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 $\mu(x)$ and of diffusion coefficient $D=1/2$, the local time $A(t)= \int_{0}^{t} d\tau \delta(X(\tau)) $ 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 $T \to +\infty$, we consider the conditioning towards the finite asymptotic local time $A_{\infty}^*<+\infty$, as well as the conditioning towards the intensive local time $a^* $ corresponding to the extensive behavior $A_T \simeq T a^*$, 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 $\mu(x)=\mu>0$; (ii) the simplest example of diffusion converging towards an equilibrium is given by the drift $\mu(x)=- \mu \, {\rm sgn}( x)$ of parameter $\mu>0$; (iii) the simplest example of recurrent diffusion that does not converge towards an equilibrium is the Brownian motion without drift $\mu=0$.