Joint distribution of two Local Times for diffusion processes with the application to the construction of various conditioned processes

For a diffusion process $X(t)$ of drift $\mu (x)$ and of diffusion coefficient $D=1\mathrm{/}2$, we study the joint distribution of the two local times $A(t)={\int }_0^{t}d\tau \delta (X(\tau ))$ and $B(t)={\int }_0^{t}d\tau \delta (X(\tau )-L)$ at positions $x=0$ and $x=L$, as well as the simpler statistics of their sum $\mathrm{\Sigma }(t)=A(t)+B(t)$. Their asymptotic statistics for large time $t\to +\mathrm{\infty }$ involves two very different cases : (i) when the diffusion process $X(t)$ is transient, the two local times $[A(t);B(t)]$ remain finite random variables $[A^{\ast }(\mathrm{\infty }),B^{\ast }(\mathrm{\infty })]$ and we analyze their limiting joint distribution ; (ii) when the diffusion process $X(t)$ is recurrent, we describe the large deviations properties of the two intensive local times $a=\frac{A(t)}{t}$ and $b=\frac{B(t)}{t}$ and of their intensive sum $\sigma =\frac{\mathrm{\Sigma }(t)}{t}=a+b$. These properties are then used to construct various conditioned processes $[X^{\ast }(t),A^{\ast }(t),B^{\ast }(t)]$ satisfying certain constraints involving the two local times, thereby generalizing our previous work [arXiv:2205.15818] concerning the conditioning with respect to a single local time $A(t)$. In particular for the infinite time horizon $T\to +\mathrm{\infty }$, we consider the conditioning towards the finite asymptotic values $[A^{\ast }(\mathrm{\infty }),B^{\ast }(\mathrm{\infty })]$ or ${\mathrm{\Sigma }}^{\ast }(\mathrm{\infty })$, as well as the conditioning towards the intensive values $[a^{\ast },b^{\ast }]$ or ${\sigma }^{\ast }$, 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 drift $\mu$.