Mixing
Does a given First Passage Time Distribution imply a single possible Fokker-Planck equation?
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Consider a 1-dimensional Continuous Markov Process $X(t)$ with fixed and constant absorbing boundaries, let's say at $pm theta$, and with starting point at $X(t=0)=0$. This setting will lead to a given First Passage Time (FPT) distribution at each boundary. Let us call the densities $g_{pm}(t,pm theta)$ respectively.
Suppose now that one could change the value of $theta$ and obtain the family of FPT distributions for all possible values of $theta$ (doesn't really matter whether one can obtain a parametric form or not). My question is, then: Would this family of distributions $g_{pm}(t,pm theta)$ imply the unicity of the underlying Fokker-Planck equation (or equivalently the form of the drift and diffusion coefficients $A(x,t)$ and $D(x,t)$)? My intuition tells me this must be the case, but I haven't found a theorem or a way to prove this claim.
Bonus question: Would this extend to a 2-dimensional case?
stochastic-processes stochastic-calculus markov-process
asked Jan 16 at 11:15
J. R. C.J. R. C.
718
$endgroup$
add a comment |
$begingroup$
Consider a 1-dimensional Continuous Markov Process $X(t)$ with fixed and constant absorbing boundaries, let's say at $pm theta$, and with starting point at $X(t=0)=0$. This setting will lead to a given First Passage Time (FPT) distribution at each boundary. Let us call the densities $g_{pm}(t,pm theta)$ respectively.
Suppose now that one could change the value of $theta$ and obtain the family of FPT distributions for all possible values of $theta$ (doesn't really matter whether one can obtain a parametric form or not). My question is, then: Would this family of distributions $g_{pm}(t,pm theta)$ imply the unicity of the underlying Fokker-Planck equation (or equivalently the form of the drift and diffusion coefficients $A(x,t)$ and $D(x,t)$)? My intuition tells me this must be the case, but I haven't found a theorem or a way to prove this claim.
Bonus question: Would this extend to a 2-dimensional case?
stochastic-processes stochastic-calculus markov-process
asked Jan 16 at 11:15
J. R. C.J. R. C.
718
$endgroup$
add a comment |
$begingroup$
Consider a 1-dimensional Continuous Markov Process $X(t)$ with fixed and constant absorbing boundaries, let's say at $pm theta$, and with starting point at $X(t=0)=0$. This setting will lead to a given First Passage Time (FPT) distribution at each boundary. Let us call the densities $g_{pm}(t,pm theta)$ respectively.
Suppose now that one could change the value of $theta$ and obtain the family of FPT distributions for all possible values of $theta$ (doesn't really matter whether one can obtain a parametric form or not). My question is, then: Would this family of distributions $g_{pm}(t,pm theta)$ imply the unicity of the underlying Fokker-Planck equation (or equivalently the form of the drift and diffusion coefficients $A(x,t)$ and $D(x,t)$)? My intuition tells me this must be the case, but I haven't found a theorem or a way to prove this claim.
Bonus question: Would this extend to a 2-dimensional case?
stochastic-processes stochastic-calculus markov-process
asked Jan 16 at 11:15
J. R. C.J. R. C.
718
$endgroup$
Consider a 1-dimensional Continuous Markov Process $X(t)$ with fixed and constant absorbing boundaries, let's say at $pm theta$, and with starting point at $X(t=0)=0$. This setting will lead to a given First Passage Time (FPT) distribution at each boundary. Let us call the densities $g_{pm}(t,pm theta)$ respectively.
Suppose now that one could change the value of $theta$ and obtain the family of FPT distributions for all possible values of $theta$ (doesn't really matter whether one can obtain a parametric form or not). My question is, then: Would this family of distributions $g_{pm}(t,pm theta)$ imply the unicity of the underlying Fokker-Planck equation (or equivalently the form of the drift and diffusion coefficients $A(x,t)$ and $D(x,t)$)? My intuition tells me this must be the case, but I haven't found a theorem or a way to prove this claim.
Bonus question: Would this extend to a 2-dimensional case?
stochastic-processes stochastic-calculus markov-process
stochastic-processes stochastic-calculus markov-process
asked Jan 16 at 11:15
J. R. C.J. R. C.
718
asked Jan 16 at 11:15
J. R. C.J. R. C.
718
asked Jan 16 at 11:15
J. R. C.J. R. C.
718
asked Jan 16 at 11:15
J. R. C.J. R. C.
718
asked Jan 16 at 11:15
J. R. C.J. R. C.
718
718
add a comment |
add a comment |
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