Stationary states for stable processes with partial resetting

We study a d-dimensional stochastic process X which arises from a Lévy process Y by partial resetting, that is, the position of the process X at a Poisson moment equals c∈(0,1) times its position right before the moment, and it develops as Y between these two consecutive moments. We focus on Y being a strictly α-stable process with α∈(0,2] having a transition density: We analyze properties of the transition density p of the process X. We establish a series representation of p. We prove its convergence as time goes to infinity (ergodicity), and we show that the limit ρY (density of the ergodic measure) can be expressed by means of the transition density of the process Y starting from zero, which results in closed concise formulae for its moments. We show that the process X reaches a nonequilibrium stationary state. Furthermore, we check that p satisfies the Fokker–Planck equation, and we confirm the harmonicity of ρY with respect to the adjoint generator. The following cases are discussed in details: Brownian motion, isotropic and d-cylindrical α-stable processes for α∈(0,2), and α-stable subordinator for α∈(0,1). We find the asymptotic behavior of p(t;x,y) as t→+∞ while (t,y) stays in a certain space-time region. For Brownian motion, we discover a phase transition, that is, a change of the asymptotic behavior of p(t;0,y) with respect to ρY(y).