# Relation between occupation probability and first passage probability

• WiFO215
In summary, the poster is asking for an answer to their question which was originally posted in a probability sub-forum. They provide a link to the post and explain that they did not receive any replies. Another user responds by summarizing the original poster's expression, explaining that it is saying a particle that arrives at position r at time t had a first arrival at r at an earlier time t' and then arrived at a relative position of 0 in the time difference t-t'. The delta function term takes into account the fact that the particle was initially at position 0. The original poster then asks if this assumption of stationarity means that the distribution does not change once the particle reaches position r, to which the responder clarifies that there
WiFO215
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.

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.

He is describing relative position zero using the same probability distribution P? So he is making the assumption there that P doesn't change once the thing reaches r? He hasn't mentioned this explicitly.

There is an implicit assumption of stationarity, i.e. the distribution depends only on time difference, not absolute time.

Okay! Thanks.

## 1. What is the relation between occupation probability and first passage probability?

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.

## 2. How are occupation probability and first passage probability calculated?

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.

## 3. What factors influence the occupation probability and first passage probability?

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.

## 4. What is the significance of studying the relation between occupation probability and first passage probability?

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.

## 5. Can the relation between occupation probability and first passage probability be applied to real-world situations?

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.

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