The concepts of Markov process in random environment,q-matrix in random environment,and q-process in random environment are introduced.The minimal q-process in random environment is constructed and the necessary and sufficient conditions for the uniqueness of q-process in random environment are given.
Let A={λk} be an infinite increasing sequence of positive integers with λk-∞，Let X={X(t),t∈R^N} be a multi-parameter fractional Brownianmotion of index α(0<α<1) in R^d，Subjct to certain hypotheses， we prove that if N<αd，then there exist positive finite constants K1 and K2 such that，with unit probability，K1≤ψ-pA(X([0,1])^N)≤ψ-PA(GrX(0,1]^N)≤K2 if and only if there exists γ>0 such that ∞↑∑↓k 1/γk^λ=∞ where ψ（s)=s^N/α(log log1/S)^N/(2α），ψ-P∧(E) is the Packing-type measure of E, X([0, 1])^N is the image and Gr X([0,1)^N)={(t,X(t)；t∈[0,1]^N} is the graph of X,resectively,We also estabish liminf type laws of the iterated logarithm for the sojoura measure of X。
