I will illustrate the concept of essential probabilistic techniques by means of examples.
(i) The "hot spots" problem. The "hot spots" conjecture of Rauch states that the the second Neumann eigenfunction in a Euclidean domain attains its maximum on the boundary of the domain. The conjecture is false at this level of generality but it has been proved to hold under various extra assumptions. The probabilistic method involved in various arguments is "coupling" of Brownian motions, i.e., running two (highly dependent) Brownian motions at the same time. The behavior of coupled Brownian motions can be used to prove some properties of the heat equation solution.
(ii) The heat equation with Neumann boundary conditions in non-cylindrical domains, i.e., in space-time domains in which that spacial domain evolves with time. The main object of this study are singularities on the boundary of the domain, if the boundary is continuous but rough. The presence of some singularities is related to certain local Brownian path properties. Some other properties of the heat solutions in non-cylindrical domains can be studied using a probabilistic technique known as the Feynman-Kac formula.
(iii) The Robin problem in fractal domains. The Robin boundary conditions are intermediate conditions between the Dirichlet and Neumann boundary conditions. They model the flow of heat or particles through a semi-permeable membrane. I will show how the properties of the heat equation solution with Robin boundary conditions can be studied using a Feynamn-Kac formula and an old jewel of probability theory, a theorem of Ray and Knight on Brownian local time.