Joint distribution of two local times for diffusion processes with the application to the construction of various conditioned processes
Alain Mazzolo, Cécile Monthus · Journal of Physics A Mathematical and Theoretical · 2023
Abstract For a diffusion process X(t) of drift μ ( x ) and of diffusion coefficient D = 1 / 2 , we study the joint distribution of the two local times A ( t ) = ∫ 0 t d τ δ ( X ( τ ) ) and B ( t ) = ∫ 0 t d τ δ ( X ( τ ) − L ) at positions x = 0 and x = L, as well as the simpler statistics of their sum Σ ( t ) = A ( t ) + B ( t ) . Their asymptotic statistics for large time t → + ∞ 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 ∗ ( ∞ ) , B ∗ ( ∞ ) ] 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 = A ( t ) t and b = B ( t ) t and of their intensive sum σ =