Supersymmetric theory of stochastic dynamics: Difference between revisions

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The [[Itô_calculus#Itô calculus for physicists|physicist's way]] to look at a [[Stochastic differential equation#Use in physics|stochastic differential equation]] is essentially a [[Non-autonomous system (mathematics)|continuous-time non-autonomous dynamical system]] that can be defined as:

<math display="block"> \dot x(t) = F(x(t))+(2\Theta)^{1/2}G_a(x(t))\xi^a(t)\equiv{\mathcal F}(\xi(t)),</math>

where <math display="inline">x\in X </math> is a point in a [[Closed manifold|closed]] [[smooth manifold]], <math display="inline">X</math>, called in dynamical systems theory a [[State-space representation|state space]] while in physics, where <math>X</math> is often a [[symplectic manifold]] with half of variables having the meaning of momenta, it is called the [[phase space]]. Further, <math> F(x)\in TX_xTX </math> is a sufficiently smooth flow [[vector field]] from the [[tangent space]] of <math> X</math> having the meaning of deterministic law of evolution, and <math> G_a \in TX, a=1, \ldots, D_\xi </math> is a set of sufficiently smooth vector fields that specify how the system is coupled to the time-dependent noise, <math>\xi(t)\in\mathbb{R}^{D_\xi}</math>, which is called [[Additive noise|additive]]/[[Multiplicative noise|multiplicative]] depending on whether <math> G_a </math>'s are independent/dependent on the position on <math>X</math>.

The randomness of the noise will be introduced later. For now, the noise is a deterministic function of time and the equation above is an [[ordinary differential equation]] (ODE) with a time-dependent flow vector field, <math>\mathcal F</math>. The solutions/trajectories of this ODE are differentiable with respect to initial conditions even for non-differentiable <math>\xi(t)</math>'s.<ref>{{Cite journal|last=Slavík|first=A.|title=Generalized differential equations: Differentiability of solutions with respect to initial conditions and parameters|journal=Journal of Mathematical Analysis and Applications|language=en|volume=402|issue=1|pages=261–274|doi=10.1016/j.jmaa.2013.01.027|year=2013|doi-access=free}}</ref> In other words, there exists a two-parameter family of noise-configuration-dependent [[diffeomorphism]]s: