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

Abstract

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.

Conditioning diffusion processes with killing rates

“…In addition, the optimal solutions K * opt ( x, t) of equation (33) and P * opt ( y, T ) of equation (39) yield that the appropriate function Q T ( x, t) reads using equations (36) and ( 42)…”

“…Stochastic bridges have been also studied for many other Markov processes, including various diffusions processes [21][22][23], discrete-time random walks and Lévy flights [24][25][26], continuous-time Markov jump processes [26], run-and-tumble trajectories [27], or processes with resetting [28]. The stochastic bridge problem has been also extended to study the conditioning with respect to some global dynamical constraint as measured by a time-additive observable of the stochastic trajectories [29][30][31][32][33].…”

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

“…The bridge problem of equation (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][31]. 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 .…”