspective Starting from the forward and backward infinitesimal generators of bilateral, timehomogeneous Markov processes, the self-adjoint Hamiltonians of the generalized Schrödinger equations are first introduced by means of suitable Doob transformations. Then, by broadening with the aid of the Dirichlet forms, the results of the Nelson stochastic mechanics, we prove that it is possible to associate bilateral, and timehomogeneous Markov processes to the wave functions stationary solutions of our generalized Schrödinger equations. Particular attention is then paid to the special case of the Lévy-Schrödinger (LS) equations and to their associated Lévy-type Markov processes, and to a few examples of Cauchy background noise. C

Explicit invariant measures for infinite dimensional SDE driven by Lévy noise with dissipative nonlinear drift I

“…For the derivation of these relations we shall use an adaptation to the case of Lévy noise of methods developed before in a diffusion setting (with Gaussian noise) which go under the name of h-transform, see [51], or "ground state transformation", see, e.g., [18] and [94]. In the finite dimensional case this has been discussed in [30], where the general case of finite dimensional Lévy noise has been considered. We relate in particular the work in [30] to work on solutions of martingale problems and relations to weak solutions of Lévy driven SDE.…”

“…In the finite dimensional case this has been discussed in [30], where the general case of finite dimensional Lévy noise has been considered. We relate in particular the work in [30] to work on solutions of martingale problems and relations to weak solutions of Lévy driven SDE. Our explicit invariant measures in finite dimensions considerable extend previously known examples in [27], [33].…”

A Class of Lévy Driven SDEs and their Explicit Invariant Measures

“…This is related to techniques known in the case of Gaussian noise as Dynkin's h-transform or, ground state transformation, see. [19].The extension to the Lévy case was initiated by [37], we give some observations and complements to this construction, stressing both its relation to the symbols discussed in Section 2.3 and the invariant measures. The discussion is then extended in Section 2.5 considering perturbed O-U-Lévy processes, defined by invariant measures and Dirichlet forms.…”

“…2.2. For this extension we follow closely [37], who were the first, to the best of our knowledge, who extended previous work on the ground state transformation for the case with Gaussian noise covered in [19] to the case of Lévy noise. Let φ be a given function on R d , such that R d φ 2 dx = 1 and φ(x) > 0, dx − a.e.…”

Levy flights in steep potential wells: Langevin modeling versus direct response to energy landscapes

“…We are not aware of any simple computation method starting from the Cauchy principal value definitions (4) or (5), compare e.g. also [41] and [42].…”