Abstract
AbstractFor a diffusion processX(t) of driftμ(x)and of diffusion coefficientD=1/2, we study the joint distribution of the two local timesA(t)=∫0tdτδ(X(τ))andB(t)=∫0tdτδ(X(τ)−L)at positionsx = 0 andx = L, as well as the simpler statistics of their sumΣ(t)=A(t)+B(t). Their asymptotic statistics for large timet→+∞involves two very different cases: (i) when the diffusion processX(t) is transient, the two local times[A(t);B(t)]remain finite random variables[A∗(∞),B∗(∞)]and we analyze their limiting joint distribution; (ii) when the diffusion processX(t) is recurrent, we describe the large deviations properties of the two intensive local timesa=A(t)tandb=B(t)tand of their intensive sumσ=Σ(t)t=a+b. These properties are then used to construct various conditioned processes[X∗(t),A∗(t),B∗(t)]satisfying certain constraints involving the two local times, thereby generalizing our previous work (Mazzolo and Monthus 2022J. Stat. Mech.103207) concerning the conditioning with respect to a single local timeA(t). In particular for the infinite time horizonT→+∞, we consider the conditioning towards the finite asymptotic values[A∗(∞),B∗(∞)]orΣ∗(∞), as well as the conditioning towards the intensive values[a∗,b∗]orσ∗, that can be compared with the appropriate ‘canonical conditioning’ based on the generating function of the local times in the regime of large deviations. This general construction is then applied to the simplest case where the unconditioned diffusion is the Brownian motion of uniform driftµ.
Subject
General Physics and Astronomy,Mathematical Physics,Modeling and Simulation,Statistics and Probability,Statistical and Nonlinear Physics