A Doob h-Transform of the Gross–Pitaevskii Hamiltonian

Abstract

We perform a Doob h-transformation of the N-body Hamiltonian for N trapped interacting Bose particles starting from the unique real strictly positive ground state. This identifies the unitarily equivalent Markov generator describing the N interacting diffusion processes. By applying the convergence results coming from the Gross-Pitaevskii (GP) scaling limit to the energy form associated with this Markov generator we obtain the asymptotic generator uniquely corresponding to the GP Hamiltonian, which has been rigorously derived from the N-body Hamiltonian by performing the same scaling limit. Our asymptotic generator is a non linear diffusion generator with a killing rate governed by the GP order parameter. The associated stochastic process properly describes the single particle motion in the Bose-Einstein condensate including the self-interaction which acts here as a probability density-dependent killing rate. Finally, in a one-dimensional setting, we discuss under which conditions the process conditioned to survival converges to a quasi-stationary distribution having density with respect to the speed measure given by the principal eigenfunction of the diffusion generator with killing.

Non-local Markovian Symmetric Forms on Infinite Dimensional Spaces I. The closability and quasi-regularity

“…In the case where the state space E is finite dimensional, much has been done on the considerations of both local and non-local Dirichlet forms and semi-Dirichlet forms (non-symmetric Dirichlet forms). The natural setting is the one where the Hilbert space, where the Dirichlet forms are defined (as quadratic forms), is L 2 (E; m), with E a general locally compact separable metric space (when E is a topological vector space, the dimension of the space is thus finite) and m a positive Radon measure on it (cf., e.g., [46,48,49,60,70,84,88], [4], [34,35] and references therein). Also, many results have been developed on the theory of general (non-symmetric) local Dirichlet forms defined on L 2 (E; m), with general topological spaces including the case of some infinite dimensional topological vector spaces, and m some Radon measures on them (cf., e.g., [3,[14][15][16][17][25][26][27][29][30][31][32]68,69,85,87] and references therein).…”

“…For the stochastic quantizations of Euclidean P(φ) 2 fields on S (R 2 → R), i.e., fields with polynomial interactions, fields with trigonometric interactions and exponential interactions on S (R 2 → R), these considerations by means of the local Dirichlet form arguments were completed by [12][13][14][15][16][17][25][26][27][29][30][31]37,59]. In this direction of the application of local Dirichlet forms, there also are corresponding considerations for measures m describing infinite particle systems (cf., e.g., [21,35,75,92] and references therein). For works on stochastic quantization of Φ 4 2 using other methods see [40,41,65] and references in [Albeverio, Ma, Röckner], [A, Kusuoka-sei 2017].…”

Mean-field limit for a class of stochastic ergodic control problems

“…is the optimal control for the problem (1.1) with cost functional (1.2) and potential V δ (see [3] for an alternative derivation of a stochastic process associated with the above cost functional). What we want to consider here is an N -particle problem converging to the solution of the optimal control ergodic problem just described.…”

Strong Kac’s chaos in the mean-field Bose–Einstein Condensation

“…The stationary probability measure P N with density ρ N can be alternatively defined as the one of the Markov diffusion process (properly) associated to the Dirichlet form ( [3,24,25,39]):…”