Natural boundaries for the Smoluchowski equation and affiliated diffusion processes.

The Schrödinger problem of deducing the microscopic dynamics from the inputoutput statistics data is known to admit a solution in terms of Markov diffusions.The uniqueness of solution is found linked to the natural boundaries respected by the underlying random motion.By choosing a reference Smoluchowski diffusion,we automatically fix the Feynman-Kac potential and the field of local accelerations it induces.We generate the family of affiliated diffusions with the same local dynamics,but different inaccessible boundaries on finite,semi-infinite and infinite domains.For each diffusion process a unique Feynman-Kac kernel is obtained by the constrained (Dirichlet boundary data) Wiener path integration.As a by-product of the discussion,we give an overview of the problem of inaccessible boundaries for the diffusion,and bring together (sometimes viewed from unexpected angles) results which are little known,and dispersed in publications from scarcely communicating areas of mathematics and physics permanent and current address:Institute of Theoretical Physics,University of Wroc law,PL-50 209 Wroc law,Poland;partially supported by the KBN grant No 20060 91 01 1 The Schrödinger problem:microscopic dynamics from the input-output statistics According to M.Kac any kind of time developement (be it deterministic or essentially probabilistic),which is analyzable in terms of probability,deserves the name of the stochastic process. Given a dynamical law of motion (for a particle as example),in many cases one can associate with it (compute or approximate the observed frequency data) a probability distribution and various mean values.In fact,it is well known that inequivalent finite difference random motion problems may give rise to the same continuous approximant (e.g. the diffusion equation representation of discrete processes).As well,in the study of nonlinear dynamical systems ,given almost any (for the purposes of our discussion ,basically one-dimensional)probability density,it is possible to construct an infinite number of deterministic finite difference equations,whose iterates are chaotic and which give rise to this a priori prescribed density. The inverse operation of deducing the detailed (possibly individual,microscopic) dynamics,which either implies or is consistent with the given probability distribution (and eventually with its own time evolution) cannot thus have a unique solution. If we disregard the detailed nature (like its chaotic ,jump process,random walk,phase space process with friction etc. implementations) of the given process,it appears that the standard Brownian motion and/or the broad class of Markovian diffusions incorporating the Wiener noise input,provide satisfactory approximations for a large variety of phenomena.It especially pertains to the explicit modelling of any unknown in detail physical process in terms of the input-output statistics (conditional probabilities and averages) of random motions with a finite time of duration. From now on ,we shall confine our attention to continuous Markov processes,whose random variable X(t), t ≥ 0 takes values on the real line R,and in particular can be restricted (constrained) to remain within the interval Λ ⊂ R,which may be finite or (semi-) infinite but basically an open set.Its boundaries ∂Λ (endpoints) will be denoted r1, r2 with −∞ ≤ r1 < r2 ≤ ∞. In the above input-output statistics context,let us invoke a probabilistic problem,ori ginally due to Schrödinger[5−7] : given two strictly positive (on an open interval) boundary probability distributions ρ0(x), ρT (x) for a process with the time of duration T ≥ 0.Can we uniquely identify the stochastic process interpolating between them ? Perhaps unexpectedly in the light of our previous comments ,the answer is known to be affirmative ,if we assume the interpolating process to be Markovian.In fact,we get here a unique Markovian diffusion,which is specified by the joint probability distribution