Let $U\subseteq (C_0[0,1];\mathbb{R})$ be an open subset of the Wiener space satisfying $0<\gamma(U)<1$; where $\gamma$ is the Wiener measure and let $W_t$ be the standard (Wiener) coordinate process. Define a new (probability) measure on $(C_0([0,1];\mathbb{R}),\mathcal{B}(C_0([0,1];\mathbb{R}))$ by $\frac{d\nu}{d\gamma}=1_U$.

  • Is the process $W_t$ under $\nu$ studied?
  • Can it be interpreted as a conditional Brownian motion?
  • What are some references, if yes.
  • $\begingroup$ He is asking whether/when a measure equivalent to the standard Wiener measure can be interpreted as a conditional BM. $\endgroup$ – R W Sep 7 at 18:12
  • $\begingroup$ Not equivalent, but (strictly) absolutely conditinuous $\endgroup$ – N00ber Sep 7 at 18:13
  • $\begingroup$ But I thought the Wiener measure is not a probability measure... $\endgroup$ – N00ber Sep 7 at 18:15
  • 1
    $\begingroup$ encyclopediaofmath.org/index.php/Wiener_measure $\endgroup$ – AIM_BLB Sep 7 at 18:15
  • 3
    $\begingroup$ Sure, you can certainly think of this process as a Brownian motion conditioned to be in $U$. I don't know that there is much to say about it at this level of generality - usually people have particular events $U$ in mind and they prove statements specific to that situation. (E.g., Brownian motion conditioned to reach level 1 before time 1.) It's also of interest to "condition" on events of measure zero, to obtain processes like Brownian bridge, but such constructions involve more technicalities. $\endgroup$ – Nate Eldredge Sep 7 at 18:58

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