Exact Asymptotics of the Density of the Transition Probability for Discontinuous Markov Processes

Forward and backward in time Cauchy problems for systems of parabolic‐type PDE with a small parameter

“…It is convenient to analyze the solution of the Hamilton-Jacobi equation by using the notion of Lagrangian manifold introduced in [18] and later in [11]. When solving Kolmogorov-Feller-type equations in the framework of our approach, this subject was studied in [1], [6], [7].…”

“…To complete the derivation of the equation for W l 1 , we must "pull" the exponential through the operator H −ε ∂ ∂ x . This operation is described in detail in [4], [7]. For the first two terms in (2.21), these transformations are trivial:…”

“…As was shown in [4], [7], [16], in the case of scalar equations and under the condition of analyticity in p (otherwise, the operator…”

“…The introduced class of functions is invariant under the resolving operator if the solution is understood in the weak sense with an exponential weight. The paper continues the study of [6].where ϕ 0 is a rapidly decreasing function and ϕ 0 , S 0 ∈ C ∞ , then the almost-everywhere solution of the Cauchy problem can be written in the form [7], [16], [19] ( 1.2) where ϕ(x, t) and S(x, t) are solutions of some first-order equations (for details, see below). *…”

“…where ϕ 0 is a rapidly decreasing function and ϕ 0 , S 0 ∈ C ∞ , then the almost-everywhere solution of the Cauchy problem can be written in the form [7], [16], [19]…”