Lévy area with a drift as a renormalization limit of Markov chains on periodic graphs

Lopusanschi, Olga; Simon, Damien

doi:10.1016/j.spa.2017.09.004

Cited by 10 publications

(12 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Periodic graphs or hidden Markov walks. Theorem 3.3 naturally generalizes the main results in Lopusanschi and Simon (2018) and Lopusanschi and Simon (2017).…”

Section: Applicationssupporting

confidence: 80%

“…4.2. Rotating drift (Lopusanschi and Simon, 2018). The following example shows that non-zero area anomaly is possible even with no speed.…”

Section: 1mentioning

confidence: 83%

“…Theorem 5.8 Simon, 2017, 2018). Let X be either an irreducible Markov chain on a periodic graph (see the definition in Lopusanschi and Simon, 2018) or an irreducible hidden Markov walk driven by a finite state Markov chain (see the definition in Lopusanschi and Simon, 2017), then the conditions of Theorem 3.3 are satisfied. Proof (Sketch): If (Y n ) n is an irreducible Markov chain on a periodic graph or an irreducible hidden Markov walk, it admit an underlying irreducible Markov chain (X n ) n on a finite state space.…”

Section: Applicationsmentioning

confidence: 99%

“…By construction, the sequence (T n ) n≥0 is strictly increasing and, as (X n ) n≥0 is irreducible, all T n are finite a.s. The increments (T n+1 − T n ) n≥0 are i.i.d., as well as the variables (Y Tn+1 − Y Tn ) n≥0 (see the proof in Lopusanschi and Simon, 2018) and, more generally,…”

Section: Applicationsmentioning

confidence: 99%

“…regeneration structure and moments of the regeneration times. Second, the discrete processes in these papers are homogeneous in space (a simple random walk on periodic graphs, see Lopusanschi and Simon, 2017, or hidden Markov walk where the jumps are independent of the current position, see Lopusanschi and Simon, 2018). In our case we allow the process to have jump distributions that are inhomogeneous in space.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Ballistic random walks in random environment as rough paths: convergence and area anomaly

Lopusanschi¹,

Orenshtein²

2021

ALEA

Self Cite

View full text Add to dashboard Cite

Annealed functional CLT in the rough path topology is proved for the standard class of ballistic random walks in random environment. Moreover, the 'area anomaly', i.e. a deterministic linear correction for the second level iterated integral of the rescaled path, is identified in terms of a stochastic area on a regeneration interval. The main theorem is formulated in more general settings, namely for any discrete process with uniformly bounded increments which admits a regeneration structure where the regeneration times have finite moments. Here the largest finite moment translates into the degree of regularity of the rough path topology. In particular, the convergence holds in the α-Hölder rough path topology for all α < 1/2 whenever all moments are finite, which is the case for the class of ballistic random walks in random environment. The latter may be compared to a special class of random walks in Dirichlet environments for which the regularity α < 1/2 is bounded away from 1/2, explicitly in terms of the corresponding trap parameter.

show abstract

“…Periodic graphs or hidden Markov walks. Theorem 3.3 naturally generalizes the main results in Lopusanschi and Simon (2018) and Lopusanschi and Simon (2017).…”

Section: Applicationssupporting

confidence: 80%

“…4.2. Rotating drift (Lopusanschi and Simon, 2018). The following example shows that non-zero area anomaly is possible even with no speed.…”

Section: 1mentioning

confidence: 83%

Section: Applicationsmentioning

confidence: 99%

Section: Applicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Ballistic random walks in random environment as rough paths: convergence and area anomaly

Lopusanschi¹,

Orenshtein²

2021

ALEA

Self Cite

View full text Add to dashboard Cite

show abstract

Time-reversibility and nonvanishing Lévy area

Gottwald,

Melbourne

2024

Nonlinearity

View full text Add to dashboard Cite

We give a complete description and clarification of the structure of the Lévy area correction to Itô/Stratonovich stochastic integrals arising as limits of time-reversible deterministic dynamical systems. In particular, we show that time-reversibility forces the Lévy area to vanish only in very specific situations that are easily classified. In the absence of such obstructions, we prove that there are no further restrictions on the Lévy area and that it is typically nonvanishing and far from negligible.

show abstract

Anomalous thermodynamics in homogenized generalized Langevin systems

Lim

2021

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

We study functionals, such as heat and work, along trajectories of a class of multi-dimensional generalized Langevin systems in various limiting situations that correspond to different level of homogenization. These are the situations where one or more of the inertial time scale(s), the memory time scale(s) and the noise correlation time scale(s) of the systems are taken to zero. We find that, unless one restricts to special situations that do not break symmetry of the Onsager matrix associated with the fast dynamics, it is generally not possible to express the effective evolution of these functionals solely in terms of trajectory of the homogenized process describing the system dynamics via the widely adopted Stratonovich convention. In fact, an anomalous term is often needed for a complete description, implying that convergence of these functionals needs more information than simply the limit of the dynamical process. We trace the origin of such impossibility to area anomaly, thereby linking the symmetry breaking and area anomaly. This hold important consequences for many nonequilibrium systems that can be modeled by generalized Langevin equations. Our convergence results hold in a strong pathwise sense.

show abstract

Lévy area with a drift as a renormalization limit of Markov chains on periodic graphs

Cited by 10 publications

References 15 publications

Ballistic random walks in random environment as rough paths: convergence and area anomaly

Ballistic random walks in random environment as rough paths: convergence and area anomaly

Time-reversibility and nonvanishing Lévy area

Anomalous thermodynamics in homogenized generalized Langevin systems

Contact Info

Product

Resources

About