Path integral for multiplicative noise: Generalized Fokker-Planck equation  and entropy production rate in stochastic processes with threshold

This paper introduces a comprehensive extension of the path-integral formalism to model stochastic processes with arbitrary multiplicative noise. To do so, the Itô diffusive process is generalized by incorporating a multiplicative noise term <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mo>[</a:mo><a:mi>η</a:mi><a:mo>(</a:mo><a:mi>t</a:mi><a:mo>)</a:mo><a:mo>]</a:mo></a:mrow></a:math> that affects the diffusive coefficient in the stochastic-differential equation. Then, using the Parisi-Sourlas method, we estimate the transition probability between states of a stochastic variable <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mo>[</b:mo><b:mi>X</b:mi><b:mo>(</b:mo><b:mi>t</b:mi><b:mo>)</b:mo><b:mo>]</b:mo></b:mrow></b:math> based on the cumulant-generating function <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mo>(</c:mo><c:msub><c:mi mathvariant="script">K</c:mi><c:mi>η</c:mi></c:msub><c:mo>)</c:mo></c:mrow></c:math> of the noise. A parameter <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>γ</e:mi><e:mo>∈</e:mo><e:mo>[</e:mo><e:mn>0</e:mn><e:mo>,</e:mo><e:mn>1</e:mn><e:mo>]</e:mo></e:mrow></e:math> is introduced to account for the type of stochastic calculation used and its effect on the Jacobian of the path-integral formalism. Next, the Feynman-Kac functional is employed to derive the Fokker-Planck equation for generalized Itô diffusive processes, addressing issues with higher-order derivatives and ensuring compatibility with known functionals such as Onsager-Machlup and Martin-Siggia-Rose-Janssen-De Dominicis in the white-noise case. The general solution for the Fokker-Planck equation is provided when the stochastic drift is proportional to the diffusive coefficient and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msub><f:mi mathvariant="script">K</f:mi><f:mi>η</f:mi></f:msub></f:math> is scale invariant. Finally, the Brownian motion (BM), the geometric Brownian motion (GBM), the Levy <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi>α</h:mi></h:math>-stable flight <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mo>[</i:mo><i:mtext>LF</i:mtext><i:mrow><i:mo>(</i:mo><i:mi>α</i:mi><i:mo>)</i:mo></i:mrow><i:mo>]</i:mo></i:math>, and the geometric Levy <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mi>α</j:mi></j:math>-stable flight <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mo>[</k:mo><k:mtext>GLF</k:mtext><k:mrow><k:mo>(</k:mo><k:mi>α</k:mi><k:mo>)</k:mo></k:mrow><k:mo>]</k:mo></k:math> are simulated with thresholds, providing analytical comparisons for the probability density, Shannon entropy, and entropy production rate. It is found that restricted BM and restricted GBM exhibit quasisteady states since the rate of entropy production never vanishes. It is also worth mentioning that in this work, the GLF(<l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mrow><l:mi>α</l:mi><l:mo>)</l:mo></l:mrow></l:math> is defined and it is shown that its solution is found without the need for Itô's lemma.