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Exact asymptotics of the density of the transition probability for discontinuous Markov processes. (English) Zbl 0963.35036

The authors study the small time asymptotics of the density p(t,x,y) of the transition probability of a Markov process ξ(t) n , ξ(0)=x n having both continuous and jump components. The forward Kolmogorov equation for the density of this process is written in the form

p t=L h p,(1)

where

L h =1 2h(A(x),)-(b,)+1 h n [exp{(β,h)}-1]μ(x,dβ)·

They construct the function Γ N (t,x,y,h) which is an asymptotic approximation of the density p and prove the convergence in the sense

max x n ,t[0,T) n exp S 1 h (Γ N -p) u (y,h) d yC N h N ·

Here S 1 =max y n [S 0 (y)+S(t,x,y))], S(t,x,y)=-lim h+0 hlnΓ N (t,x,y,h), and S 0 (y) is a smooth given function. To this end they apply the Hamilton approach developed by V. Maslov and a special representation of the Dirac delta-function.

In Section 2 the authors construct a formal asymptotics V N to the symbol V of the exact solution to the equation (1) rewritten in the Hamilton form

-hV t+Hx , - h xV=0

with initial data V(t,x,y)| t=0 =exp[-(x-y-z) 2 2h], z n being a parameter and H(x,p)=(A(x)p,p)+(b(x),p)+ n [exp{(β,p)}-1]μ(x,dβ). The asymptotics V N is constructed in the form

V N =exp[-Φ h](ϕ 0 +hϕ 1 +h N ϕ N )

where Φ is proved to be a solution to the Hamilton-Jacobi equation

Φ t+H(x, x)V=0

with initial data Φ| t=0 =(x-y-z) 2 and ϕ k ,k=0,,N satisfy corresponding transport equations. The expression for the leading term of the exponential asymptotics (as h0) for V N is derived as well. In Section 3 the authors study the behavior of the function S(t,x,y) as t0 and illustrate the results by several examples. The last section deals with the justification of the constructed asymptotics.

MSC:
35C20Asymptotic expansions of solutions of PDE
35S10Initial value problems for pseudodifferential operators
60J35Transition functions, generators, resolvents