Optimal Resetting Brownian Bridges

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

doi:10.48550/arxiv.2201.01994

Cited by 4 publications

(4 citation statements)

References 77 publications

(98 reference statements)

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“…In most cases, this quantity is not known analytically but when it is at our disposal, Doob's technique has been successfully applied to various kinds of conditioned processes [1,11,[15][16][17]. This approach continues to attract intense research efforts, mostly aimed at extending the range of applicability of the Doob theory, for instance towards discrete-time constrained random walks and Lévy flights [18,19], run-and-tumble trajectories [20], processes with resetting [21], or non-intersecting Brownian bridges [22].…”

Section: Introductionmentioning

confidence: 99%

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

Monthus¹,

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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Section: Introductionmentioning

confidence: 99%

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

Monthus¹,

Mazzolo²

2022

Preprint

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“…Let us also mention the conditioning in the presence of killing rates [3,[21][22][23][24][25][26][27][28] or when the killing occurs only via an absorbing boundary condition [29][30][31][32]. Note that stochastic bridges have been studied for many other Markov processes, including various diffusions processes [33][34][35], discretetime random walks and Lévy flights [36][37][38], continuous-time Markov jump processes [38], run-and-tumble trajectories [39], or processes with resetting [40].…”

Section: Introductionmentioning

confidence: 99%

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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“…In the field of diffusion processes, the basic example of the Brownian bridge has been extended to many other conditioning constraints, including the Brownian excursion [12,13], the Brownian meander [14], the taboo process [15][16][17][18][19][20], or non-intersecting Brownian bridges [21]. Besides diffusion processes, the stochastic bridges have been studied for many other Markov processes, including discrete-time random walks and Lévy flights [22][23][24], continuous-time Markov jump processes [24], run-and-tumble trajectories [25], or processes with resetting [26].…”

Section: Introductionmentioning

confidence: 99%

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

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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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Optimal Resetting Brownian Bridges

Cited by 4 publications

References 77 publications

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

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

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

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

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