×

zbMATH — the first resource for mathematics

Variational processes and stochastic versions of mechanics. (English) Zbl 0623.60102
[Abridged review. The complete version is available on demand.]
In this paper the theory of Bernstein processes as developed by Fortet, Beurling, and Jamison is applied to the stochastic mechanics of Nelson. Let X(t) be a process valued in \(M={\mathbb{R}}^ d\). The forward derivative and dispersion are defined by \[ DX(t)\equiv \lim_{\Delta t\to 0}E[\Delta X(t)/\Delta t| {\mathcal P}_ t]\quad and\quad CX(t)\equiv \lim_{\Delta t\to 0}E[\Delta X(t)\Delta X(t)| {\mathcal P}_ t] \] (a truncation factor being omitted in the latter conditional expectation) where the past \({\mathcal P}_ t\) of t is generated by \(\{\) X(s): \(s\leq t\}\). If in particular, X is Markov, both DX(t)\(\equiv b(X(t),t)\) and CX(t)\(\equiv C(X(t),t)\) are determined by the forward transition (density) function p(s,x,t,y) by \[ b(x,s)\equiv \lim_{\Delta s\to 0}(\Delta s)^{-1}\int_{{\mathbb{B}}(x,\epsilon)}(y-x)p(s,x,s+\Delta s,y)dy \] and replacing (y-x) by (y-x)(y-x) for C(x,s).
Reversing the sense of time we have the backward derivative, \(D_*X(s)\), and the dispersion, \(C_*X(s)\); which equal to \(b_*(X(s),s)\) and \(C_*(X(s),s)\) when X is Markov, the past being replaced by the future \({\mathcal F}_ s\). The backward transition density is now \(p_*(s,x,t,y)\equiv p(s,x,ty)\rho (x,s)/\rho (y,t)\) if \(\rho\) (x,s) is the density of X(s).
The current velocity v and the osmotic velocity u are defined by \(v=(b+b_*)/2\) and \(u=(b-b_*)/2\), respectively, and the continuity equation \(\partial_ t\rho +div j=0\) is satisfied for \(j=\rho v.\)
A Bernstein process is a process X for which \({\mathcal P}(t)\wedge {\mathcal F}(s)\) is independent of \({\mathcal P}(s)\vee {\mathcal F}(t)\) conditioned by \(\{\) X(s),X(t)\(\}\), whenever \(s<t\). From a Bernstein transition (density) function \(\beta\) (s,x;t,y;u,z) \((a\leq s<t<u\leq b)\) and a probability measure m on \(M^ 2\), there is a canonical Bernstein process (X(t), \(t\in [a,b]=I)\) such that: \[ E[X(t)\in dy| \quad {\mathcal P}(s)\vee {\mathcal F}(u)] = \beta (s,X(s),\quad t,y,u,X(u))dy \] and (X(a),X(b)) is distributed as m. In general this is not Markov, even when \(\beta =\hat k\) is induced by a Markov transition.
Reviewer: Yang Weizhe

MSC:
60J60 Diffusion processes
70H25 Hamilton’s principle
60J35 Transition functions, generators and resolvents
81S40 Path integrals in quantum mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1103/PhysRev.150.1079 · doi:10.1103/PhysRev.150.1079
[2] DOI: 10.1007/BF01224827 · Zbl 0558.60059 · doi:10.1007/BF01224827
[3] DOI: 10.1063/1.525006 · doi:10.1063/1.525006
[4] DOI: 10.1016/0022-1236(81)90079-3 · Zbl 0482.60063 · doi:10.1016/0022-1236(81)90079-3
[5] DOI: 10.1007/BF00669792 · Zbl 0563.60098 · doi:10.1007/BF00669792
[6] Schrödinger E., Ann. Inst. H. Poincaré 2 pp 269– (1932)
[7] DOI: 10.1103/PhysRevA.33.1532 · doi:10.1103/PhysRevA.33.1532
[8] DOI: 10.1007/BF00532864 · Zbl 0326.60033 · doi:10.1007/BF00532864
[9] Fortet R., J. Math. Pures Appl. pp 83– (1940)
[10] DOI: 10.2307/1969644 · Zbl 0047.09303 · doi:10.2307/1969644
[11] DOI: 10.1063/1.1666536 · Zbl 0291.60028 · doi:10.1063/1.1666536
[12] DOI: 10.1007/BF01442148 · Zbl 0398.93068 · doi:10.1007/BF01442148
[13] DOI: 10.1103/PhysRevD.27.1774 · doi:10.1103/PhysRevD.27.1774
[14] DOI: 10.2307/1970151 · Zbl 0091.13001 · doi:10.2307/1970151
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.