Conditional probabilities in multiplicative noise processes

We address the calculation of transition probabilities in multiplicative noise stochastic differential equations using a path integral approach. We show the equivalence between the conditional probability and the propagator of a quantum particle with variable mass. Introducing a time reparametrization, we are able to transform the problem of multiplicative noise fluctuations into an equivalent additive one. We illustrate the method by showing the explicit analytic computation of the conditional probability of a harmonic oscillator in a nonlinear multiplicative environment.

Figure 1

Solution of the Fokker-Planck equation, Eqs. (2.9) and (2.10), for the model given by Eqs. (5.1) and (5.2) with initial condition $P(x,0)=\delta \left(x\right)$ and the following values of the parameters: $\omega =1$, $\lambda =0.1$, $\sigma =0.5$, and $\alpha =0$.

Figure 2

${\mathrm{\Gamma }}^2\left(T\right)$ computed from Eq. (5.16) and from Eq. (5.17) for $\alpha =0$, $\sigma =0.5$, $\lambda =0.1$. Both curves coincide within the graphic precision.

Figure 3

Comparison of the variance ${\mathrm{\Gamma }}^2\left(T\right)$, computed with Eq. (5.16), with the exact numerical evaluation ${\mathrm{\Gamma }}_{\mathrm{ex}}^2\left(T\right)$. The solid line is ${\mathrm{\Gamma }}^2\left(T\right)$, while ${\mathrm{\Gamma }}_{\mathrm{ex}}^2\left(T\right)$ is depicted by the dotted line. We used the following parameters for both panels: $\omega =1$, $\sigma =1.5$, $\lambda =0.1$. (a) was computed in the Itô prescription, $\alpha =0$, while (b) was calculated in the Hänggi-Klimontovich prescription, $\alpha =1$.

Figure 4

$\mathrm{\Delta }\mathrm{\Gamma }=|{\mathrm{\Gamma }}_{\mathrm{ex}}^2\left(T\right)-{\mathrm{\Gamma }}^2\left(T\right)|$ for the parameters $\omega =1$, $\sigma =1.5$, $\lambda =0.1$. The dot-dashed line was computed in the Itô interpretation $\alpha =0$, the solid line, in the Stratonovich prescription $\alpha =1/2$, while the dashed line was computed in the Hänggi-Klimontovich prescription $\alpha =1$.