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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 $p\left(t,x,y\right)$ of the transition probability of a Markov process $\xi \left(t\right)\in {ℝ}^{n}$, $\xi \left(0\right)=x\in {ℝ}^{n}$ having both continuous and jump components. The forward Kolmogorov equation for the density of this process is written in the form

$\frac{\partial p}{\partial t}={L}_{h}p,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where

${L}_{h}=\frac{1}{2}h\left(A\left(x\right)\nabla ,\nabla \right)-\left(b,\nabla \right)+\frac{1}{h}{\int }_{{ℝ}^{n}}\left[exp\left\{\left(\beta ,h\nabla \right)\right\}-1\right]\mu \left(x,d\beta \right)·$

They construct the function ${{\Gamma }}_{N}\left(t,x,y,h\right)$ which is an asymptotic approximation of the density $p$ and prove the convergence in the sense

$\underset{x\in {ℝ}^{n},t\in \left[0,T\right)}{max}\left|{\int }_{{ℝ}^{n}}exp\left\{\frac{{S}_{1}}{h}\right\}\left({{\Gamma }}_{N}-p\right)u\left(y,h\right)\phantom{\rule{0.166667em}{0ex}}dy\right|\le {C}_{N}{h}_{N}·$

Here ${S}_{1}={max}_{y\in {ℝ}^{n}}\left[{S}_{0}\left(y\right)+S\left(t,x,y\right)\right)\right]$, $S\left(t,x,y\right)=-{lim}_{h\to +0}hln{{\Gamma }}_{N}\left(t,x,y,h\right)$, and ${S}_{0}\left(y\right)$ 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

$-h\frac{\partial V}{\partial t}+H\left(x,-h\frac{\partial }{\partial x}\right)V=0$

with initial data ${V\left(t,x,y\right)|}_{t=0}=exp\left[-\frac{{\left(x-y-z\right)}^{2}}{2h}\right],$ $z\in {ℝ}^{n}$ being a parameter and $H\left(x,p\right)=\left(A\left(x\right)p,p\right)+\left(b\left(x\right),p\right)+{\int }_{{ℝ}^{n}}^{}\left[exp\left\{\left(\beta ,p\right)\right\}-1\right]\mu \left(x,d\beta \right)$. The asymptotics ${V}_{N}$ is constructed in the form

${V}_{N}=exp\left[\frac{-{\Phi }}{h}\right]\left({\phi }_{0}+h{\phi }_{1}+\cdots {h}^{N}{\phi }_{N}\right)$

where ${\Phi }$ is proved to be a solution to the Hamilton-Jacobi equation

$\frac{\partial {\Phi }}{\partial t}+H\left(x,\frac{\partial }{\partial x}\right)V=0$

with initial data ${{\Phi }|}_{t=0}=\frac{\left(x-y-z\right)}{2}$ and ${\phi }_{k},k=0,\cdots ,N$ satisfy corresponding transport equations. The expression for the leading term of the exponential asymptotics (as $h\to 0$) for ${V}_{N}$ is derived as well. In Section 3 the authors study the behavior of the function $S\left(t,x,y\right)$ as $t\to 0$ and illustrate the results by several examples. The last section deals with the justification of the constructed asymptotics.

##### MSC:
 35C20 Asymptotic expansions of solutions of PDE 35S10 Initial value problems for pseudodifferential operators 60J35 Transition functions, generators, resolvents