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 ) = ∫ 0 t d τ δ ( X ( τ ) ) at the origin x = 0 is one of the most important time-additive 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 <?CDATA ${A}_{\infty }^{\ast } A ∞ * &lt; + ∞ , as well as the conditioning towards the intensive local time a* corresponding to the extensive behavior A T ≃ Ta*, 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 &gt; 0; (ii) the simplest example of diffusion converging towards an equilibrium is given by the drift μ(x) = −μ sgn(x) of parameter μ &gt; 0; (iii) the simplest example of recurrent diffusion that does not converge towards an equilibrium is the Brownian motion without drift μ = 0.

Joint distribution of two local times for diffusion processes with the application to the construction of various conditioned processes

“…We have discussed the asymptotic behavior for large time t → +∞ : (i) when the diffusion process X(t) is transient, the two local times [A(t); B(t)] remain finite random variables [A * (∞), B * (∞)] and we have analyzed their limiting joint distribution ; (ii) when the diffusion process X(t) is recurrent, we have described the large deviations properties of the two intensive local times a = A (t) t and b = B(t) t and of the intensive sum σ = Σ(t) t = a + b. We have then used these properties to construct various conditioned processes [X * (t), A * (t), B * (t)] satisfying certain constraints involving the two local times, thereby generalizing our previous work [106] concerning the conditioning with respect to the single local time A(t). In particular for the infinite time horizon T → +∞, we have considered the conditioning towards the finite asymptotic values [A * (∞), B * (∞)] or Σ * (∞), as well as the conditioning towards the intensive values [a * , b * ] or σ * , that we have compared in appendix A with the appropriate 'canonical conditioning' based on the generating function of the local times in the regime of large deviations.…”

“…However, as in quantum mechanics where delta impurities are well-known to be much simpler than smoother potentials, the delta function in equation ( 2) is actually a huge technical simplification with respect to the arbitrary general additive observable of equation (5). Indeed, the exact Dyson equation associated to a single delta impurity allows to analyze the statistics of a single local time in terms of the properties of the propagator G t (x|x 0 ) of the diffusion process X(t) alone, as recalled in detail in [106]. In the present paper, it will be thus interesting to use similarly the exact Dyson equation associated to two single delta impurities in order to characterize the joint statistics of the two Local Times A(t) = A x=0 (t) and B(t) = A x=L (t) at positions x = 0 and x = L A(t) ≡ ˆt 0 dτ δ(X(τ ))…”

“…In section 6, we construct various conditioned joint processes [X * (t), A * (t), B * (t)] satisfying certain constraints involving the two local times, thereby generalizing our previous work [106] concerning the conditioning with respect to the single local time A(t). The conditioned Bridge towards the given local times [A * T , B * T ] at the finite time horizon T can be constructed as follows: the conditioned process [X * (t), A * (t), B * (t)] satisfies the following Ito SDE system analog to equation ( 7)…”

“…When the diffusion process X(t) is recurrent, it is more appropriate to introduce the two intensive local times of equation (11) and to analyze their large deviations properties thereby generalizing the previous works concerning the single intensive local time a [2,106].…”

“…In this section, the goal is to construct conditioned joint processes [X * (t), A * (t), B * (t)] satisfying certain constraints involving the two local times, thereby generalizing our previous work [106] concerning the conditioning with respect to the single local time A(t).…”