The authors study diffusion processes on a locally compact separable metric measure space \((M,d,\mu)\) equipped with a \(\sigma\)-finite Borel measure \(\mu\). The processes are generated by local Dirichlet forms. The main result of the paper is to provide off-diagonal upper bounds for the heat kernel (transition probability density) \(p_t(x,y)\) where \(t\geq 0\) and \(x\), \(y\) are from the state space (up to a negligible set \(N\)). The heat kernel bounds are of the type \[ p_t(x,y) \leq F_t(x,y) \exp\left(-\gamma \left[ t^{-1}d^\beta(x,y)\right]^{1/(\beta-1)}\right) \] where \(\gamma >0\), \(\beta >1\) (\(\beta=1\) is the Gaussian case, \(\beta>2\) appears in connection with diffusions on fractals) and a positive function \(F_t(x,y)\). Typically, \(F_t(x,y)\) is a power function in \(t\) or given by the \(\mu\)-volume of balls around \(x\) and \(y\) with radius \(t^{1/\beta}\).
The main result of the paper states that one can get global heat kernel estimates by restricting the process to an open subset \(U\subset M\) of diameter less or equal than \(R\). More precisely, assume that the following conditions hold

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for \((t,x,y), (s,z,w)\in (0,R^\beta]\times U\times U\), \(s\leq t\) one has \[ \frac{F_s(z,w)}{F_t(x,y)} \leq c_F s^{-\alpha_F}\max\{t,d^\beta(x,z), d^\beta(y,w)\}^{\alpha_F}, \]

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for \((t,x)\in (0,R^\beta)\times (U\setminus N)\) and any Borel set \(A\subset U\) \[ \mathbb{P}^x(X_t\in A, \tau_U > t) \leq \int_A F_t(x,y)\,\mu(dy), \]

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for \((x,r)\in (U\setminus N)\times (0,R)\) such that \(B_r(x)\subset U\) and \(t>0\) \[ \mathbb{P}^x(\tau_{B_r(x)}\leq t) \leq \exp\left[-\gamma r^{\beta/(\beta-1)} t^{-1/(\beta-1)}\right] \]

then the heat kernel has a density \(p_t(x,y)\), \((t,x,y)\in (0,\infty)\times (M \setminus N)\times V\), \(V\) is a certain open neighbourhood of \(U\), and it satisfies a heat kernel estimate of the type indicated above for all (!) \(x\in M\setminus N\) and all \(y\in V\).
The proof uses two main ingredients, a “multiple Dynkin-Hunt formula”, i.e. a formula for the transition semigroup \((P_t)_t\) of the type \[ P_tu(x) = P_t^Uu(x) + \sum_{n=1}^\infty\mathbb{E}^x\left[\mathbf{1}_{\sigma_n\leq t} P_{t-\sigma_n}^Uu(X_{\sigma_n})\right] \] where \((P_t^U)_t\) is the transition semigroup of the process killed upon leaving \(U\) (“part process”) and \(\sigma_n\) is the random time of the \(n\)th re-entering of the original (non-killed) process \(X\) into the set \(U\). In the final section, the authors provide good exit probability estimates for diffusions as needed in the third condition above.