# Markov Processes

For a Markov process defined by a matrix `T`

where `T`

is the operator such that `Tf = E[df]`

`stationary_distribution(T)`

returns its stationary distribution`feynman_kac_backward(T, t, ψ, f, V)`

returns the solution of the PDE`u_t(x, t) + T u - V(x, t) u + f(x, t) = 0`

with`u(x, T) = ψ(x)`

Moreoveor,

`generator(DiffusionProcess(x, μ, σ))`

creates the transition matrix of a diffusive process with drift`μ(x)`

and volatility`σ(x)`

with reflecting boundaries.

# Additive Functionals

For an additive functional `m`

defined by a function `ξ -> T(ξ)`

where `T`

is the operator such that `T f= E[d(e^(ξm)f)]`

`cgf(f)`

returns the long run scaled CGF of`m`

`tail_index(f)`

returns the tail index of the stationary distribution of`e^m`

Moreover,

`generator(AdditiveFunctional(DiffusionProcess(x, μ, σ), μm, σm)`

creates the function`ξ -> T(ξ)`

for the additive functional with drift`μm(x)`

and volatility`σm(x)`

## Related Packages

- SimpleDifferentialOperators contains more general tools to define operators with different boundary counditions. In contrast, InfinitesimalGenerators always assumes reflecting boundaries.