Conditioning diffusion processes with killing rates

Abstract

When the unconditioned process is a diffusion submitted to a space-dependent killing rate k ( x → ) , various conditioning constraints can be imposed for a finite time horizon T. We first analyze the conditioned process when one imposes both the surviving distribution at time T and the killing-distribution for the intermediate times t ∈ [0, T]. When the conditioning constraints are less-detailed than these full distributions, we construct the appropriate conditioned processes via the optimization of the dynamical large deviations at level 2.5 in the presence of the conditioning constraints that one wishes to impose. Finally, we describe various conditioned processes for the infinite horizon T → +∞. This general construction is then applied to two illustrative examples in order to generate stochastic trajectories satisfying various types of conditioning constraints: the first example concerns the pure diffusion in dimension d with the quadratic killing rate k ( x → ) = γ x → 2 , while the second example is the Brownian motion with uniform drift submitted to the delta killing rate k(x) = kδ(x) localized at the origin x = 0.

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

“…Over the years, many different conditioning constraints have been considered, including the Brownian excursion [26,27], the Brownian meander [28], the taboo processes [29][30][31][32][33][34], or nonintersecting Brownian bridges [35]. Let us also mention the conditioning in the presence of killing rates [18,[36][37][38][39][40][41][42][43] or when the killing occurs only via an absorbing boundary condition [44][45][46][47]. Stochastic bridges have been also studied for many other Markov processes, including various diffusions processes [48][49][50], discrete-time random walks and Lévy flights [51][52][53], continuous-time Markov jump processes [53], run-and-tumble trajectories [54], or processes with resetting [55].…”

Conditioning diffusion processes with respect to the local time at the origin

“…In particular, the rate function I(a) can be computed from the knowledge of Ĝs (0|0) via equation (106) using the saddle-point value s a determined by equation ( 103). As it should for consistency, equations ( 103) and (106) correspond to the reciprocal quasi-Legendre transform of equation (100). Simple examples will be given in equations ( 189), (191)-(193), as well as in equations ( 228), ( 247), ( 251) and (252).…”

“…allows to recover the free propagator of the Brownian motion with alternating drift [28] as it should. The probability Π(A, t|x 0 , A 0 , t 0 ) of equation ( 65) can be obtained via the integration of the joint propagator P(x, A, t|x 0 , A 0 , t 0 ) of equation (239) over the final position x…”

“…Among the different conditioned diffusions that have been constructed besides the basic example of the Brownian Bridge, one can cite the Brownian excursion [11,12], the Brownian meander [13], the taboo processes [14][15][16][17][18][19], or non-intersecting Brownian bridges [20]. Let us also mention the conditioning in the presence of killing rates [3,[21][22][23][24][25][26][27][28] or when the killing occurs only via an absorbing boundary condition [29][30][31][32]. Note that stochastic bridges have been studied for many other Markov processes, including various diffusions processes [33][34][35], discrete-time random walks and Lévy flights [36][37][38], continuous-time Markov jump processes [38], run-and-tumble trajectories [39], or processes with resetting [40].…”

Conditioning two diffusion processes with respect to their first-encounter properties

“…In particular, this formula has been applied to impose an arbitrary final distribution E * (C f , T) of C f at some fixed horizon T [5,78] or to analyze the conditioning with respect to the firstpassage-time properties of one-dimensional diffusions at some level a [5,78,79], as well as the conditioning of diffusion processes with killing rates [80].…”