mathman said:All the expression is saying is that a particle arriving at r at time t will have had a first arrival at r at time t' (which may = t) ahd then arrive at a relative position of 0 in time t-t'. The delta function term simply says that P(0,0) reflects the fact that it was at 0 to start with.
The occupation probability is the likelihood that a given state or position will be occupied by a system over a period of time. The first passage probability is the likelihood that a system will pass through a particular state for the first time. The relation between these two probabilities is that the occupation probability directly affects the first passage probability. As the occupation probability increases, the first passage probability also increases.
Occupation probability is calculated by dividing the time spent in a particular state by the total time. First passage probability is calculated by dividing the number of times a system passes through a particular state for the first time by the total number of passages.
The occupation probability and first passage probability are influenced by the initial conditions of the system, the dynamics of the system, and any external forces or perturbations acting on the system.
Studying the relation between occupation probability and first passage probability allows us to understand the behavior and dynamics of complex systems. It also has practical applications in fields such as physics, chemistry, biology, and economics.
Yes, the relation between occupation probability and first passage probability can be applied to real-world situations. It can be used to model and predict the behavior of various systems, such as the movement of particles in a gas, the spread of diseases in a population, or the fluctuations of stock prices in the market.