please show all work and explain

Problem 1 Consider the heat PDE in d = 3: u = 4.4 tu(0,1) = u(x) = (1) 2 + sin() (i) Show that no (RR). (ii) Write down the unique solution to (1) as an expectation involving a 3-dimensional Brownian motion B = (B1, B2, Bs) that starts at = (*1,12,13) under P. (iii) Write down your answer as an integral over 2. (iv) Write down your answer as an integral over Rd for all +20. Consider the heat PDE in d = 3: = u(0,2) = 10:0) = il = u0 (1) e-(3+2) 2 + sin(x) (i) Show that uo C. (R3; R). (ii) Write down the unique solution to (1) as an expectation involving a 3-dimensional Brownian motion B = (B1, B2, B3) that starts at 2 = (21,12,13) under pr. (ii) Write down your answer as an integral over 2. (iv) Write down your answer as an integral over Rd for all t2 0. Problem 1 Consider the heat PDE in d = 3: u = 4.4 tu(0,1) = u(x) = (1) 2 + sin() (i) Show that no (RR). (ii) Write down the unique solution to (1) as an expectation involving a 3-dimensional Brownian motion B = (B1, B2, Bs) that starts at = (*1,12,13) under P. (iii) Write down your answer as an integral over 2. (iv) Write down your answer as an integral over Rd for all +20. Consider the heat PDE in d = 3: = u(0,2) = 10:0) = il = u0 (1) e-(3+2) 2 + sin(x) (i) Show that uo C. (R3; R). (ii) Write down the unique solution to (1) as an expectation involving a 3-dimensional Brownian motion B = (B1, B2, B3) that starts at 2 = (21,12,13) under pr. (ii) Write down your answer as an integral over 2. (iv) Write down your answer as an integral over Rd for all t2 0