Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1005conditionlim W (x, T ; y, 0) = δ(x − y) .T →0 +Moreover, it follows that the solution of the diffusion equation for a generalinitial condition is given by∫ρ(x ′′ , t ′′ ) = W (x ′′ , t ′′ ; x ′ , t ′ ) ρ(x ′ , t ′ ) dx ′ .Iteration of this equation N times, with ɛ = (t ′′ − t ′ )/(N + 1), leads to theequationρ(x ′′ , t ′′ ) = N ′ ∫∫· · ·e −(1/2νɛ) P N∏Nl=0 (x l+1−x l ) 2l=1dx l ρ(x ′ , t ′ ) dx ′ ,where x N+1 ≡ x ′′ and x 0 ≡ x ′ . This equation features the imaginary timepropagator for a free particle of unit mass as given formally as∫∫W (x ′′ , t ′′ ; x ′ , t ′ ) = N D[x] e −(1/2ν) ẋ 2 dt ,where N denotes a formal normalization factor.The similarity of this expression with the Feynman path integral [forV (x) = 0] is clear, but there is a profound difference between these equations.In the former (Feynman) case the underlying measure is only finitelyadditive, while in the latter (Wiener) case the continuum limit actually definesa genuine measure, i.e., a countably additive measure on paths, whichis a version of the famous Wiener measure. In particular,∫W (x ′′ , t ′′ ; x ′ , t ′ ) = dµ ν W (x),where µ ν W denotes a measure on continuous paths x(t), t′ ≤ t ≤ t ′′ , forwhich x(t ′′ ) ≡ x ′′ and x(t ′ ) ≡ x ′ . Such a measure is said to be a pinnedWiener measure, since it specifies its path values at two time points, i.e.,at t = t ′ and at t = t ′′ > t ′ .We note that Brownian motion paths have the property that with probabilityone they are concentrated on continuous paths. However, it is alsotrue that the time derivative of a Brownian path is almost nowhere defined,which means that, with probability one, ẋ(t) = ±∞ for all t.