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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

60J60 Diffusion processes
70H25 Hamilton’s principle
60J35 Transition functions, generators and resolvents
81S40 Path integrals in quantum mechanics
Full Text: DOI
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