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The large-time and vanishing-noise limits for entropy production in nondegenerate diffusions

Renaud Raquépas

Abstract

We investigate the behaviour of a family of entropy production functionals associated to stochastic differential equations of the form $\mathrm{d} X_s = -\nabla V(X_s) \, \mathrm{d} s + b(X_s) \, \mathrm{d} s + \sqrt{2ε} \, \mathrm{d} W_s $, where $b$ is a globally Lipschitz nonconservative vector field keeping the system out of equilibrium, with emphasis on the large-time limit and then the vanishing-noise limit. Different members of the family correspond to different choices of boundary terms. Our analysis yields a law of large numbers and a local large deviation principle which does not depend on the choice of boundary terms and which exhibits a Gallavotti--Cohen symmetry. We use techniques from the theory of semigroups and from semiclassical analysis to reduce the description of the asymptotic behaviour of the functional to the study of the leading eigenvalue of a quadratic approximation of a deformation of the infinitesimal generator near critical points of $V$.

The large-time and vanishing-noise limits for entropy production in nondegenerate diffusions

Abstract

We investigate the behaviour of a family of entropy production functionals associated to stochastic differential equations of the form , where is a globally Lipschitz nonconservative vector field keeping the system out of equilibrium, with emphasis on the large-time limit and then the vanishing-noise limit. Different members of the family correspond to different choices of boundary terms. Our analysis yields a law of large numbers and a local large deviation principle which does not depend on the choice of boundary terms and which exhibits a Gallavotti--Cohen symmetry. We use techniques from the theory of semigroups and from semiclassical analysis to reduce the description of the asymptotic behaviour of the functional to the study of the leading eigenvalue of a quadratic approximation of a deformation of the infinitesimal generator near critical points of .

Paper Structure

This paper contains 14 sections, 31 theorems, 168 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setup, definitions and preliminary results
  3. Assumptions on the equation and immediate consequences
  4. Time reversal and the canonical entropy production functional
  5. Generalised entropy production functionals
  6. Definition and the weak law of large numbers
  7. Assumptions on the boundary terms and the initial condition
  8. A representation for the moment-generating function
  9. Large deviations in the large-time limit
  10. The linear case
  11. The rate function in the vanishing-noise limit
  12. Convergence in the proof of Proposition \ref{['prop:lim-t-eps']}
  13. Properties of the deformed generators
  14. Standard probabilistic consequences of our assumptions

Key Result

Proposition 2.2

Under Assumptions (L0) and (L1), if $\lambda$ and the Lebesgue measure are mutually absolutely continuous, then $\mathcal{P}_t^{\lambda,\epsilon}$ and $\mathcal{P}_t^{\lambda,\epsilon} \circ \Theta^{-1}$ are mutually absolutely continuous and for $\mathcal{P}_t^{\lambda,\epsilon}$-almost all $\gamma \in \mathcal{C}_t$, where $\mu_0^\epsilon$ is defined by eq:eq-meas. Moreover, for all $\alpha \i

Figures (3)

  • Figure 1: We consider a polynomial potential $V : \mathbf{R}^2 \to \mathbf{R}$ with a global maximum in $x_1 = (x_1^1,0)$, a saddle point in $x_2 = (x_2^1,0)$ and a local minimum in $x_3 = (x_3^1,0)$. On the left: the profile of $V$ for $x^2 \equiv 0$ as well as a nonconservative vector field $b$ which is stationary in all those critical points superimposed on a contour plot of $V$. On the right: $e_1$ and $e_3$ from \ref{['eq:ej-as-trace']} are plotted as functions of $\alpha$; $e_2$ lies below the visible region.
  • Figure 2: In the case of the function $e$ on the left, obtained by taking the maximum of $e_1$, $e_2$ and $e_3$ in Figure \ref{['fig:cgf-sketch']}, the jump in the derivative from $-\mathfrak{m}_3$ to $-\mathfrak{m}_1$ at the origin causes $e_*$ to vanish on the interval $[\mathfrak{m}_1,\mathfrak{m}_3]$. The Legendre transform $e_*$ is sketched on the right.
  • Figure 3: The orange region enclosed in the solid contours is the set of values allowed in Lemma \ref{['lem:link-I-Abb']} computed for $(k_b,h_b) = (0.33,0.75), (0.33,1.5)$ and $(0.49,1.5)$ --- from left to right.

Theorems & Definitions (65)

  • Remark 2.1
  • Proposition 2.2
  • proof : Proof of Proposition \ref{['prop:RN']}
  • Corollary 2.3
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 55 more