Estimates for the probability that Itô processes remain near a path

Let W=(Wi)i∈NW=(Wi)i∈N be an infinite dimensional Brownian motion and (Xt)t≥0(Xt)t≥0 a continuous adaptednn-dimensional process. Set τR=inf{t:|Xt−xt|≥Rt}τR=inf{t:|Xt−xt|≥Rt}, where xt,t≥0xt,t≥0 is a RnRn-valued deterministic differentiable curve and Rt>0,t≥0Rt>0,t≥0 a time-dependent radius. We assume that, up to τRτR, the process XX solves the following (not necessarily Markov) SDESDE:Xt∧τR=x+∑j=1∞∫0t∧τRσj(s,ω,Xs)dWsj+∫0t∧τRb(s,ω,Xs)ds. Under local conditions on the coefficients, we obtain lower bounds for P(τR≥T)P(τR≥T) as well as estimates for distribution functions and expectations. These results are discussed in the elliptic and log-normal frameworks. An example of a diffusion process that satisfies the weak Hörmander condition is also given.