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consider a complete Riemannian manifold $M$ with heat kernel $p_M$ and let $U\subset M$ be an open set. Let $W_{x,t}^{y}$ be the Wiener measure associated to the Brownian motion starting at $x$ and ending at $y$ after time 't'. Consider the following function:

$$U\ni y \mapsto E_{t}^{x,y}\left( 1_{\{ t<\tau_U \}} \right)\in [0,1],$$

where $\tau_U$ is the first exit time from $U$. I am wondering if it is known whether this function is continuous or not? Is there a reference where I can find this proven in the general case described above?

I would very appreciate any help!

Best wishes

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  • $\begingroup$ Is $t$ a constant here? And can you clarify the definition of $W_x^y$? $\endgroup$
    – Nate Eldredge
    Commented Nov 19, 2016 at 14:57
  • $\begingroup$ Yes, $t>0$ is constant but arbitrary. $\endgroup$
    – Denilson Orr
    Commented Nov 19, 2016 at 15:07
  • $\begingroup$ So is $W_x^y$ supposed to be the Wiener measure corresponding to a Brownian motion conditioned to start at $x$ and be at $y$ at time $t$? Or something else? Also, then doesn't this function take its values in $[0,1]$? $\endgroup$
    – Nate Eldredge
    Commented Nov 19, 2016 at 15:14
  • $\begingroup$ Yes, exactly. I mean the Wiener measure corresponding to a Brownian motion conditioned to start at x and be at y at time t. $\endgroup$
    – Denilson Orr
    Commented Nov 19, 2016 at 16:32

2 Answers 2

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As the transition densities for the Wiener measure are continuous, weak continuity results of a type that may be what you seek can be found in http://projecteuclid.org/euclid.aop/1298669175 ["Markovian bridges: Weak continuity and pathwise constructions" by L. Chaumont and G. Uribe Bravo].

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  • $\begingroup$ Thank you very much for your help. This is a very interesting article. However, it seems as if weak continuity is not enough in order to get the continuity of the function $y\mapsto W_{x,t}^y(A)$ for fixed $A,t,x$. Am I missing something? $\endgroup$
    – Denilson Orr
    Commented Nov 22, 2016 at 14:56
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    $\begingroup$ Be advised that the proof of Theorem 5 in the Chaumont and Uribe Bravo paper has an error. This is mentioned at the end of arxiv.org/pdf/1101.4184v3.pdf $\endgroup$
    – HMPanzo
    Commented Nov 22, 2016 at 20:07
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I am somewhat uncertain about what you mean but $F_t$, but I think the answer is no regardless. Take the simplest case of Brownian bridge, You must be excluding events like $A = \lbrace W_t = y_0 \rbrace$ for which your function is clearly discontinuous from $F_t$, but if you will allow $A_i = \lbrace W_{t_i} < y_0 \rbrace $ for any $ t_i < t$ to be in $F_t$, then you will have to allow $A_i$ i.o., which will be similarly discontinuous.

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