The Markov processes of Schr�dinger

Monge’s problem with a quadratic cost by the zero-noise limit of h-path processes

“…The h-path process for √ εW (·) in C([0, 1]) with an initial distribution P 0 and a terminal one P 1,ε is the unique weak solution to the following (see [14]): for t ∈ [0, 1],…”

“…If we assume (A.2) p 1 (x) := P 1 (dx)/dx exists, then we can define h ε (t, x), X ε (t) and b ε (s, x) in the same way as in (1.7)-(1.9), respectively, by replacing (∫ 0,ε , ∫ 1,ε ) by (∫ 0,ε , ∫ 1,ε ) (see [14]). …”

A stochastic control approach to reciprocal diffusion processes

“…It is shown in [18] and two measures/20 and/2r, what is the minimum eneroy control u* for which x u* evolves from/20 to/2r? This problem is rigorously stated and solved in this paper.…”

“…Defining h,(x, t)= ~ q(t, x, T, z)f~(z)dz and using a special Harnack inequality [2], it can be shown that h. ~ h uniformily on K x [0, T] for any compact K. Therefore h(x, t) is in C 2' I(R" x [0, T)). For further details see [18] and references therein.…”

Reciprocal Processes: A Stochastic Analysis Approach

“…Many fundamental reciprocal properties were given by Jamison in a series of articles [Jam70], [Jam74], [Jam75], first in the context of Gaussian processes. Contributions to a physical interpretation and to the development of a stochastic calculus adjusted to reciprocal diffusions have been made by Zambrini and various co-authors in their interest of creating a Euclidean version of quantum mechanics (see [CZ91], [TZ97] and the monograph [CZ03]).…”