Time Reversal of Diffusions

Haussmann, U. G.; Pardoux, Étienne

doi:10.1214/aop/1176992362

Cited by 175 publications

(169 citation statements)

References 15 publications

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“…For a clear discussion of time-reversal in the context of Markovian equations of motion, see Refs. [57,58]. Although the action functional in Eq.…”

Section: The Time-reversal Transformationmentioning

confidence: 99%

Dynamical symmetries of Markov processes with multiplicative white noise

Aron¹,

Barci²,

Cugliandolo³

et al. 2016

J. Stat. Mech.

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We analyse various properties of stochastic Markov processes with multiplicative white noise. We take a single-variable problem as a simple example, and we later extend the analysis to the LandauLifshitz-Gilbert equation for the stochastic dynamics of a magnetic moment. In particular, we focus on the non-equilibrium transfer of angular momentum to the magnetization from a spin-polarised current of electrons, a technique which is widely used in the context of spintronics to manipulate magnetic moments. We unveil two hidden dynamical symmetries of the generating functionals of these Markovian multiplicative white-noise processes. One symmetry only holds in equilibrium and we use it to prove generic relations such as the fluctuation-dissipation theorems. Out of equilibrium, we take profit of the symmetry-breaking terms to prove fluctuation theorems. The other symmetry yields strong dynamical relations between correlation and response functions which can notably simplify the numerical analysis of these problems. Our construction allows us to clarify some misconceptions on multiplicative white-noise stochastic processes that can be found in the literature. In particular, we show that a first-order differential equation with multiplicative white noise can be transformed into an additive-noise equation, but that the latter keeps a non-trivial memory of the discretisation prescription used to define the former.

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“…For a clear discussion of time-reversal in the context of Markovian equations of motion, see Refs. [57,58]. Although the action functional in Eq.…”

Section: The Time-reversal Transformationmentioning

confidence: 99%

Dynamical symmetries of Markov processes with multiplicative white noise

Aron¹,

Barci²,

Cugliandolo³

et al. 2016

J. Stat. Mech.

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“…We call X reversible if Y (t) := X(1 − t) is a solution of the same stochastic differential equation (with exactly the same coefficients b and σ as X but a different Brownian motion). From a general result by Haussmann and Pardoux [13] it follows that Y is a diffusion process that satisfies the equation…”

Section: Dx(t) = B(x(t))dt + σ(X(t))dw (T) mentioning

confidence: 99%

Maximal regularity for gradient systems with boundary degeneracy

Cannarsa¹,

Prato²,

Metafune³

et al. 2015

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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We study a class of elliptic operators L that degenerate at the boundary of a bounded open set O ⊂ R d and possess a symmetrizing invariant measure µ. Such operators are associated with diffusion processes in O which are invariant for time reversal. After showing that the corresponding elliptic equation λϕ − Lϕ = f has a unique weak solution for any λ > 0 and f ∈ L 2 (O, µ), we obtain new results for the characterization of the domain of L.

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“…Markov process in the sense that the time-reversed process on any interval [0, T ] has the same dynamics as Z; see [19]. Furthermore, Assumption (A3) follows if it can be shown that Z does not explode to the boundary of E and [5,27].…”

Section: Is a Reversingmentioning

confidence: 99%

“…Standard results (e.g. [19]) in the theory of time reversal imply that ζ is a diffusion in its own filtration, and identify the corresponding coefficients. In order to deal with χ , we return to the definition of χ T and define yet one more process…”

Section: The Distribution Of (Z 0 X 0 ) Via Diffusion Time Reversalmentioning

confidence: 99%

Continuous-time perpetuities and time reversal of diffusions

Kardaras

Robertson

2016

Finance Stoch

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We consider the problem of estimating the joint distribution of a continuous-time perpetuity and the underlying factors which govern the cash flow rate, in an ergodic Markovian model. Two approaches are used to obtain the distribution. The first identifies a partial differential equation for the conditional cumulative distribution function of the perpetuity given the initial factor value, which under certain conditions ensures the existence of a density for the perpetuity. The second (and more general) approach, using techniques of time reversal, identifies the joint law as the stationary distribution of an ergodic multidimensional diffusion. This latter approach allows efficient use of Monte Carlo simulation, as the distribution is obtained by sampling a single path of the reversed process.

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Time Reversal of Diffusions

Cited by 175 publications

References 15 publications

Dynamical symmetries of Markov processes with multiplicative white noise

Dynamical symmetries of Markov processes with multiplicative white noise

Maximal regularity for gradient systems with boundary degeneracy

Continuous-time perpetuities and time reversal of diffusions

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