# Conditioned Brownian motion?

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.
• He is asking whether/when a measure equivalent to the standard Wiener measure can be interpreted as a conditional BM. – R W Sep 7 at 18:12
• Not equivalent, but (strictly) absolutely conditinuous – N00ber Sep 7 at 18:13
• But I thought the Wiener measure is not a probability measure... – N00ber Sep 7 at 18:15
• encyclopediaofmath.org/index.php/Wiener_measure – AIM_BLB Sep 7 at 18:15
• 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. – Nate Eldredge Sep 7 at 18:58