Supersymmetric theory of stochastic dynamics: Difference between revisions

Revision as of 22:09, 15 July 2025

Revision as of 14:37, 18 July 2025

Line 80:

<ref name=":19">{{Cite journal|last=Ruelle|first=D.|date=1990-12-01|title=An extension of the theory of Fredholm determinants|journal=Publications Mathématiques de l'Institut des Hautes Études Scientifiques|language=en|volume=72|issue=1|pages=175–193|doi=10.1007/bf02699133|s2cid=121869096|issn=0073-8301|url=http://www.numdam.org/item/PMIHES_1990__72__175_0/}}</ref>

-- the [[dynamical systems theory]] counterpart of the stochastic evolution operator of the theory of SDEs and/or the Parisi-Sourlas approach. For [[Gaussian noise|Gaussian]] [[White noise|white noise]], <math> \langle \xi^a(t) \rangle_{\text{noise}} =0, \langle\xi^a(t)\xi^b(t')\rangle_{\text{noise}} = \delta^{ab}\delta(t-t')</math>..., the GTO is

<math display="block" > \hat{\mathcal M }_{tt'} = \langle \hat M(\xi)_{t't}^*\rangle_{\text{noise}} = e^{-(t-t')\hat H},. </math>

withHere, the ''infinitesimal'' GTO, oris the stochastic evolution operator in the [[Stratonovich integral|Stratonovich interpretation]] in the traditional approach to SDEs,<ref>

<ref>{{Cite journal

| last = Ruelle

| first = David

| title = Ergodic theory of chaos and strange attractors

| journal = Reviews of Modern Physics

| volume = 57

| issue = 3

| pages = 617–656

| date = July 1985

| doi = 10.1103/RevModPhys.57.617

| bibcode = 1985RvMP...57..617E

}}</ref>

<math display="block" > \hat H = \hat L_F - \Theta \hat L_{G_a}\hat L_{G_a},</math>

where <math> \hat L_F</math> is the [[Lie derivative]] along the vector field specified in the subscript. Its fundamental mathematical meaning -- the pullback averaged over noise -- ensures that GTO is unique. It corresponds to [[Stratonovich integral|Stratonovich interpretation]] in the traditional approach to SDEs.<ref>

{{cite book

| last = Kunita

Line 119 ⟶ 105:

}}

</ref>

<ref>{{Cite journal

| last = Ruelle

| first = David

| title = Ergodic theory of chaos and strange attractors

| journal = Reviews of Modern Physics

| volume = 57

| issue = 3

| pages = 617–656

| date = July 1985

| doi = 10.1103/RevModPhys.57.617

| bibcode = 1985RvMP...57..617E

}}</ref>

<math display="block" > \hat H = \hat L_F - \Theta \hat L_{G_a}\hat L_{G_a},</math>

where <math> \hat L_F</math> is the [[Lie derivative]] along the vector field specified in the subscript. ItsNote, however, that its fundamental mathematical meaning -- the pullback averaged over noise -- ensures that GTO is unique. It corresponds to(see [[Stratonovich#Physical integralPerspective#Operator representation|StratonovichOperator interpretationrepresentation]] in the traditional approach to SDEsbelow).<ref>

=== Topological supersymmetry ===