The product of spaces Φ × D is considered, where Φ is the set of all continuous, nondecreasing functions ϕ:[0,∞)→(0,∞), ϕ(0)=0, ϕ(t)→∞(t→∞), and D is the set of all right continuous functions ξ:(0,∞)→X; here X is some metric space. Two mappings are defined: the first is the projection q(ϕ,ξ)=ξ, and the second is the change of time U(ϕ,ξ)=ξoϕ. The following equivalence relation is defined on D: $$\xi _1 \sim \xi _2 \Leftrightarrow \exists _{\varphi _1 , \varphi _1 } \in \Phi :\xi _1 ^\circ \varphi _1 = \xi _2 ^\circ \varphi _2 $$ . Letℳ be the set of all equivalence classes, and let L be the mapping ξ4∼ξ2, Lξ is called the curve corresponding to ξ. The following theorem is proved: two stochastic processes with probability measures P1 and P2 on D possess identical random curves (i.e.,P1oL−1=P2oL−1) if and only if there exist two changes of time (i.e., probability measures Q1 and Q2 on ϕ×D for which P1=Q1oq−1, P2=Q2oq−1 which take these two processes into a process with measure\(\tilde P\)(i.e., Q1ou−1=Q2ou−1,=∼P) If (Px1)x∈X and (Px2)x∈X are two families of probability measures for which Px1oL−1=Px2oL−1∀x∈X then for each x e X the corresponding measures QX1 andQX2 can be found in the following manner. The set of regenerative times of the family\(\left( {\tilde P_x } \right)_{x \in X} \) contains all stopping times which are simultaneously regenerative times of the families (px1)x∈X and (Px2)x∈X and possess a certain special property of first intersection.