Generating stochastic trajectories with global dynamical constraints

Bruyne, Benjamin De; Majumdar, Satya N.; Orland, Henri; Schehr, Grégory

doi:10.1088/1742-5468/ac3e70

Cited by 14 publications

(22 citation statements)

References 77 publications

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“…As described in detail in [43,44], the corresponding conditioned process [X * (t), A * (t)] then satisfies the Ito system analog to Eq. 33…”

Section: Construction Of Conditioned Processes Involving the Local Timementioning

confidence: 99%

“…In particular, the conditioning on the area has been studied via various methods for Brownian processes or bridges [41] and for Ornstein-Uhlenbeck bridges [42]. The conditioning on the area and on other time-additive observables has been then analyzed both for the Brownian motion and for discretetime random walks [43]. This approach has been generalized [44] to various types of discrete-time or continuous-time Markov processes, while the time-additive observable can involve both the time spent in each configuration and the increments of the Markov process.…”

Section: Introductionmentioning

confidence: 99%

“…belongs to the interval 0 ≤ O [a,b] (t) ≤ t. The conditioning with respect to the occupation time of the interval [a = 0, b = +∞[ has been studied recently for the Brownian motion without drift [43], while the canonical conditioning with respect to the occupation time has been analyzed for various settings [73,74].…”

Section: Introductionmentioning

confidence: 99%

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Conditioning diffusion processes with respect to the local time at the origin

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Monthus²

2022

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When the unconditioned process is a diffusion process X(t) of drift µ(x) and of diffusion coefficient D = 1/2, the local time A(t) = t 0 dτ δ(X(τ )) at the origin x = 0 is one of the most important timeadditive observable. We construct various conditioned processes [X * (t), A * (t)] involving the local time A * (T ) at the time horizon T . When the horizon T is finite, we consider the conditioning towards the final position X * (T ) and towards the final local time A * (T ), as well as the conditioning towards the final local time A * (T ) alone without any condition on the final position X * (T ). In the limit of the infinite time horizon T → +∞, we consider the conditioning towards the finite asymptotic local time A * ∞ < +∞, as well as the conditioning towards the intensive local time a * corresponding to the extensive behavior AT T a * , that can be compared with the appropriate 'canonical conditioning' based on the generating function of the local time in the regime of large deviations. This general construction is then applied to generate various constrained stochastic trajectories for three unconditioned diffusions with different recurrence/transience properties : (i) the simplest example of transient diffusion corresponds to the uniform strictly positive drift µ(x) = µ > 0; (ii) the simplest example of diffusion converging towards an equilibrium is given by the drift µ(x) = −µ sgn(x) of parameter µ > 0; (iii) the simplest example of recurrent diffusion that does not converge towards an equilibrium is the Brownian motion without drift µ = 0.

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“…As described in detail in [43,44], the corresponding conditioned process [X * (t), A * (t)] then satisfies the Ito system analog to Eq. 33…”

Section: Construction Of Conditioned Processes Involving the Local Timementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Conditioning diffusion processes with respect to the local time at the origin

Mazzolo¹,

Monthus²

2022

Preprint

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“…In particular, the conditioning on the area has been studied via various methods for Brownian processes or bridges [23] and for Ornstein-Uhlenbeck bridges [24]. The conditioning on the area and on other time-additive observables has been then analyzed both for the Brownian motion and for discrete-time random walks [25]. This approach has been generalized recently [26] to various types of discretetime or continuous-time Markov processes, while the time-additive observable can involve both the time spent in each configuration and the increments of the Markov process.…”

Section: Introductionmentioning

confidence: 99%

Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

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Mazzolo²

2022

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When the unconditioned process is a diffusion living on the half-line x ∈] − ∞, a[ in the presence of an absorbing boundary condition at position x = a, we construct various conditioned processes corresponding to finite or infinite horizon. When the time horizon is finite T < +∞, the conditioning consists in imposing the probability P * (y, T ) to be surviving at time T and at the position y ∈ ] − ∞, a[, as well as the probability γ * (Ta) to have been absorbed at the previous time Ta ∈ [0, T ]. When the time horizon is infinite T = +∞, the conditioning consists in imposing the probability γ * (Ta) to have been absorbed at the time Ta ∈ [0, +∞[, whose normalization [1−S * (∞)] determines the conditioned probability S * (∞) ∈ [0, 1] of forever-survival. This case of infinite horizon T = +∞ can be thus reformulated as the conditioning of diffusion processes with respect to their first-passagetime properties at position a. This general framework is applied to the explicit case where the unconditioned process is the Brownian motion with uniform drift µ in order to generate stochastic trajectories satisfying various types of conditioning constraints. Finally, we describe the links with the dynamical large deviations at Level 2.5 and the stochastic control theory.

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“…The bridge problem of Eq. 1 can be also adapted to analyze the conditioning with respect to some global dynamical constraint as measured by a time-additive observable A of the stochastic trajectories: the idea is then to consider the bridge formula for the joint process (C, A) instead of the configuration C alone [27][28][29][30]. This 'microcanonical conditioning', where the time-additive observable is constrained to reach a given value after the finite time window T is the counterpart of the 'canonical conditioning' based on generating functions of additive observables that has been much studied recently in the field of non-equilibrium Markov processes .…”

Section: Introductionmentioning

confidence: 99%

Conditioning two diffusion processes with respect to their first-encounter properties

Mazzolo,

Monthus

2022

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We consider two independent identical diffusion processes that annihilate upon meeting in order to study their conditioning with respect to their first-encounter properties. For the case of finite horizon T < +∞, the maximum conditioning consists in imposing the probability P * (x, y, T ) that the two particles are surviving at positions x and y at time T , as well as the probability γ * (z, t) of annihilation at position z at the intermediate times t ∈ [0, T ]. The adaptation to various conditioning constraints that are less-detailed than these full distributions is analyzed via the optimization of the appropriate relative entropy with respect to the unconditioned processes. For the case of infinite horizon T = +∞, the maximum conditioning consists in imposing the first-encounter probability γ * (z, t) at position z at all finite times t ∈ [0, +∞[, whose normalization [1 − S * (∞)] determines the conditioned probability S * (∞) ∈ [0, 1] of forever-survival. This general framework is then applied to the explicit cases where the unconditioned processes are respectively two Brownian motions, two Ornstein-Uhlenbeck processes, or two tanh-drift processes, in order to generate stochastic trajectories satisfying various types of conditioning constraints. Finally, the link with the stochastic control theory is described via the optimization of the dynamical large deviations at Level 2.5 in the presence of the conditioning constraints that one wishes to impose.

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Generating stochastic trajectories with global dynamical constraints

Abstract: We propose a method to exactly generate Brownian paths x c (t) that are constrained to return to the origin at some future time t f , with a given fixed area A f … Show more

Cited by 14 publications

References 77 publications

Conditioning diffusion processes with respect to the local time at the origin

Conditioning diffusion processes with respect to the local time at the origin

Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

Conditioning two diffusion processes with respect to their first-encounter properties

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