By M.M. Rao
Provides formerly unpublished fabric at the basic rules and houses of Orlicz series and serve as areas. Examines the pattern direction habit of stochastic strategies.
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Extra resources for Applications of Orlicz spaces
Thus, while Brownian motion is completely random, diffusion is not exactly as random as Brownian motion. For example, diffusion does not occur in a homogeneous medium where there is no concentration gradient. Thus, Brownian motion may be considered a probabilistic model of diffusion in a homogeneous medium. Consider a physical system with state xðtÞ; t $ 0. The behavior of the system when an input wðtÞ; t $ 0, is presented to it is governed by a differential equation of the following form that gives the rate of change of the state: dxðtÞ 5 aðxðtÞ; tÞ 1 bðxðtÞ; tÞwðtÞ t $ 0 dt ð3:1Þ 52 Markov Processes for Stochastic Modeling where the functions a and b depend on the system properties.
Since the interarrival times in a Poisson process are exponentially distributed, let TX be the random variable that denotes the interarrival time in the fXðtÞ; t $ 0g process and let TY be the random variable that denotes the interarrival time in the fYðtÞ; t $ 0g process. Thus, we are interested in the computing P½TX , TY , where fTX ðxÞ 5 λX e2λX x ; x $ 0 and fTY ðyÞ 5 λY e2λY y ; y $ 0. 4. 4 Partitioning the regions around the line TX 5 TY. TY TX Y =T T X < TY T X > TY TX 44 Markov Processes for Stochastic Modeling Thus, P½TX , TY 5 ðN ðN fTX TY ðx; yÞdy dx 5 x50 y5x 5 ðN λX e2λX x e2λY x dx 5 x50 5 ðN ðN x50 ðN λX λY e2λX x e2λY y dy dx y5x λX e2ðλX 2λY Þx dx x50 λX λX 1 λY Another way to derive this result is by considering events that occur within the small time interval ½t; t 1 Δt.
G. If Tr is a stopping time, then we have fTr 5 ng 5 fX1 6¼ r; . ; Xn21 6¼ r; Xn 5 rg Introduction to Markov Processes 53 For example, if we define the recurrence time of state i, fi ðnÞ, as the conditional probability that given that the process is presently in state i, the first time it will return to state i occurs in exactly n transitions, then we have that fTi 5 ng 5 fX0 5 i; X1 6¼ i; . ; Xn21 6¼ i; Xn 5 ig Similarly, if we define the first passage time between state i and state j, fij ðnÞ, as the conditional probability that given that the process is presently in state i, the first time it will enter state j occurs in exactly n transitions, then we have that fTij 5 ng 5 fX0 5 i; X1 6¼ j; .